Table Of Content

Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   6-1 Operations on Functions   Find (f + g)(x), (f – g)(x), (x),and 2.    (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. SOLUTION:     1.      SOLUTION:               ANSWER:     ANSWER:     For each pair of functions, find and , if they exist. State the domain and range for each composed function.     3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} 2.    SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . eSolutionsManual-PoweredbyCognero So,  is undefined. Page1       Therefore:         ANSWER:   is undefined;     ANSWER:   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. For each pair of functions, find and , if   they exist. State the domain and range for each composed function.     Therefore: 3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is undefined; So,  is undefined.     Find and , if they exist. State the domain and range for each composed   function. Therefore:   5.    SOLUTION:       ANSWER:     is undefined;   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)}   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   ANSWER:   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.   6.      Therefore: SOLUTION:     ANSWER:     is undefined;   Find and , if they exist. State the domain and range for each composed function.     5.  ANSWER:     SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan ANSWER:   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   6.  ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college SOLUTION:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      ANSWER:   SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any 9.  restrictions in domain or range.     SOLUTION:   8.    SOLUTION:                   ANSWER:     ANSWER:         10.    9.  SOLUTION:     SOLUTION:               ANSWER:       ANSWER:       11.        SOLUTION:   10.    SOLUTION:     ANSWER:         ANSWER:     12.  SOLUTION:       11.    SOLUTION:   ANSWER:       ANSWER:   13.    SOLUTION:       12.  SOLUTION:     ANSWER:     ANSWER:   14.    SOLUTION:     13.    SOLUTION:       ANSWER:     ANSWER:     15.    SOLUTION:     14.    SOLUTION:     ANSWER:     ANSWER:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined   after any number of years?  b. What is the difference in the populations of the 15.  two cities?      SOLUTION:   SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if   they exist. State the domain and range for each ANSWER:   composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10     b. (f - g)(x) = 25x + 40 ANSWER:       For each pair of functions, find and , if they exist. State the domain and range for each composed function. 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)} 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)}   g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} SOLUTION:     SOLUTION:   Therefore:   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     ANSWER:   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined. 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   ANSWER:   is The range of is not a subset of the domain of undefined; .   So,  is undefined. The range of is not a subset of the domain of 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} .   So,  is undefined. SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is undefined;  is undefined.   So,  is undefined. 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:     ANSWER:   The range of is not a subset of the domain of is undefined; .   So,  is undefined.   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} ANSWER:     ; is undefined. SOLUTION:     The range of is not a subset of the domain of 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} . g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   So,  is undefined. SOLUTION:     The range of is not a subset of the domain of ANSWER:   is undefined; . .   So,  is undefined.   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)} ANSWER:     ; is undefined. SOLUTION:     24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . So,  is undefined. . The range of is not a subset of the domain of So,  is undefined.   . So,  is undefined. ANSWER:     ; is undefined.   ANSWER:   is undefined;  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)}   g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)} SOLUTION:     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} is undefined;  is undefined. g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     ANSWER:     ANSWER:     Find and , if they exist. State the domain and range for each composed   function.   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} 27.  g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     28.  ANSWER:     SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers}   ANSWER:   29.    , D = {all real numbers}, R = {all SOLUTION:   even numbers} , D = {all real numbers}, R = {all odd numbers}   28.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   30.  , D = {all real numbers}, R = {all   real numbers} SOLUTION:   , D = {all real numbers}, R = {all real numbers}   29.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –14} real numbers} , D = {all real numbers}, R = {y | y   ≥ –10} ANSWER:     31.  , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all SOLUTION:   real numbers}   30.      SOLUTION:     , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} ANSWER:     32.  , D = {all real numbers}, R = {y | y   ≥ –14} , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –10}   31.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} ANSWER:     33.  , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –17}   32.      SOLUTION:       , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0.875} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 6.5} real numbers}     ANSWER:   34.    , D = {all real numbers}, R = {y | y ≥ 0.875} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ 6.5}   33.      SOLUTION:     , D = {all real numbers}, R = {y | y   ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     ANSWER:   35.  , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all SOLUTION:   real numbers}   34.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0}   ANSWER:   36. FINANCE A ceramics store manufactures and sells , D = {all real numbers}, R = {y | y coffee mugs. The revenue r(x) from the sale of x ≥ 1} coffee mugs is given by r(x) = 6.5x. Suppose the , D = {all real numbers}, R = {y | y function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850. ≥ 0}     a. Write the profit function.   35.  b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:     b.   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     a. P(x) = 5.75x – 1850   ANSWER:   b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900   , D = {all real numbers}, R = {y | y 37. CCSS SENSE-MAKING Ms. Smith wants to buy ≥ 0} an HDTV, which is on sale for 35% off the original , D = {all real numbers}, R = {y | y price of $2299. The sales tax is 6.25%. ≥ 0}     a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x). 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x b. Which composition of functions represents the coffee mugs is given by r(x) = 6.5x. Suppose the price of the HDTV, or ? Explain function for the cost of manufacturing x coffee mugs your reasoning. is c(x) = 0.75x + 1850.     c. How much will Ms. Smith pay for the HDTV? a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee SOLUTION:   mugs. a. Let p(x) be the price after the discount where x is   the original price. Discount = 35%(x) = 0.35x SOLUTION:   Therefore: a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. Let t(x) be the price after the sales tax. So: Sales tax = 6.25%(x) = 0.0625x Therefore:   b.   b. Since = , either function represents the price.   c.  ANSWER:   Substitute x = 2299. a. P(x) = 5.75x – 1850   b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900     Therefore, Mr. Smith will pay $1587.75 for HDTV.   37. CCSS SENSE-MAKING Ms. Smith wants to buy ANSWER:   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. a. p(x) = 0.65x; t(x) = 1.0625x     a. Write two functions representing the price after b. Since = , either function the discount p(x) and the price after sales tax t(x). represents the price.     b. Which composition of functions represents the c. $1587.75 price of the HDTV, or ? Explain   your reasoning.   Perform each operation if f(x) = x2 + x – 12 and c. How much will Ms. Smith pay for the HDTV? g(x) = x – 3. State the domain of the resulting   function.   SOLUTION:   38.  a. Let p(x) be the price after the discount where x is   the original price. Discount = 35%(x) = 0.35x SOLUTION:   Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x D = {all real numbers} Therefore:   ANSWER:   2   (f – g)(x) = x – 9; D = {all real numbers} b. Since = , either function   represents the price.   39.  c.    SOLUTION:   Substitute x = 2299. D = {all real numbers}     ANSWER:   Therefore, Mr. Smith will pay $1587.75 for HDTV. ; D = {all real   numbers} ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x   40.  b. Since = , either function represents the price.     SOLUTION:   c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function.   38.    SOLUTION:     ANSWER:   D = {all real numbers}   ANSWER:     2 (f – g)(x) = x – 9; D = {all real numbers} 2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x +   8, find each value.   39.  41.      SOLUTION:   SOLUTION:   D = {all real numbers} Substitute x = –2.   ANSWER:     ; D = {all real numbers} ANSWER:     25   40.  42.      SOLUTION:   SOLUTION:   Substitute x = 3.     ANSWER:   –69 ANSWER:     43.      SOLUTION:   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   41.      Substitute x = –5.   SOLUTION:   Substitute x = –2.   ANSWER:     483   ANSWER:   25 44.      SOLUTION:   42.    SOLUTION:     Substitute x = 2. Substitute x = 3.     ANSWER:   –69     ANSWER:   43.  –1     SOLUTION:   45.    SOLUTION:     Substitute x = –5.     Substitute x = –3.     ANSWER:     483   ANSWER:   –5 44.      46.  SOLUTION:     SOLUTION:     Substitute x = 2.     Substitute x = 9.       ANSWER:   ANSWER:   –1 2303     45.  47.      SOLUTION:   SOLUTION:       Substitute x = –3. ANSWER:     –30a + 5   48.      SOLUTION:   ANSWER:   –5   46.      SOLUTION:   ANSWER:   2 5a + 70a + 240     Substitute x = 9. 49.      SOLUTION:     ANSWER:   2303     ANSWER:   47.  2 –10a + 10a + 1     SOLUTION:   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     ANSWER:   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid. –30a + 5     c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid. 48.      d. VERBAL Describe the relationship among the SOLUTION:   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   ANSWER:   2 5a + 70a + 240     b. 49.    SOLUTION:     ANSWER:     2 –10a + 10a + 1 c.   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the d. Sample answer: For each value of x, the vertical graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). distance between the graph of g(x) and the x-axis is   the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) SOLUTION:   (x). a.   ANSWER:   a.   b. b.   c. c. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs 51. EMPLOYMENT The number of women and men of f(x) and (f + g)(x) and between f(x) and (f – g) age 16 and over employed each year in the United (x). States can be modeled by the following equations,   where x is the number of years since 1994 and y is the number of people in thousands. ANSWER:   women: y =1086.4x + 56,610 a. men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is b. the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference c. in the number of men and women employed in the U.S.   ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S.   If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x2 – 2x d. Sample answer: For each value of x, the vertical + 1, find each value. distance between the graph of g(x) and the x-axis is   the same as the vertical distance between the graphs 52.  of f(x) and (f + g)(x) and between f(x) and (f – g) (x).     SOLUTION:   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United   States can be modeled by the following equations, Substitute x = 3. where x is the number of years since 1994 and y is   the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of ANSWER:   men and women employed in the United States during this time. –180     b. If f is the function for the number of men, and g is the function for the number of women, what does ( f 53.  – g)(x) represent?     SOLUTION:   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   Substitute x = 1.     b. The function represents the difference in the number of men and women employed in the U.S.     ANSWER:   a. y = 2085.6x +123,060 ANSWER:     0 b. The function represents the difference in the   number of men and women employed in the U.S.   54.  2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value.   SOLUTION:   52.    SOLUTION:     Substitute x = 3.     ANSWER:     ANSWER:   –180 55.      SOLUTION:   53.    SOLUTION:       Substitute x = 1. ANSWER:     1   56.      ANSWER:   SOLUTION:   0   54.      SOLUTION:   ANSWER:   –33   57.      SOLUTION:   ANSWER:       55.    ANSWER:   256 SOLUTION:     58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.     a. Tabular Make a table showing values for ANSWER:     ,  , , and 1 .      56.  b. Graphical Use a graphing calculator to   graph  ,  , , and SOLUTION:   on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate ANSWER:   plane. –33     e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x). 57.      SOLUTION:   SOLUTION:   a.   ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .        b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .     b. d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:     a.       b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.      2.  SOLUTION:     SOLUTION:                 ANSWER:     ANSWER:     2.      For each pair of functions, find and , if SOLUTION:   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined.       Therefore:   ANSWER:       ANSWER:   is undefined; 6-1 Operations on Functions     For each pair of functions, find and , if 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} they exist. State the domain and range for each g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} composed function.     SOLUTION:   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined. The range of is not a subset of the domain of   . So,  is undefined.   Therefore:     Therefore:   ANSWER:   is undefined;   Find and , if they exist. State   the domain and range for each composed function.     5.  ANSWER:   is undefined;   SOLUTION:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:     Therefore:   eSolutionsManual-PoweredbyCognero Page2   6.  ANSWER:   is undefined;     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:       7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either ANSWER:   before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are   taken before or after taxes are withheld? Explain.   SOLUTION:   6.  Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college SOLUTION:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     7. CCSS MODELING Dora has 8% of her earnings 8.  deducted from her paycheck for a college savings plan. She can choose to take the deduction either   before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate SOLUTION:   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.    SOLUTION:                 ANSWER:       ANSWER:     9.      SOLUTION:     10.      SOLUTION:               ANSWER:     ANSWER:         10.      11.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:         11.  12.    SOLUTION:   SOLUTION:     ANSWER:   ANSWER:       13.  12.    SOLUTION:   SOLUTION:       ANSWER:   ANSWER:       14.  13.      SOLUTION:   SOLUTION:       ANSWER:     ANSWER:       14.    15.  SOLUTION:     SOLUTION:       ANSWER:   ANSWER:       15.  16. POPULATION In a particular county, the population of the two largest cities can be modeled   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is SOLUTION:   in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each   composed function.   16. POPULATION In a particular county, the 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} population of the two largest cities can be modeled g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is in thousands. SOLUTION:   a. What is the population of the two cities combined after any number of years?  Therefore: b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)}   g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   ANSWER:   SOLUTION:     Therefore: 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:       ANSWER:   ANSWER:       18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)}   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of .   So,  is undefined. ANSWER:     is ANSWER:   undefined; is undefined;  is undefined.     21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)} ANSWER:     is SOLUTION:   undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined. The range of is not a subset of the domain of   . 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} So,  is undefined. g = {(1, –4), (2, –3), (3, –2), (4, –1)}     ANSWER:   SOLUTION:   ; is undefined. The range of is not a subset of the domain of   . 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} So,  is undefined. g = {(4, –3), (3, –7), (9, 8), (–4, –4)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     ANSWER:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     Find and , if they exist. State the domain and range for each composed function. 28.      27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers} ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers}     29.  28.      SOLUTION:   SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     29.  30.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –14} real numbers} , D = {all real numbers}, R = {y | y   ≥ –10}   30.  31.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –10} , D = {all real numbers}, R = {y | y   ≥ –17}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –10} , D = {all real numbers}, R = {y | y   ≥ –17}   31.  32.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   32.  33.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5} , D = {all real numbers}, R = {all   real numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     34.  35.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0}   35.  36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the SOLUTION:   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:   , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y b. ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0} a. P(x) = 5.75x – 1850     b. P(500) = $1025; P(1000) = $3900; P(5000) = 36. FINANCE A ceramics store manufactures and sells $26,900 coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs 37. CCSS SENSE-MAKING Ms. Smith wants to buy is c(x) = 0.75x + 1850. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. a. Write the profit function.     a. Write two functions representing the price after b. Find the profit on 500, 1000, and 5000 coffee the discount p(x) and the price after sales tax t(x). mugs.     b. Which composition of functions represents the price of the HDTV, or ? Explain SOLUTION:   a. The profit function P(x) is given by your reasoning.    where r(x) is the revenue c. How much will Ms. Smith pay for the HDTV? function and c(x) is the cost function.   So: SOLUTION:   a. Let p(x) be the price after the discount where x is   the original price. b. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore:   ANSWER:   b. Since = , either function a. P(x) = 5.75x – 1850 represents the price.     b. P(500) = $1025; P(1000) = $3900; P(5000) = c.  $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%.   Substitute x = 2299. a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     b. Which composition of functions represents the Therefore, Mr. Smith will pay $1587.75 for HDTV.   price of the HDTV, or ? Explain your reasoning. ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x c. How much will Ms. Smith pay for the HDTV?     b. Since = , either function SOLUTION:   represents the price. a. Let p(x) be the price after the discount where x is   the original price. c. $1587.75 Discount = 35%(x) = 0.35x   Therefore: 2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting Let t(x) be the price after the sales tax. function. Sales tax = 6.25%(x) = 0.0625x   Therefore: 38.      SOLUTION:   b. Since = , either function represents the price.   c.  D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   Substitute x = 2299. 39.      Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price.   ANSWER:   c. $1587.75 ; D = {all real   numbers}   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. 40.    38.      SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     ANSWER:   39.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. D = {all real numbers}     41.  ANSWER:     ; D = {all real SOLUTION:   numbers}   40.  Substitute x = –2.   SOLUTION:     ANSWER:   25   42.    SOLUTION:     ANSWER:   Substitute x = 3.     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + ANSWER:   8, find each value. –69     41.    43.    SOLUTION:   SOLUTION:   Substitute x = –2.   Substitute x = –5.     ANSWER:   25     42.    ANSWER:   483 SOLUTION:     44.    Substitute x = 3. SOLUTION:     ANSWER:   –69     Substitute x = 2.   43.    SOLUTION:       ANSWER:   Substitute x = –5. –1     45.    SOLUTION:     ANSWER:   483     44.  Substitute x = –3.     SOLUTION:     ANSWER:   –5     Substitute x = 2.   46.    SOLUTION:       ANSWER:   Substitute x = 9. –1     45.    SOLUTION:     ANSWER:   2303     47.  Substitute x = –3.     SOLUTION:       ANSWER:   –5 ANSWER:     –30a + 5   46.    48.  SOLUTION:     SOLUTION:     Substitute x = 9.     ANSWER:   2 5a + 70a + 240     ANSWER:   49.  2303     SOLUTION:   47.    SOLUTION:       ANSWER:   2 –10a + 10a + 1 ANSWER:     –30a + 5   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x. 48.      a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x). SOLUTION:     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the ANSWER:   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). 2   5a + 70a + 240   SOLUTION:   a. 49.    SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) c. on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) b. (x).   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is ANSWER:   the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) a. (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is b. the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   c. SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is ANSWER:   the same as the vertical distance between the graphs a. y = 2085.6x +123,060 of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). b. The function represents the difference in the   number of men and women employed in the U.S.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United 2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x States can be modeled by the following equations, + 1, find each value. where x is the number of years since 1994 and y is the number of people in thousands.   women: y =1086.4x + 56,610 52.  men: y = 999.2x +66,450     a. Write a function that models the total number of SOLUTION:   men and women employed in the United States during this time.     Substitute x = 3. b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. ANSWER:   The total number of men and women employed is –180 given by   53.      b. The function represents the difference in the number of men and women employed in the SOLUTION:   U.S.   ANSWER:   a. y = 2085.6x +123,060     Substitute x = 1. b. The function represents the difference in the   number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value.     ANSWER:   52.  0     SOLUTION:   54.    Substitute x = 3.     SOLUTION:   ANSWER:   –180     53.  ANSWER:     SOLUTION:     55.      SOLUTION:   Substitute x = 1.     ANSWER:     0 ANSWER:     1   54.  56.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   –33     57.  55.      SOLUTION:   SOLUTION:     ANSWER:     256 ANSWER:     1 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and 56.  2  if f(x) = x + 1 and g(x) = x – 3.     SOLUTION:   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to   graph  ,  , , and ANSWER:   on the same coordinate plane. –33     c. Verbal Explain the relationship between 57.   and  .     SOLUTION:   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   ANSWER:   a. 256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .       d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   b. a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:         Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any ANSWER:   restrictions in domain or range.   1.    SOLUTION:       2.    SOLUTION:             ANSWER:         ANSWER:   2.    SOLUTION:     For each pair of functions, find and , if   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.       Therefore: ANSWER:         ANSWER:   For each pair of functions, find and , if is undefined; they exist. State the domain and range for each composed function.     3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)}   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.     Therefore:   Therefore:     ANSWER:   is undefined;     ANSWER:   Find and , if they exist. State is undefined; the domain and range for each composed function.     5.  4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.       Therefore:     ANSWER:   ANSWER:   is undefined; 6-1 Operations on Functions     Find and , if they exist. State the domain and range for each composed 6.  function.     SOLUTION:   5.    SOLUTION:         ANSWER:     ANSWER:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate 6.  is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are   taken before or after taxes are withheld? Explain. SOLUTION:     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college ANSWER:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only eSolutionsManual-PoweredbyCognero $62.70 will go to her college plan and $166.25 wiPlla ggeo 3 to taxes.     Find (f + g)(x), (f – g)(x), (x),and 7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   (x) for each f(x) and g(x). Indicate any plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable restrictions in domain or range. income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are 8.  taken before or after taxes are withheld? Explain.     SOLUTION:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. ANSWER:     8.    SOLUTION:       9.      SOLUTION:           ANSWER:           ANSWER:   9.    SOLUTION:         10.      SOLUTION:         ANSWER:         ANSWER:       10.    SOLUTION:       11.    SOLUTION:         ANSWER:     ANSWER:       11.    SOLUTION:     12.  SOLUTION:     ANSWER:   ANSWER:     12.    SOLUTION:   13.    SOLUTION:     ANSWER:     ANSWER:     13.      SOLUTION:   14.    SOLUTION:       ANSWER:     ANSWER:     14.    SOLUTION:     15.    SOLUTION:     ANSWER:     ANSWER:     15.    SOLUTION:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   ANSWER:     b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10     b. (f - g)(x) = 25x + 40 16. POPULATION In a particular county, the   population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is For each pair of functions, find and , if the number of years since 2000 and the population is they exist. State the domain and range for each in thousands. composed function.   a. What is the population of the two cities combined 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} after any number of years?  g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   b. What is the difference in the populations of the two cities?  SOLUTION:     SOLUTION:   Therefore: a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:     Therefore: ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   ANSWER:   SOLUTION:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   The range of is not a subset of the domain of ANSWER:   .   So,  is undefined. 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     ANSWER:   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} is g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)} undefined;     SOLUTION:   The range of is not a subset of the domain of 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} .   So,  is undefined. SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   is undefined; . is   undefined; 22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)} 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)}   g = {(3, –9), (7, 2), (8, –6), (12, 0)} SOLUTION:     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} ANSWER:     is undefined; . SOLUTION:     22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined. ANSWER:     ; is undefined. ANSWER:     ; is undefined. 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)}   g = {(1, –4), (2, –3), (3, –2), (4, –1)} 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)}   g = {(4, –3), (3, –7), (9, 8), (–4, –4)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   The range of is not a subset of the domain of 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} . g = {(–1, 5), (3, –4), (6, 4), (10, 8)} So,  is undefined.     SOLUTION:   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   ANSWER:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all   even numbers} , D = {all real numbers}, R = {all ANSWER:   odd numbers} ANSWER:     Find and , if they exist. State , D = {all real numbers}, R = {all the domain and range for each composed even numbers} function. , D = {all real numbers}, R = {all   odd numbers}   27.    28.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all even numbers} ANSWER:   , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all 28.  real numbers}     SOLUTION:   29.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all 29.  real numbers} , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   30.    SOLUTION:       , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –10} real numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers}   30.  , D = {all real numbers}, R = {y | y ≥ –14}   , D = {all real numbers}, R = {y | y ≥ –10} SOLUTION:     31.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17} , D = {all real numbers}, R = {y | y   ≥ –14} , D = {all real numbers}, R = {y | y ANSWER:   ≥ –10}   , D = {all real numbers}, R = {y | y 31.  ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} SOLUTION:     32.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 6.5} ≥ –11}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ –17}   32.  , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} SOLUTION:     33.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 6.5}   , D = {all real numbers}, R = {all real numbers} 33.  , D = {all real numbers}, R = {all real numbers}     SOLUTION:   34.    SOLUTION:         , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}   , D = {all real numbers}, R = {all real numbers} ANSWER:     , D = {all real numbers}, R = {y | y 34.  ≥ 1} , D = {all real numbers}, R = {y | y   ≥ 0} SOLUTION:     35.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     35.  , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.     a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   , D = {all real numbers}, R = {y | y a. The profit function P(x) is given by ≥ 0}  where r(x) is the revenue , D = {all real numbers}, R = {y | y function and c(x) is the cost function. ≥ 0} So:   ANSWER:     b. , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the ANSWER:   function for the cost of manufacturing x coffee mugs a. P(x) = 5.75x – 1850 is c(x) = 0.75x + 1850.     b. P(500) = $1025; P(1000) = $3900; P(5000) = a. Write the profit function. $26,900     b. Find the profit on 500, 1000, and 5000 coffee mugs. 37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. SOLUTION:     a. The profit function P(x) is given by a. Write two functions representing the price after  where r(x) is the revenue the discount p(x) and the price after sales tax t(x). function and c(x) is the cost function.   So: b. Which composition of functions represents the price of the HDTV, or ? Explain your reasoning.     b. c. How much will Ms. Smith pay for the HDTV?   SOLUTION:   a. Let p(x) be the price after the discount where x is the original price. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. ANSWER:   Sales tax = 6.25%(x) = 0.0625x a. P(x) = 5.75x – 1850 Therefore:   b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900     b. Since = , either function 37. CCSS SENSE-MAKING Ms. Smith wants to buy represents the price. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. c.    a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the price of the HDTV, or ? Explain Substitute x = 2299. your reasoning.   c. How much will Ms. Smith pay for the HDTV?     Therefore, Mr. Smith will pay $1587.75 for HDTV.   SOLUTION:   a. Let p(x) be the price after the discount where x is ANSWER:   the original price. a. p(x) = 0.65x; t(x) = 1.0625x Discount = 35%(x) = 0.35x   Therefore: b. Since = , either function represents the price.   Let t(x) be the price after the sales tax. c. $1587.75 Sales tax = 6.25%(x) = 0.0625x Therefore:   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting   function. b. Since = , either function   represents the price. 38.      c.  SOLUTION:   Substitute x = 2299. D = {all real numbers}   ANSWER:     2 Therefore, Mr. Smith will pay $1587.75 for HDTV. (f – g)(x) = x – 9; D = {all real numbers}     ANSWER:   39.  a. p(x) = 0.65x; t(x) = 1.0625x     b. Since = , either function SOLUTION:   represents the price.   c. $1587.75   D = {all real numbers} Perform each operation if f(x) = x2 + x – 12 and   g(x) = x – 3. State the domain of the resulting ANSWER:   function.   ; D = {all real 38.  numbers}     SOLUTION:   40.    SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   39.    SOLUTION:     ANSWER:   D = {all real numbers}     ANSWER:   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + ; D = {all real 8, find each value. numbers}     41.    40.  SOLUTION:     SOLUTION:   Substitute x = –2.   ANSWER:   25     42.  ANSWER:     SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + Substitute x = 3. 8, find each value.   41.      SOLUTION:   ANSWER:   –69   43.  Substitute x = –2.   SOLUTION:     ANSWER:   25     Substitute x = –5.   42.    SOLUTION:     ANSWER:   Substitute x = 3. 483   44.      ANSWER:   SOLUTION:   –69   43.    SOLUTION:     Substitute x = 2.     Substitute x = –5.     ANSWER:   –1     ANSWER:   45.  483     SOLUTION:   44.    SOLUTION:     Substitute x = –3.     Substitute x = 2.     ANSWER:   –5   46.      SOLUTION:   ANSWER:   –1   45.      Substitute x = 9.   SOLUTION:       Substitute x = –3.   ANSWER:   2303   47.      ANSWER:   SOLUTION:   –5   46.      SOLUTION:   ANSWER:   –30a + 5   48.      Substitute x = 9.   SOLUTION:       ANSWER:   2303 ANSWER:     2 5a + 70a + 240   47.    49.  SOLUTION:     SOLUTION:     ANSWER:   –30a + 5     48.  ANSWER:   2   –10a + 10a + 1   SOLUTION:   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid. ANSWER:     2 5a + 70a + 240 c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   49.  d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     SOLUTION:   SOLUTION:   a.   ANSWER:     b. 2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     c. d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a.   c. b. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). b.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time. c.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs b. The function represents the difference of f(x) and (f + g)(x) and between f(x) and (f – g) in the number of men and women employed in the (x). U.S.     51. EMPLOYMENT The number of women and men ANSWER:   age 16 and over employed each year in the United a. y = 2085.6x +123,060 States can be modeled by the following equations,   where x is the number of years since 1994 and y is b. The function represents the difference in the the number of people in thousands. number of men and women employed in the U.S. women: y =1086.4x + 56,610   men: y = 999.2x +66,450   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x a. Write a function that models the total number of + 1, find each value. men and women employed in the United States   during this time.   52.  b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f SOLUTION:   – g)(x) represent?     SOLUTION:   Substitute x = 3. a. Add the functions.   The total number of men and women employed is given by   ANSWER:   b. The function represents the difference –180 in the number of men and women employed in the   U.S.   53.  ANSWER:     a. y = 2085.6x +123,060   SOLUTION:   b. The function represents the difference in the number of men and women employed in the U.S.   If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x2 – 2x   + 1, find each value. Substitute x = 1.     52.    SOLUTION:       ANSWER:   Substitute x = 3. 0     54.    ANSWER:   SOLUTION:   –180   53.    SOLUTION:     ANSWER:     Substitute x = 1.     55.    SOLUTION:     ANSWER:   0   54.      ANSWER:   SOLUTION:   1   56.    SOLUTION:     ANSWER:       55.  ANSWER:     –33 SOLUTION:     57.    SOLUTION:     ANSWER:   1     ANSWER:   56.  256     SOLUTION:   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and   .  ANSWER:   –33     b. Graphical Use a graphing calculator to graph  ,  , , and 57.    on the same coordinate plane. SOLUTION:     c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate plane. ANSWER:     256 e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and SOLUTION:   a. 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.     e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   SOLUTION:   a.   b.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:         Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. ANSWER:     1.    SOLUTION:       2.    SOLUTION:             ANSWER:         ANSWER:   2.    SOLUTION:       For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.       Therefore: ANSWER:         For each pair of functions, find and , if ANSWER:   they exist. State the domain and range for each is undefined; composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.     Therefore:   Therefore:     ANSWER:   is undefined;     ANSWER:   Find and , if they exist. State is undefined; the domain and range for each composed function.     5.  4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.       Therefore:     ANSWER:   ANSWER:   is undefined;     Find and , if they exist. State the domain and range for each composed 6.  function.     5.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable 6.  income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain. SOLUTION:     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college ANSWER:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go 6-1 Operations on Functions to taxes.     7. CCSS MODELING Dora has 8% of her earnings Find (f + g)(x), (f – g)(x), (x),and deducted from her paycheck for a college savings plan. She can choose to take the deduction either   (x) for each f(x) and g(x). Indicate any before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate restrictions in domain or range. is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain. 8.      SOLUTION:   Either way, she will have $228.95 taken from her SOLUTION:   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range.   ANSWER:   8.    SOLUTION:       9.      SOLUTION:   eSolutionsManual-PoweredbyCognero Page4         ANSWER:           ANSWER:   9.    SOLUTION:         10.      SOLUTION:         ANSWER:         ANSWER:       10.    SOLUTION:       11.    SOLUTION:         ANSWER:     ANSWER:       11.    SOLUTION:     12.  SOLUTION:     ANSWER:   ANSWER:     12.    SOLUTION:   13.    SOLUTION:     ANSWER:     ANSWER:     13.      SOLUTION:   14.    SOLUTION:       ANSWER:     ANSWER:     14.    SOLUTION:     15.    SOLUTION:     ANSWER:     ANSWER:     15.    SOLUTION:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   ANSWER:     b. The difference in the populations of the cities is given by:   ANSWER:     a. (f + g)(x) = 375x + 10   16. POPULATION In a particular county, the b. (f - g)(x) = 25x + 40 population of the two largest cities can be modeled   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is For each pair of functions, find and , if in thousands. they exist. State the domain and range for each composed function. a. What is the population of the two cities combined   after any number of years?  17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} b. What is the difference in the populations of the   two cities?    SOLUTION:   SOLUTION:   a. The population of the cities after x years is the Therefore: sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}     ANSWER:   SOLUTION:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of .   So,  is undefined. 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     ANSWER:   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} is g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   undefined;   SOLUTION:   The range of is not a subset of the domain of 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} .   So,  is undefined. SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   is undefined; . is   undefined;   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)} 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)}   g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} ANSWER:     is undefined; . SOLUTION:     22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.   ANSWER:   ; is undefined. ANSWER:     ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)} 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)}   g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   The range of is not a subset of the domain of 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} . g = {(–1, 5), (3, –4), (6, 4), (10, 8)} So,  is undefined.     SOLUTION:   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   ANSWER:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all   even numbers} ANSWER:   , D = {all real numbers}, R = {all odd numbers} ANSWER:     Find and , if they exist. State , D = {all real numbers}, R = {all the domain and range for each composed even numbers} function. , D = {all real numbers}, R = {all   odd numbers} 27.      28.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   even numbers} ANSWER:   , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all real numbers} 28.  , D = {all real numbers}, R = {all real numbers}     SOLUTION:   29.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all 29.  real numbers} , D = {all real numbers}, R = {all   real numbers} SOLUTION:     30.    SOLUTION:       , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –10} real numbers}   , D = {all real numbers}, R = {all real numbers} ANSWER:     30.  , D = {all real numbers}, R = {y | y ≥ –14}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –10}   31.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ –14}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ –10}   31.  , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} SOLUTION:     32.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 6.5} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} ANSWER:     32.  , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 6.5}   33.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} ANSWER:     , D = {all real numbers}, R = {all real numbers} 33.  , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   34.    SOLUTION:         , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0} , D = {all real numbers}, R = {all   real numbers} ANSWER:     , D = {all real numbers}, R = {y | y 34.  ≥ 1} , D = {all real numbers}, R = {y | y   ≥ 0} SOLUTION:     35.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0} ANSWER:     35.  , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs   is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   , D = {all real numbers}, R = {y | y a. The profit function P(x) is given by ≥ 0}  where r(x) is the revenue , D = {all real numbers}, R = {y | y function and c(x) is the cost function. ≥ 0} So:   ANSWER:     b. , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the ANSWER:   function for the cost of manufacturing x coffee mugs a. P(x) = 5.75x – 1850 is c(x) = 0.75x + 1850.     b. P(500) = $1025; P(1000) = $3900; P(5000) = a. Write the profit function. $26,900     b. Find the profit on 500, 1000, and 5000 coffee mugs. 37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. SOLUTION:     a. The profit function P(x) is given by a. Write two functions representing the price after  where r(x) is the revenue the discount p(x) and the price after sales tax t(x). function and c(x) is the cost function.   So: b. Which composition of functions represents the price of the HDTV, or ? Explain your reasoning.     b. c. How much will Ms. Smith pay for the HDTV?   SOLUTION:   a. Let p(x) be the price after the discount where x is the original price. Discount = 35%(x) = 0.35x Therefore: ANSWER:   Let t(x) be the price after the sales tax. a. P(x) = 5.75x – 1850 Sales tax = 6.25%(x) = 0.0625x   Therefore: b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900     b. Since = , either function 37. CCSS SENSE-MAKING Ms. Smith wants to buy represents the price. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. c.    a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the price of the HDTV, or ? Explain Substitute x = 2299. your reasoning.   c. How much will Ms. Smith pay for the HDTV?     Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     a. Let p(x) be the price after the discount where x is ANSWER:   the original price. Discount = 35%(x) = 0.35x a. p(x) = 0.65x; t(x) = 1.0625x Therefore:   b. Since = , either function represents the price. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x c. $1587.75 Therefore:   2 Perform each operation if f(x) = x + x – 12 and   g(x) = x – 3. State the domain of the resulting b. Since = , either function function.   represents the price.   38.  c.    SOLUTION:   Substitute x = 2299. D = {all real numbers}     ANSWER:   Therefore, Mr. Smith will pay $1587.75 for HDTV. (f – g)(x) = x2 – 9; D = {all real numbers}     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x 39.      b. Since = , either function SOLUTION:   represents the price.   c. $1587.75   D = {all real numbers} 2 Perform each operation if f(x) = x + x – 12 and   g(x) = x – 3. State the domain of the resulting function. ANSWER:     ; D = {all real 38.  numbers}     SOLUTION:   40.    SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   39.    SOLUTION:     ANSWER:   D = {all real numbers}     ANSWER:   ; D = {all real If f(x) = 5x, g(x) = –2x + 1, and h(x) = x2 + 6x + numbers} 8, find each value.     41.  40.    SOLUTION:     SOLUTION:   Substitute x = –2.   ANSWER:   25     ANSWER:   42.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. Substitute x = 3.   41.      SOLUTION:   ANSWER:   –69   43.  Substitute x = –2.   SOLUTION:     ANSWER:   25     Substitute x = –5.   42.    SOLUTION:     ANSWER:   Substitute x = 3. 483   44.      ANSWER:   SOLUTION:   –69   43.    SOLUTION:     Substitute x = 2.     Substitute x = –5.     ANSWER:   –1     ANSWER:   45.  483     SOLUTION:   44.    SOLUTION:     Substitute x = –3.     Substitute x = 2.     ANSWER:   –5   46.      ANSWER:   SOLUTION:   –1   45.      Substitute x = 9. SOLUTION:       Substitute x = –3.     ANSWER:   2303   47.      ANSWER:   SOLUTION:   –5   46.      SOLUTION:   ANSWER:   –30a + 5   48.    Substitute x = 9.     SOLUTION:       ANSWER:   2303 ANSWER:     2 5a + 70a + 240   47.    49.  SOLUTION:     SOLUTION:     ANSWER:   –30a + 5     48.  ANSWER:     2 –10a + 10a + 1 SOLUTION:     50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) ANSWER:   on the same coordinate grid. 2   5a + 70a + 240 c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   49.  d. VERBAL Describe the relationship among the   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   SOLUTION:   a.   ANSWER:     –10a2 + 10a + 1 b.   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the c. graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a.   c. b. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). b.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States c. during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is   the same as the vertical distance between the graphs b. The function represents the difference of f(x) and (f + g)(x) and between f(x) and (f – g) in the number of men and women employed in the (x). U.S.     51. EMPLOYMENT The number of women and men ANSWER:   age 16 and over employed each year in the United a. y = 2085.6x +123,060 States can be modeled by the following equations,   where x is the number of years since 1994 and y is the number of people in thousands. b. The function represents the difference in the women: y =1086.4x + 56,610 number of men and women employed in the U.S. men: y = 999.2x +66,450     2 a. Write a function that models the total number of If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x men and women employed in the United States + 1, find each value. during this time.     52.  b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent? SOLUTION:     SOLUTION:     Substitute x = 3. a. Add the functions.   The total number of men and women employed is given by   ANSWER:   b. The function represents the difference –180 in the number of men and women employed in the   U.S.   53.  ANSWER:     a. y = 2085.6x +123,060   SOLUTION:   b. The function represents the difference in the number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value. Substitute x = 1.     52.    SOLUTION:       ANSWER:   Substitute x = 3. 0     54.    ANSWER:   –180 SOLUTION:     53.    SOLUTION:     ANSWER:     Substitute x = 1.     55.      SOLUTION:   ANSWER:   0   54.      SOLUTION:   ANSWER:   1   56.    SOLUTION:     ANSWER:       55.  ANSWER:     –33 SOLUTION:     57.    SOLUTION:     ANSWER:   1     ANSWER:   56.  256     SOLUTION:   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for     ,  , , and ANSWER:   .  –33     b. Graphical Use a graphing calculator to 57.  graph  ,  , , and   on the same coordinate plane. SOLUTION:     c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate plane. ANSWER:     256 e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x). 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and SOLUTION:   a. 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.     e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   SOLUTION:   a.   b.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range. 2.      1.  SOLUTION:     SOLUTION:                   ANSWER:   ANSWER:       For each pair of functions, find and , if they exist. State the domain and range for each composed function.   2.  3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.       Therefore:         ANSWER:     is undefined; ANSWER:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:     The range of is not a subset of the domain of For each pair of functions, find and , if . they exist. State the domain and range for each So,  is undefined. composed function.     3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     Therefore: SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:     is undefined;     Therefore: Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of ANSWER:   . So,  is undefined.       6.  Therefore:   SOLUTION:     ANSWER:   is undefined;     Find and , if they exist. State the domain and range for each composed function.   5.      ANSWER:   SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     ANSWER:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go 6.  to taxes.     ANSWER:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     ANSWER:   8.    SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan ANSWER:   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   9.  8.    SOLUTION:     SOLUTION:                     ANSWER:   ANSWER:   6-1 Operations on Functions       9.  10.      SOLUTION:   SOLUTION:               ANSWER:       ANSWER:         11.      10.  SOLUTION:     SOLUTION:   eSolutionsManual-PoweredbyCognero Page5     ANSWER:       ANSWER:     12.    SOLUTION:     11.    SOLUTION:   ANSWER:     ANSWER:     13.    SOLUTION:     12.  SOLUTION:       ANSWER:   ANSWER:     14.    SOLUTION:     13.    SOLUTION:       ANSWER:     ANSWER:     15.      SOLUTION:   14.    SOLUTION:     ANSWER:     ANSWER:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is   in thousands. a. What is the population of the two cities combined 15.  after any number of years?    b. What is the difference in the populations of the two cities?  SOLUTION:     SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10     b. (f - g)(x) = 25x + 40   ANSWER:   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}     SOLUTION:   16. POPULATION In a particular county, the population of the two largest cities can be modeled Therefore: by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40     For each pair of functions, find and , if ANSWER:   they exist. State the domain and range for each composed function.     17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:   SOLUTION:   Therefore:   ANSWER:       19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} ANSWER:   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   The range of is not a subset of the domain of 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} .   So,  is undefined. SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is So,  is undefined. undefined; The range of is not a subset of the domain of   . So,  is undefined. 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)}   g = {(3, –9), (7, 2), (8, –6), (12, 0)}   ANSWER:   SOLUTION:   is undefined;  is undefined.   The range of is not a subset of the domain of . 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} So,  is undefined. g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:     ANSWER:   is undefined;   The range of is not a subset of the domain of . 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} So,  is undefined.     SOLUTION:   ANSWER:   The range of is not a subset of the domain of ; is undefined.   . So,  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   The range of is not a subset of the domain of . 22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)} So,  is undefined.     SOLUTION:   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   The range of is not a subset of the domain of SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. .   So,  is undefined. The range of is not a subset of the domain of ANSWER:   ; is undefined. .   So,  is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} ANSWER:     is undefined;  is undefined.   SOLUTION:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined.   ANSWER:     is undefined;  is undefined.   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)} 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)}   g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   SOLUTION:       ANSWER:   ANSWER:       Find and , if they exist. State 26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} the domain and range for each composed function. g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     ANSWER:   28.      SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all odd numbers} real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all 29.  even numbers} , D = {all real numbers}, R = {all   odd numbers} SOLUTION:     28.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all real numbers} 30.  , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   29.    SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {y | y ≥ –14} ANSWER:   , D = {all real numbers}, R = {y | y ≥ –10}   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all 31.  real numbers}     SOLUTION:   30.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}     , D = {all real numbers}, R = {y | y ANSWER:   ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   , D = {all real numbers}, R = {y | y ≥ –14} 32.  , D = {all real numbers}, R = {y | y ≥ –10}     SOLUTION:   31.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     , D = {all real numbers}, R = {y | y ≥ –11} ANSWER:   , D = {all real numbers}, R = {y | y ≥ –17}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y 33.  ≥ –17}     SOLUTION:   32.    SOLUTION:         , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {y | y ≥ 0.875} ANSWER:   , D = {all real numbers}, R = {y | y ≥ 6.5} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {y | y 34.  ≥ 0.875} , D = {all real numbers}, R = {y | y   ≥ 6.5} SOLUTION:     33.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0}   , D = {all real numbers}, R = {all real numbers} 35.  , D = {all real numbers}, R = {all real numbers}     SOLUTION:   34.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}     , D = {all real numbers}, R = {y | y ANSWER:   ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 1} , D = {all real numbers}, R = {y | y 36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x ≥ 0} coffee mugs is given by r(x) = 6.5x. Suppose the   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   35.  a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee SOLUTION:   mugs.   SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function.   So:   b.   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:   ANSWER:   a. P(x) = 5.75x – 1850   b. P(500) = $1025; P(1000) = $3900; P(5000) = , D = {all real numbers}, R = {y | y $26,900 ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} 37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x a. Write two functions representing the price after coffee mugs is given by r(x) = 6.5x. Suppose the the discount p(x) and the price after sales tax t(x). function for the cost of manufacturing x coffee mugs   is c(x) = 0.75x + 1850. b. Which composition of functions represents the   price of the HDTV, or ? Explain a. Write the profit function. your reasoning.     b. Find the profit on 500, 1000, and 5000 coffee c. How much will Ms. Smith pay for the HDTV? mugs.     SOLUTION:   SOLUTION:   a. Let p(x) be the price after the discount where x is a. The profit function P(x) is given by the original price.  where r(x) is the revenue Discount = 35%(x) = 0.35x function and c(x) is the cost function. Therefore: So: Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x b. Therefore:   b. Since = , either function represents the price.   c.  ANSWER:   a. P(x) = 5.75x – 1850   b. P(500) = $1025; P(1000) = $3900; P(5000) = Substitute x = 2299. $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original Therefore, Mr. Smith will pay $1587.75 for HDTV. price of $2299. The sales tax is 6.25%.     ANSWER:   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x). a. p(x) = 0.65x; t(x) = 1.0625x     b. Which composition of functions represents the b. Since = , either function price of the HDTV, or ? Explain represents the price. your reasoning.     c. $1587.75 c. How much will Ms. Smith pay for the HDTV?     2 Perform each operation if f(x) = x + x – 12 and SOLUTION:   g(x) = x – 3. State the domain of the resulting a. Let p(x) be the price after the discount where x is function. the original price.   Discount = 35%(x) = 0.35x 38.  Therefore:   SOLUTION:   Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore: D = {all real numbers}     b. Since = , either function ANSWER:   represents the price. 2 (f – g)(x) = x – 9; D = {all real numbers}     c.  39.    SOLUTION:   Substitute x = 2299.   D = {all real numbers} Therefore, Mr. Smith will pay $1587.75 for HDTV.     ANSWER:   ANSWER:   ; D = {all real a. p(x) = 0.65x; t(x) = 1.0625x   numbers}   b. Since = , either function represents the price.   40.  c. $1587.75     SOLUTION:   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function.   38.    SOLUTION:     D = {all real numbers} ANSWER:     ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     39.  If f(x) = 5x, g(x) = –2x + 1, and h(x) = x2 + 6x + 8, find each value.     SOLUTION:   41.    SOLUTION:   D = {all real numbers}   ANSWER:   Substitute x = –2. ; D = {all real numbers}     ANSWER:   40.  25     SOLUTION:   42.    SOLUTION:   Substitute x = 3.     ANSWER:   ANSWER:   –69     43.  2   If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. SOLUTION:     41.    SOLUTION:     Substitute x = –5.   Substitute x = –2.     ANSWER:   ANSWER:   25 483     44.  42.      SOLUTION:   SOLUTION:   Substitute x = 3.   Substitute x = 2.     ANSWER:   –69   43.      ANSWER:   SOLUTION:   –1   45.      Substitute x = –5. SOLUTION:       Substitute x = –3.     ANSWER:   483     44.    ANSWER:   –5 SOLUTION:     46.    SOLUTION:     Substitute x = 2.     Substitute x = 9.     ANSWER:   –1     ANSWER:   2303 45.      SOLUTION:   47.    SOLUTION:     Substitute x = –3.     ANSWER:   –30a + 5     48.  ANSWER:     –5   SOLUTION:   46.    SOLUTION:     ANSWER:     2 5a + 70a + 240 Substitute x = 9.     49.    SOLUTION:     ANSWER:   2303   47.      ANSWER:   SOLUTION:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.     a. TABULAR Make a table showing values for f ANSWER:   (x), g(x), ( f + g) (x), and ( f –g)(x). –30a + 5     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid. 48.      c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid. SOLUTION:     d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   ANSWER:   2 5a + 70a + 240   49.    b.   SOLUTION:     ANSWER:   2 –10a + 10a + 1     50. MULTIPLE REPRESENTATIONS Let f(x) = x2 c. and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     d. Sample answer: For each value of x, the vertical SOLUTION:   distance between the graph of g(x) and the x-axis is a. the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a.   b. b.   c. c. d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is d. Sample answer: For each value of x, the vertical the same as the vertical distance between the graphs distance between the graph of g(x) and the x-axis is of f(x) and (f + g)(x) and between f(x) and (f – g) the same as the vertical distance between the graphs (x). of f(x) and (f + g)(x) and between f(x) and (f – g)   (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United ANSWER:   States can be modeled by the following equations, a. where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States b. during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by c.   b. The function represents the difference in the number of men and women employed in the U.S.   ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S. d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x2 – 2x of f(x) and (f + g)(x) and between f(x) and (f – g) + 1, find each value. (x).     52.  51. EMPLOYMENT The number of women and men   age 16 and over employed each year in the United SOLUTION:   States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands.   women: y =1086.4x + 56,610 Substitute x = 3. men: y = 999.2x +66,450     a. Write a function that models the total number of men and women employed in the United States during this time.   ANSWER:   b. If f is the function for the number of men, and g is –180 the function for the number of women, what does ( f   – g)(x) represent?   53.  SOLUTION:   a. Add the functions.   The total number of men and women employed is SOLUTION:   given by   b. The function represents the difference   in the number of men and women employed in the Substitute x = 1. U.S.     ANSWER:   a. y = 2085.6x +123,060     b. The function represents the difference in the ANSWER:   number of men and women employed in the U.S.   0   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value. 54.    52.      SOLUTION:   SOLUTION:     Substitute x = 3.     ANSWER:   ANSWER:   –180     55.  53.      SOLUTION:   SOLUTION:     Substitute x = 1.     ANSWER:   1     56.  ANSWER:   0     SOLUTION:   54.    SOLUTION:     ANSWER:   –33     57.    ANSWER:   SOLUTION:     55.      SOLUTION:   ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and   2  if f(x) = x + 1 and g(x) = x – 3. ANSWER:     1 a. Tabular Make a table showing values for     ,  , , and 56.  .      SOLUTION:   b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .     ANSWER:   d. Graphical Use a graphing calculator to graph [f –33 ◦ g](x), and [g ◦ f](x) on the same coordinate   plane.   57.  e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x). SOLUTION:     SOLUTION:   a.   ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and     on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate   plane. b.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:   a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     1.    2.  SOLUTION:     SOLUTION:                 ANSWER:     ANSWER:     2.      For each pair of functions, find and , if SOLUTION:   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined.       Therefore:     ANSWER:     ANSWER:   is undefined;     For each pair of functions, find and , if 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} they exist. State the domain and range for each g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} composed function.     SOLUTION:   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   The range of is not a subset of the domain of .   So,  is undefined. Therefore:     Therefore:   ANSWER:   is undefined;   Find and , if they exist. State the domain and range for each composed   function.     5.  ANSWER:   is undefined;   SOLUTION:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:     Therefore:     6.  ANSWER:   is undefined;     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable ANSWER:   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are   taken before or after taxes are withheld? Explain.   SOLUTION:   6.  Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the SOLUTION:   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and ANSWER:     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     7. CCSS MODELING Dora has 8% of her earnings 8.  deducted from her paycheck for a college savings plan. She can choose to take the deduction either   before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate SOLUTION:   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.    SOLUTION:                 ANSWER:       ANSWER:     9.        SOLUTION:   10.      SOLUTION:               ANSWER:     ANSWER:           10.    11.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:   6-1 O  perations on Functions     11.  12.    SOLUTION:   SOLUTION:     ANSWER:   ANSWER:       13.  12.    SOLUTION:   SOLUTION:     eSolutionsManual-PoweredbyCognero Page6   ANSWER:   ANSWER:       14.  13.    SOLUTION:     SOLUTION:       ANSWER:     ANSWER:       14.    15.  SOLUTION:     SOLUTION:       ANSWER:   ANSWER:       16. POPULATION In a particular county, the 15.  population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is SOLUTION:   in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each   composed function.   16. POPULATION In a particular county, the 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} population of the two largest cities can be modeled g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is in thousands. SOLUTION:   a. What is the population of the two cities combined after any number of years?  Therefore: b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)}   g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} ANSWER:     SOLUTION:     Therefore: 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:       ANSWER:   ANSWER:       18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of   . So,  is undefined. ANSWER:     is ANSWER:   undefined; is undefined;  is undefined.     21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   ANSWER:   is SOLUTION:   undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined. The range of is not a subset of the domain of   . 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} So,  is undefined. g = {(1, –4), (2, –3), (3, –2), (4, –1)}     ANSWER:   SOLUTION:   ; is undefined. The range of is not a subset of the domain of   . So,  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     is undefined;  is undefined.   ANSWER:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     Find and , if they exist. State the domain and range for each composed 28.  function.     27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers} ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers}     29.  28.      SOLUTION:   SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     29.  30.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –14} real numbers} , D = {all real numbers}, R = {y | y   ≥ –10}   30.  31.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     31.  32.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   32.  33.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5} , D = {all real numbers}, R = {all   real numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     34.  35.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0}   35.  36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the SOLUTION:   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:   , D = {all real numbers}, R = {y | y ≥ 0}   b. , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0} a. P(x) = 5.75x – 1850     b. P(500) = $1025; P(1000) = $3900; P(5000) = 36. FINANCE A ceramics store manufactures and sells $26,900 coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs 37. CCSS SENSE-MAKING Ms. Smith wants to buy is c(x) = 0.75x + 1850. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. a. Write the profit function.     a. Write two functions representing the price after b. Find the profit on 500, 1000, and 5000 coffee the discount p(x) and the price after sales tax t(x). mugs.     b. Which composition of functions represents the price of the HDTV, or ? Explain SOLUTION:   your reasoning. a. The profit function P(x) is given by    where r(x) is the revenue c. How much will Ms. Smith pay for the HDTV? function and c(x) is the cost function.   So: SOLUTION:   a. Let p(x) be the price after the discount where x is   the original price. b. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore:   ANSWER:   b. Since = , either function a. P(x) = 5.75x – 1850 represents the price.     b. P(500) = $1025; P(1000) = $3900; P(5000) = c.  $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. Substitute x = 2299.   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     Therefore, Mr. Smith will pay $1587.75 for HDTV. b. Which composition of functions represents the   price of the HDTV, or ? Explain your reasoning. ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x c. How much will Ms. Smith pay for the HDTV?     b. Since = , either function SOLUTION:   represents the price. a. Let p(x) be the price after the discount where x is   the original price. c. $1587.75 Discount = 35%(x) = 0.35x   Therefore: Perform each operation if f(x) = x2 + x – 12 and g(x) = x – 3. State the domain of the resulting Let t(x) be the price after the sales tax. function.   Sales tax = 6.25%(x) = 0.0625x Therefore: 38.    SOLUTION:     b. Since = , either function represents the price.   c.  D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   Substitute x = 2299. 39.      Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price.   ANSWER:   c. $1587.75 ; D = {all real   numbers}   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. 40.      38.    SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     ANSWER:   39.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. D = {all real numbers}     41.    ANSWER:   ; D = {all real SOLUTION:   numbers}   40.  Substitute x = –2.   SOLUTION:     ANSWER:   25   42.    SOLUTION:     ANSWER:   Substitute x = 3.     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + ANSWER:   8, find each value. –69     41.  43.      SOLUTION:   SOLUTION:   Substitute x = –2.   Substitute x = –5.     ANSWER:   25     42.    ANSWER:   483 SOLUTION:     44.    Substitute x = 3. SOLUTION:     ANSWER:   –69     Substitute x = 2.   43.    SOLUTION:       ANSWER:   Substitute x = –5. –1     45.    SOLUTION:     ANSWER:   483     Substitute x = –3. 44.      SOLUTION:     ANSWER:   –5     Substitute x = 2.   46.    SOLUTION:       Substitute x = 9. ANSWER:     –1   45.    SOLUTION:     ANSWER:   2303     47.  Substitute x = –3.     SOLUTION:       ANSWER:   –5 ANSWER:     –30a + 5   46.    48.    SOLUTION:   SOLUTION:     Substitute x = 9.     ANSWER:   2 5a + 70a + 240     49.  ANSWER:   2303     SOLUTION:   47.    SOLUTION:     ANSWER:     2 –10a + 10a + 1 ANSWER:     –30a + 5   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x. 48.      a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x). SOLUTION:     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). ANSWER:     2 5a + 70a + 240   SOLUTION:   a. 49.    SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) c. on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) b. (x).   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs ANSWER:   of f(x) and (f + g)(x) and between f(x) and (f – g) a. (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is b. the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   c. SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S.   d. Sample answer: For each value of x, the vertical ANSWER:   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs a. y = 2085.6x +123,060 of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). b. The function represents the difference in the   number of men and women employed in the U.S.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United 2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x States can be modeled by the following equations, + 1, find each value. where x is the number of years since 1994 and y is   the number of people in thousands. women: y =1086.4x + 56,610 52.  men: y = 999.2x +66,450     SOLUTION:   a. Write a function that models the total number of men and women employed in the United States during this time.     Substitute x = 3. b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   ANSWER:   a. Add the functions. The total number of men and women employed is –180 given by   53.      b. The function represents the difference SOLUTION:   in the number of men and women employed in the U.S.   ANSWER:   a. y = 2085.6x +123,060     Substitute x = 1. b. The function represents the difference in the   number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value.   ANSWER:   52.  0     SOLUTION:   54.      Substitute x = 3.   SOLUTION:   ANSWER:   –180     53.  ANSWER:     SOLUTION:     55.      SOLUTION:   Substitute x = 1.       ANSWER:   0 ANSWER:     1   54.  56.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   –33     57.  55.      SOLUTION:   SOLUTION:     ANSWER:     256   ANSWER:   1 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and 56.  2  if f(x) = x + 1 and g(x) = x – 3.     SOLUTION:   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and   on the same coordinate plane. ANSWER:   –33     c. Verbal Explain the relationship between  and  . 57.      d. Graphical Use a graphing calculator to graph [f SOLUTION:   ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   ANSWER:   a. 256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .       d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   b. a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.      SOLUTION:   2.      SOLUTION:               ANSWER:     ANSWER:     2.      SOLUTION:   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.       Therefore:   ANSWER:       ANSWER:   is undefined;     For each pair of functions, find and , if they exist. State the domain and range for each 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} composed function.   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} 3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.     Therefore:   Therefore:   ANSWER:   is undefined;     Find and , if they exist. State the domain and range for each composed   function.   ANSWER:   5.  is undefined;     SOLUTION:   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:     The range of is not a subset of the domain of . So,  is undefined.       Therefore: ANSWER:       ANSWER:   is undefined; 6.      Find and , if they exist. State SOLUTION:   the domain and range for each composed function.   5.      SOLUTION:       ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings ANSWER:   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   6.  SOLUTION:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan SOLUTION:   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. 7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either 8.  before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are SOLUTION:   taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and ANSWER:     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.      SOLUTION:               ANSWER:       ANSWER:     9.    SOLUTION:       10.      SOLUTION:           ANSWER:         ANSWER:       10.        SOLUTION:   11.    SOLUTION:         ANSWER:     ANSWER:         11.    12.  SOLUTION:   SOLUTION:     ANSWER:   ANSWER:       12.  13.    SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   6-1 Operations on Functions     13.  14.      SOLUTION:   SOLUTION:       ANSWER:     ANSWER:     14.      SOLUTION:   15.    SOLUTION:   eSolutionsManual-PoweredbyCognero Page7   ANSWER:     ANSWER:       15.  16. POPULATION In a particular county, the   population of the two largest cities can be modeled SOLUTION:   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:     ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40     For each pair of functions, find and , if they exist. State the domain and range for each 16. POPULATION In a particular county, the composed function. population of the two largest cities can be modeled   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} the number of years since 2000 and the population is g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} in thousands.   a. What is the population of the two cities combined SOLUTION:   after any number of years?  b. What is the difference in the populations of the Therefore: two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}     SOLUTION:   ANSWER:   Therefore:   18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:       ANSWER:   18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} SOLUTION:   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   is is undefined;  is undefined. undefined;     20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   is undefined; .     22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} ANSWER:   g = {(5, –4), (4, –3), (–1, 2), (2, 3)} is   undefined; SOLUTION:     21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   The range of is not a subset of the domain of ; is undefined. .   So,  is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)} ANSWER:     ; is undefined. SOLUTION:     The range of is not a subset of the domain of 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} . g = {(4, –3), (3, –7), (9, 8), (–4, –4)} So,  is undefined.   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} ANSWER:   g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} ANSWER:   g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all even numbers}   , D = {all real numbers}, R = {all odd numbers} Find and , if they exist. State   the domain and range for each composed function.   28.  27.      SOLUTION:   SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all odd numbers} real numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers} , D = {all real numbers}, R = {all   real numbers}   28.  29.      SOLUTION:   SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all   real numbers} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all   real numbers}   29.    30.  SOLUTION:     SOLUTION:       , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14}   , D = {all real numbers}, R = {y | y ≥ –10} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14}   , D = {all real numbers}, R = {y | y ≥ –10} 30.      31.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –14}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} ANSWER:     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17}   31.    32.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 0.875}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 6.5}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} 32.      33.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 6.5} real numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5} , D = {all real numbers}, R = {all   real numbers}   33.  34.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1}   , D = {all real numbers}, R = {y | y ≥ 0}   34.    35.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y 35.  ≥ 0}     36. FINANCE A ceramics store manufactures and sells SOLUTION:   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee mugs.   SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function.   So: , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0} b.   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     a. P(x) = 5.75x – 1850 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x b. P(500) = $1025; P(1000) = $3900; P(5000) = coffee mugs is given by r(x) = 6.5x. Suppose the $26,900 function for the cost of manufacturing x coffee mugs   is c(x) = 0.75x + 1850.   37. CCSS SENSE-MAKING Ms. Smith wants to buy a. Write the profit function. an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%.     b. Find the profit on 500, 1000, and 5000 coffee mugs. a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     SOLUTION:   b. Which composition of functions represents the a. The profit function P(x) is given by price of the HDTV, or ? Explain  where r(x) is the revenue your reasoning. function and c(x) is the cost function.   So: c. How much will Ms. Smith pay for the HDTV?   SOLUTION:     a. Let p(x) be the price after the discount where x is b. the original price. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore: ANSWER:     a. P(x) = 5.75x – 1850   b. Since = , either function b. P(500) = $1025; P(1000) = $3900; P(5000) = represents the price. $26,900     c.  37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%.   a. Write two functions representing the price after Substitute x = 2299. the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the   price of the HDTV, or ? Explain Therefore, Mr. Smith will pay $1587.75 for HDTV. your reasoning.     c. How much will Ms. Smith pay for the HDTV? ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x   SOLUTION:   b. Since = , either function a. Let p(x) be the price after the discount where x is represents the price. the original price.   Discount = 35%(x) = 0.35x Therefore: c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and Let t(x) be the price after the sales tax. g(x) = x – 3. State the domain of the resulting Sales tax = 6.25%(x) = 0.0625x Therefore: function.   38.      b. Since = , either function SOLUTION:   represents the price.   c.  D = {all real numbers}   ANSWER:   Substitute x = 2299. (f – g)(x) = x2 – 9; D = {all real numbers}     39.  Therefore, Mr. Smith will pay $1587.75 for HDTV.     SOLUTION:   ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   b. Since = , either function represents the price. D = {all real numbers}     c. $1587.75 ANSWER:     ; D = {all real 2 Perform each operation if f(x) = x + x – 12 and numbers} g(x) = x – 3. State the domain of the resulting   function.   40.  38.      SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     39.  ANSWER:     SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + D = {all real numbers} 8, find each value.     ANSWER:   41.  ; D = {all real   numbers} SOLUTION:     40.    Substitute x = –2. SOLUTION:     ANSWER:   25   42.      SOLUTION:   ANSWER:   Substitute x = 3.   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x +   8, find each value.   ANSWER:   –69 41.      SOLUTION:   43.    SOLUTION:   Substitute x = –2.     Substitute x = –5. ANSWER:     25   42.      SOLUTION:   ANSWER:   483   44.  Substitute x = 3.   SOLUTION:     ANSWER:   –69     43.  Substitute x = 2.     SOLUTION:       Substitute x = –5.   ANSWER:   –1   45.      SOLUTION:   ANSWER:   483   44.      Substitute x = –3.   SOLUTION:     ANSWER:     Substitute x = 2. –5     46.    SOLUTION:     ANSWER:     –1 Substitute x = 9.     45.    SOLUTION:     ANSWER:   2303     Substitute x = –3.   47.    SOLUTION:     ANSWER:   –5     ANSWER:   46.  –30a + 5     SOLUTION:   48.    SOLUTION:     Substitute x = 9.     ANSWER:     5a2 + 70a + 240   ANSWER:   2303 49.      47.  SOLUTION:     SOLUTION:       ANSWER:   ANSWER:   –30a + 5 –10a2 + 10a + 1     48.  50. MULTIPLE REPRESENTATIONS Let f(x) = x2   and g(x) = x.   SOLUTION:   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   ANSWER:   d. VERBAL Describe the relationship among the 2 5a + 70a + 240 graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     SOLUTION:   49.  a.   SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is b. the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   d. Sample answer: For each value of x, the vertical ANSWER:   distance between the graph of g(x) and the x-axis is a. the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United b. States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent? c.   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the d. Sample answer: For each value of x, the vertical U.S.   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs ANSWER:   of f(x) and (f + g)(x) and between f(x) and (f – g) (x). a. y = 2085.6x +123,060     b. The function represents the difference in the 51. EMPLOYMENT The number of women and men number of men and women employed in the U.S. age 16 and over employed each year in the United   States can be modeled by the following equations, where x is the number of years since 1994 and y is 2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x the number of people in thousands. + 1, find each value. women: y =1086.4x + 56,610   men: y = 999.2x +66,450 52.    a. Write a function that models the total number of   men and women employed in the United States SOLUTION:   during this time.   b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f Substitute x = 3. – g)(x) represent?     SOLUTION:   a. Add the functions. The total number of men and women employed is ANSWER:   given by –180     b. The function represents the difference 53.  in the number of men and women employed in the   U.S. SOLUTION:     ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the   number of men and women employed in the U.S. Substitute x = 1.     2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value.     52.    ANSWER:   0 SOLUTION:       54.  Substitute x = 3.     SOLUTION:   ANSWER:   –180   53.      ANSWER:   SOLUTION:     55.      Substitute x = 1.   SOLUTION:     ANSWER:   0     ANSWER:   54.  1     56.  SOLUTION:     SOLUTION:     ANSWER:     ANSWER:   –33     55.  57.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   256 1     58. MULTIPLE REPRESENTATIONS You will 56.  explore ,  , , and   2  if f(x) = x + 1 and g(x) = x – 3. SOLUTION:     a. Tabular Make a table showing values for   ,  , , and .      b. Graphical Use a graphing calculator to graph  ,  , , and ANSWER:   –33 on the same coordinate plane.     57.  c. Verbal Explain the relationship between    and  . SOLUTION:     d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   ANSWER:   SOLUTION:   256   a. 58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .     d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:     a. b.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     1.    2.  SOLUTION:     SOLUTION:                 ANSWER:     ANSWER:     2.      For each pair of functions, find and , if they exist. State the domain and range for each SOLUTION:   composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined.     Therefore:       ANSWER:     ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} For each pair of functions, find and , if g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} they exist. State the domain and range for each   composed function.   SOLUTION:   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} The range of is not a subset of the domain of g = {(9, 11), (6, 15), (10, 13), (5, 8)} .   So,  is undefined. SOLUTION:     The range of is not a subset of the domain of .   So,  is undefined. Therefore:       Therefore: ANSWER:   is undefined;   Find and , if they exist. State the domain and range for each composed   function.     5.  ANSWER:     is undefined; SOLUTION:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:     Therefore:     6.  ANSWER:     is undefined; SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable ANSWER:   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her 6.  paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college   plan and $152.95 will go to taxes. If she takes the SOLUTION:   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and ANSWER:     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     8.  7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable SOLUTION:   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.    SOLUTION:                 ANSWER:       ANSWER:     9.        SOLUTION:   10.      SOLUTION:                 ANSWER:   ANSWER:           10.    11.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:         12.  11.    SOLUTION:   SOLUTION:   ANSWER:     ANSWER:       13.    12.  SOLUTION:   SOLUTION:       ANSWER:   ANSWER:       14.    13.  SOLUTION:     SOLUTION:         ANSWER:   ANSWER:       14.  15.    SOLUTION:     SOLUTION:       ANSWER:   ANSWER:   6-1 Operations on Functions     16. POPULATION In a particular county, the 15.  population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is in thousands. SOLUTION:   a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.     16. POPULATION In a particular county, the 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} population of the two largest cities can be modeled g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is SOLUTION:   in thousands. a. What is the population of the two cities combined Therefore: after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations. eSolutionsManual-PoweredbyCognero Page8   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.     17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} ANSWER:     SOLUTION:     Therefore: 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:       ANSWER:   ANSWER:       18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is   undefined; ANSWER:     is undefined;  is undefined.   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)}   g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   ANSWER:   SOLUTION:   is undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   The range of is not a subset of the domain of SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. .   So,  is undefined. ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined.   ANSWER:   ; is undefined.   The range of is not a subset of the domain of . 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} So,  is undefined. g = {(1, –4), (2, –3), (3, –2), (4, –1)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of ; is undefined. .   So,  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} The range of is not a subset of the domain of g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   . So,  is undefined. SOLUTION:     ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     is undefined;  is undefined. ANSWER:     25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:     , D = {all real numbers}, R = {all ANSWER:   even numbers} , D = {all real numbers}, R = {all odd numbers}     Find and , if they exist. State the domain and range for each composed 28.  function.     27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers} ANSWER:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers}   29.  28.    SOLUTION:     SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     30.  29.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     30.  31.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     31.  32.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –11} ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 6.5}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –11} ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 6.5}     32.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0.875} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 6.5} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 1} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 1} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0} real numbers}     35.  34.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0}   35.  36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs SOLUTION:   is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:   , D = {all real numbers}, R = {y | y   ≥ 0} b. , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:   , D = {all real numbers}, R = {y | y a. P(x) = 5.75x – 1850 ≥ 0}     b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the 37. CCSS SENSE-MAKING Ms. Smith wants to buy function for the cost of manufacturing x coffee mugs an HDTV, which is on sale for 35% off the original is c(x) = 0.75x + 1850. price of $2299. The sales tax is 6.25%.     a. Write the profit function. a. Write two functions representing the price after   the discount p(x) and the price after sales tax t(x). b. Find the profit on 500, 1000, and 5000 coffee   mugs. b. Which composition of functions represents the   price of the HDTV, or ? Explain SOLUTION:   your reasoning. a. The profit function P(x) is given by    where r(x) is the revenue c. How much will Ms. Smith pay for the HDTV? function and c(x) is the cost function.   So: SOLUTION:   a. Let p(x) be the price after the discount where x is the original price.   Discount = 35%(x) = 0.35x b. Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore:   b. Since = , either function ANSWER:   represents the price. a. P(x) = 5.75x – 1850     c.  b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. Substitute x = 2299.   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     Therefore, Mr. Smith will pay $1587.75 for HDTV. b. Which composition of functions represents the   price of the HDTV, or ? Explain ANSWER:   your reasoning.   a. p(x) = 0.65x; t(x) = 1.0625x   c. How much will Ms. Smith pay for the HDTV?   b. Since = , either function represents the price. SOLUTION:     a. Let p(x) be the price after the discount where x is c. $1587.75 the original price.   Discount = 35%(x) = 0.35x Therefore: 2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x 38.  Therefore:   SOLUTION:     b. Since = , either function represents the price.   c.  D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   Substitute x = 2299. 39.      SOLUTION:   Therefore, Mr. Smith will pay $1587.75 for HDTV.   ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price. ANSWER:     ; D = {all real c. $1587.75 numbers}     2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting 40.  function.     38.  SOLUTION:     SOLUTION:   D = {all real numbers}   ANSWER:   (f – g)(x) = x2 – 9; D = {all real numbers}     ANSWER:   39.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   D = {all real numbers} 41.      ANSWER:   SOLUTION:   ; D = {all real numbers}   Substitute x = –2. 40.      SOLUTION:   ANSWER:   25   42.    SOLUTION:     ANSWER:   Substitute x = 3.     2 ANSWER:   If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + –69 8, find each value.     41.  43.      SOLUTION:   SOLUTION:   Substitute x = –2.   Substitute x = –5.     ANSWER:   25     42.  ANSWER:     483 SOLUTION:     44.    SOLUTION:   Substitute x = 3.   ANSWER:     –69 Substitute x = 2.     43.    SOLUTION:     ANSWER:     Substitute x = –5. –1     45.    SOLUTION:     ANSWER:   483     Substitute x = –3. 44.      SOLUTION:     ANSWER:   –5     Substitute x = 2. 46.      SOLUTION:       Substitute x = 9. ANSWER:     –1   45.      SOLUTION:   ANSWER:   2303   47.    Substitute x = –3.     SOLUTION:       ANSWER:   ANSWER:   –5 –30a + 5     46.  48.      SOLUTION:   SOLUTION:     Substitute x = 9.     ANSWER:   2 5a + 70a + 240     49.  ANSWER:     2303   SOLUTION:   47.    SOLUTION:     ANSWER:     2 –10a + 10a + 1 ANSWER:     –30a + 5   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   48.  a. TABULAR Make a table showing values for f   (x), g(x), ( f + g) (x), and ( f –g)(x). SOLUTION:     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). ANSWER:     2 5a + 70a + 240 SOLUTION:     a. 49.    SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     c. b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). b.   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs ANSWER:   of f(x) and (f + g)(x) and between f(x) and (f – g) a. (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. b. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   c. a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S.   d. Sample answer: For each value of x, the vertical ANSWER:   distance between the graph of g(x) and the x-axis is a. y = 2085.6x +123,060 the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) b. The function represents the difference in the (x). number of men and women employed in the U.S.     51. EMPLOYMENT The number of women and men 2 age 16 and over employed each year in the United If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x States can be modeled by the following equations, + 1, find each value. where x is the number of years since 1994 and y is   the number of people in thousands. 52.  women: y =1086.4x + 56,610   men: y = 999.2x +66,450   SOLUTION:   a. Write a function that models the total number of men and women employed in the United States   during this time. Substitute x = 3.     b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   ANSWER:   a. Add the functions. –180 The total number of men and women employed is   given by 53.      b. The function represents the difference SOLUTION:   in the number of men and women employed in the U.S.   ANSWER:     a. y = 2085.6x +123,060 Substitute x = 1.     b. The function represents the difference in the number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value.   ANSWER:   52.  0     SOLUTION:   54.      Substitute x = 3. SOLUTION:     ANSWER:   –180     ANSWER:   53.    SOLUTION:     55.      SOLUTION:   Substitute x = 1.       ANSWER:   0 ANSWER:     1   54.  56.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   –33     57.    55.    SOLUTION:   SOLUTION:     ANSWER:     256   ANSWER:   1 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and 56.   if f(x) = x2 + 1 and g(x) = x – 3.     SOLUTION:   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and   on the same coordinate plane. ANSWER:   –33     c. Verbal Explain the relationship between  and  . 57.      d. Graphical Use a graphing calculator to graph [f SOLUTION:   ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   a. ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .       d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     b. SOLUTION:   a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:     2.  Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any SOLUTION:   restrictions in domain or range.   1.      SOLUTION:               ANSWER:     ANSWER:     For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   2.  The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.     Therefore:           ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} ANSWER:   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     For each pair of functions, find and , if   they exist. State the domain and range for each Therefore: composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of is undefined;   . So,  is undefined. Find and , if they exist. State   the domain and range for each composed function.     Therefore: 5.    SOLUTION:         ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} ANSWER:   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:     The range of is not a subset of the domain of . 6.  So,  is undefined.     SOLUTION:     Therefore:     ANSWER:   is undefined;   Find and , if they exist. State the domain and range for each composed   function.   ANSWER:   5.      SOLUTION:   7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the ANSWER:   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Either way, she will have $228.95 taken from her 6.  paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only SOLUTION:   $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range.   8.      SOLUTION:   ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes. ANSWER:     ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   Find (f + g)(x), (f – g)(x), (x),and 9.    (x) for each f(x) and g(x). Indicate any   SOLUTION:   restrictions in domain or range.   8.      SOLUTION:                 ANSWER:     ANSWER:       10.      SOLUTION:   9.    SOLUTION:               ANSWER:       ANSWER:       11.    SOLUTION:       10.    SOLUTION:     ANSWER:         ANSWER:     12.  SOLUTION:       11.    SOLUTION:   ANSWER:       13.  ANSWER:     SOLUTION:       12.  SOLUTION:     ANSWER:     14.  ANSWER:     SOLUTION:     13.    SOLUTION:       ANSWER:     ANSWER:     15.    SOLUTION:     14.    SOLUTION:     ANSWER:       ANSWER:   16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?    b. What is the difference in the populations of the two cities?    15.  SOLUTION:     a. The population of the cities after x years is the sum of the individual populations. SOLUTION:     b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each   composed function.   ANSWER:   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   a. (f + g)(x) = 375x + 10 ANSWER:     6-1 Ob.p (efr -a tgi)o(nxs) =o n2 5Fxu +n c4t0i ons     For each pair of functions, find and , if 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} they exist. State the domain and range for each g = {(6, 8), (–12, –5), (0, 5), (5, 1)} composed function.     SOLUTION:   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of .   So,  is undefined. The range of is not a subset of the domain of 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} . g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   So,  is undefined.   SOLUTION:   eSolutionsManual-PoweredbyCog nero ANSWER:   Page9 is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)}   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)} ANSWER:     is SOLUTION:   undefined; The range of is not a subset of the domain of   . 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} So,  is undefined. g = {(3, –9), (7, 2), (8, –6), (12, 0)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is So,  is undefined. undefined;     ANSWER:   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} ; is undefined. g = {(3, –9), (7, 2), (8, –6), (12, 0)}     SOLUTION:   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is undefined; .     ANSWER:   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} ; is undefined. g = {(5, –4), (4, –3), (–1, 2), (2, 3)}     SOLUTION:   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.     ANSWER:   ANSWER:   ; is undefined. is undefined;  is undefined.     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. The range of is not a subset of the domain of .   So,  is undefined. 26. f = {(12, –3), (9, –2), (8, –1), (6, 3)}   g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)} ANSWER:     is undefined;  is undefined. SOLUTION:     25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     ANSWER:       ANSWER:   Find and , if they exist. State the domain and range for each composed function.     27.  26. f = {(12, –3), (9, –2), (8, –1), (6, 3)}   g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:   SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   28.      ANSWER:   SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   27.      , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:   , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers}   , D = {all real numbers}, R = {all odd numbers} 29.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   28.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     30.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   29.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   31.    , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   30.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     ANSWER:   32.    , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   31.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –11} ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 6.5}     ANSWER:   33.    , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   32.      SOLUTION:     , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     33.    SOLUTION:     , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y   ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y   ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0} , D = {all real numbers}, R = {all   real numbers}   35.  ANSWER:     , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   34.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     36. FINANCE A ceramics store manufactures and sells ANSWER:   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the , D = {all real numbers}, R = {y | y function for the cost of manufacturing x coffee mugs ≥ 1} is c(x) = 0.75x + 1850. , D = {all real numbers}, R = {y | y   ≥ 0} a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee mugs. 35.      SOLUTION:   a. The profit function P(x) is given by SOLUTION:    where r(x) is the revenue function and c(x) is the cost function. So:   b.     , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:   , D = {all real numbers}, R = {y | y a. P(x) = 5.75x – 1850 ≥ 0}     b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900 ANSWER:     37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original , D = {all real numbers}, R = {y | y price of $2299. The sales tax is 6.25%. ≥ 0}   , D = {all real numbers}, R = {y | y a. Write two functions representing the price after ≥ 0} the discount p(x) and the price after sales tax t(x).     b. Which composition of functions represents the 36. FINANCE A ceramics store manufactures and sells price of the HDTV, or ? Explain coffee mugs. The revenue r(x) from the sale of x your reasoning. coffee mugs is given by r(x) = 6.5x. Suppose the   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850. c. How much will Ms. Smith pay for the HDTV?     a. Write the profit function. SOLUTION:     a. Let p(x) be the price after the discount where x is b. Find the profit on 500, 1000, and 5000 coffee the original price. mugs. Discount = 35%(x) = 0.35x   Therefore: SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x function and c(x) is the cost function. Therefore: So:     b. Since = , either function b. represents the price.   c.  Substitute x = 2299. ANSWER:   a. P(x) = 5.75x – 1850     b. P(500) = $1025; P(1000) = $3900; P(5000) = Therefore, Mr. Smith will pay $1587.75 for HDTV. $26,900     ANSWER:   37. CCSS SENSE-MAKING Ms. Smith wants to buy a. p(x) = 0.65x; t(x) = 1.0625x an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. b. Since = , either function   represents the price. a. Write two functions representing the price after   the discount p(x) and the price after sales tax t(x). c. $1587.75     b. Which composition of functions represents the price of the HDTV, or ? Explain 2 Perform each operation if f(x) = x + x – 12 and your reasoning. g(x) = x – 3. State the domain of the resulting   function. c. How much will Ms. Smith pay for the HDTV?     38.  SOLUTION:     a. Let p(x) be the price after the discount where x is SOLUTION:   the original price. Discount = 35%(x) = 0.35x Therefore: D = {all real numbers} Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x Therefore: ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     b. Since = , either function 39.  represents the price.     c.  SOLUTION:   D = {all real numbers} Substitute x = 2299.   ANSWER:     ; D = {all real Therefore, Mr. Smith will pay $1587.75 for HDTV. numbers}     ANSWER:   40.  a. p(x) = 0.65x; t(x) = 1.0625x     b. Since = , either function represents the price. SOLUTION:     c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function.   38.      SOLUTION:   ANSWER:   D = {all real numbers}     ANSWER:   If f(x) = 5x, g(x) = –2x + 1, and h(x) = x2 + 6x + 2 8, find each value. (f – g)(x) = x – 9; D = {all real numbers}     41.  39.      SOLUTION:   SOLUTION:   Substitute x = –2. D = {all real numbers}     ANSWER:   ANSWER:   ; D = {all real 25 numbers}     42.  40.      SOLUTION:   SOLUTION:   Substitute x = 3.   ANSWER:   –69     ANSWER:   43.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.     Substitute x = –5. 41.      SOLUTION:     Substitute x = –2. ANSWER:   483     44.  ANSWER:     25   SOLUTION:   42.    SOLUTION:     Substitute x = 2.   Substitute x = 3.   ANSWER:     –69 ANSWER:     –1   43.    45.  SOLUTION:     SOLUTION:     Substitute x = –5.     Substitute x = –3.       ANSWER:   ANSWER:   483   –5   44.  46.      SOLUTION:   SOLUTION:       Substitute x = 9. Substitute x = 2.         ANSWER:   2303 ANSWER:     –1   47.    45.  SOLUTION:     SOLUTION:     ANSWER:     –30a + 5 Substitute x = –3.     48.    SOLUTION:     ANSWER:   –5   46.      ANSWER:   SOLUTION:   2 5a + 70a + 240   49.      Substitute x = 9.   SOLUTION:       ANSWER:   2303 ANSWER:     2 –10a + 10a + 1   47.    50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x. SOLUTION:     a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x)   on the same coordinate grid. ANSWER:     –30a + 5 c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   48.  d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     SOLUTION:   SOLUTION:   a.   ANSWER:     2 b. 5a + 70a + 240   49.    SOLUTION:       c. ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid. d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is d. VERBAL Describe the relationship among the the same as the vertical distance between the graphs graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). SOLUTION:     a. ANSWER:   a.   b. b. c.   c. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).     d. Sample answer: For each value of x, the vertical 51. EMPLOYMENT The number of women and men distance between the graph of g(x) and the x-axis is age 16 and over employed each year in the United the same as the vertical distance between the graphs States can be modeled by the following equations, of f(x) and (f + g)(x) and between f(x) and (f – g) where x is the number of years since 1994 and y is (x). the number of people in thousands.   women: y =1086.4x + 56,610 men: y = 999.2x +66,450 ANSWER:     a. a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent? b.   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S. c.   ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value.   d. Sample answer: For each value of x, the vertical 52.  distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) SOLUTION:   (x).     51. EMPLOYMENT The number of women and men Substitute x = 3. age 16 and over employed each year in the United   States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   ANSWER:   a. Write a function that models the total number of –180 men and women employed in the United States   during this time.   53.  b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent? SOLUTION:     SOLUTION:   a. Add the functions. The total number of men and women employed is given by   Substitute x = 1.     b. The function represents the difference in the number of men and women employed in the U.S.     ANSWER:   ANSWER:   0 a. y = 2085.6x +123,060     b. The function represents the difference in the 54.  number of men and women employed in the U.S.     2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x SOLUTION:   + 1, find each value.   52.    SOLUTION:       Substitute x = 3. ANSWER:       55.  ANSWER:   –180     SOLUTION:   53.    SOLUTION:     ANSWER:     1 Substitute x = 1.     56.    SOLUTION:     ANSWER:   0   54.      ANSWER:   SOLUTION:   –33   57.    SOLUTION:     ANSWER:       ANSWER:   55.  256     SOLUTION:   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and   ANSWER:   .  1     b. Graphical Use a graphing calculator to graph  ,  , , and 56.    on the same coordinate plane. SOLUTION:     c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate   plane.   ANSWER:   e.  Verbal Explain the relationship between [f ◦ g] –33 (x), and [g ◦ f](x).     SOLUTION:   57.  a.   SOLUTION:     ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and   .      b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between    and  . b.   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   a.       b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:     Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     1.    SOLUTION:     ANSWER:           2.    SOLUTION:     ANSWER:         2.      SOLUTION:     ANSWER:         For each pair of functions, find and , if   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of . So,  is undefined.     Therefore:   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is undefined; So,  is undefined.       4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} Therefore: g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.         Therefore: ANSWER:   is undefined;     ANSWER:   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} is undefined; g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   Find and , if they exist. State The range of is not a subset of the domain of the domain and range for each composed function. .   So,  is undefined. 5.        SOLUTION:   Therefore:   ANSWER:     is undefined;   Find and , if they exist. State the domain and range for each composed function.     ANSWER:   5.      SOLUTION:   6.    SOLUTION:         ANSWER:       ANSWER:   6.    SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college ANSWER:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings Either way, she will have $228.95 taken from her plan. She can choose to take the deduction either paycheck. If she takes the college savings plan before taxes are withheld, which reduces her taxable deduction before taxes, $76 will go to her college income, or after taxes are withheld. Dora’s tax rate plan and $152.95 will go to taxes. If she takes the is 17.5%. If her pay before taxes and deductions is college savings plan deduction after taxes, only $950, will she save more money if the deductions are $62.70 will go to her college plan and $166.25 will go taken before or after taxes are withheld? Explain. to taxes.     SOLUTION:   Find (f + g)(x), (f – g)(x), (x),and Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   (x) for each f(x) and g(x). Indicate any deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the restrictions in domain or range. college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes. 8.      ANSWER:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     8.    SOLUTION:     ANSWER:           9.    SOLUTION:     ANSWER:         9.      SOLUTION:         ANSWER:           10.      SOLUTION:     ANSWER:       10.          SOLUTION:   ANSWER:       11.        SOLUTION:     ANSWER:       11.      SOLUTION:   ANSWER:     12.  SOLUTION:     ANSWER:     12.  SOLUTION:   ANSWER:     13.    SOLUTION:   ANSWER:       13.      ANSWER:   SOLUTION:       14.    SOLUTION:     ANSWER:     14.    SOLUTION:     ANSWER:     15.    SOLUTION:     ANSWER:     15.      SOLUTION:   ANSWER:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined   after any number of years?  ANSWER:   b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.     16. POPULATION In a particular county, the b. The difference in the populations of the cities is population of the two largest cities can be modeled given by: by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands.   a. What is the population of the two cities combined ANSWER:   after any number of years?  a. (f + g)(x) = 375x + 10   b. What is the difference in the populations of the b. (f - g)(x) = 25x + 40 two cities?      SOLUTION:   For each pair of functions, find and , if a. The population of the cities after x years is the they exist. State the domain and range for each sum of the individual populations. composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}     b. The difference in the populations of the cities is SOLUTION:   given by: Therefore:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   6-1 Operations on Functions is undefined;  is undefined.     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:   is undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; eSolutionsManual-PoweredbyCognero Page10   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}     ANSWER:   SOLUTION:   is undefined; . The range of is not a subset of the domain of   . 22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} So,  is undefined. g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is undefined; .     ANSWER:   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} ; is undefined. g = {(5, –4), (4, –3), (–1, 2), (2, 3)}     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} SOLUTION:   g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined. The range of is not a subset of the domain of   . 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} So,  is undefined. g = {(4, –3), (3, –7), (9, 8), (–4, –4)}     ANSWER:   SOLUTION:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.     ANSWER:   ANSWER:   ; is undefined. is undefined;  is undefined.     24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(1, –4), (2, –3), (3, –2), (4, –1)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     ANSWER:     Find and , if they exist. State   the domain and range for each composed , D = {all real numbers}, R = {all function. even numbers}   , D = {all real numbers}, R = {all odd numbers} 27.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   28.      , D = {all real numbers}, R = {all SOLUTION:   even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all 28.  real numbers}     SOLUTION:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   29.      , D = {all real numbers}, R = {all real numbers} SOLUTION:   , D = {all real numbers}, R = {all real numbers}   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {all 29.  real numbers} , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   30.      , D = {all real numbers}, R = {all real numbers} SOLUTION:   , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {y | y 30.  ≥ –14} , D = {all real numbers}, R = {y | y   ≥ –10} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {y | y   ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   31.    , D = {all real numbers}, R = {y | y   ≥ –14} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}     , D = {all real numbers}, R = {y | y 31.  ≥ –11} , D = {all real numbers}, R = {y | y   ≥ –17} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   32.    , D = {all real numbers}, R = {y | y   ≥ –11} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}     , D = {all real numbers}, R = {y | y 32.  ≥ 0.875} , D = {all real numbers}, R = {y | y   ≥ 6.5} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {y | y   ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   33.    , D = {all real numbers}, R = {y | y   ≥ 0.875} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     , D = {all real numbers}, R = {all 33.  real numbers} , D = {all real numbers}, R = {all   real numbers} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers}   34.      , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     34.  , D = {all real numbers}, R = {y | y ≥ 1}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1}   , D = {all real numbers}, R = {y | y ≥ 0}   35.      , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     35.  , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the , D = {all real numbers}, R = {y | y function for the cost of manufacturing x coffee mugs ≥ 0} is c(x) = 0.75x + 1850. , D = {all real numbers}, R = {y | y   ≥ 0} a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee ANSWER:   mugs.   SOLUTION:   , D = {all real numbers}, R = {y | y a. The profit function P(x) is given by ≥ 0}  where r(x) is the revenue , D = {all real numbers}, R = {y | y function and c(x) is the cost function. ≥ 0} So:   36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x b. coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.   SOLUTION:   ANSWER:   a. The profit function P(x) is given by a. P(x) = 5.75x – 1850  where r(x) is the revenue   function and c(x) is the cost function. b. P(500) = $1025; P(1000) = $3900; P(5000) = So: $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original b. price of $2299. The sales tax is 6.25%.   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the price of the HDTV, or ? Explain your reasoning.   c. How much will Ms. Smith pay for the HDTV? ANSWER:     a. P(x) = 5.75x – 1850   SOLUTION:   b. P(500) = $1025; P(1000) = $3900; P(5000) = a. Let p(x) be the price after the discount where x is $26,900 the original price.   Discount = 35%(x) = 0.35x Therefore: 37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x a. Write two functions representing the price after Therefore: the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the   price of the HDTV, or ? Explain b. Since = , either function your reasoning. represents the price.     c. How much will Ms. Smith pay for the HDTV? c.    SOLUTION:   a. Let p(x) be the price after the discount where x is the original price. Discount = 35%(x) = 0.35x Therefore: Substitute x = 2299. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x Therefore, Mr. Smith will pay $1587.75 for HDTV. Therefore:   ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x     b. Since = , either function b. Since = , either function represents the price.   represents the price. c.    c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. Substitute x = 2299.   38.      Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price.   ANSWER:   c. $1587.75 (f – g)(x) = x2 – 9; D = {all real numbers}     2 Perform each operation if f(x) = x + x – 12 and 39.  g(x) = x – 3. State the domain of the resulting function.     SOLUTION:   38.    SOLUTION:   D = {all real numbers}   ANSWER:   D = {all real numbers} ; D = {all real   numbers} ANSWER:     2 (f – g)(x) = x – 9; D = {all real numbers}   40.  39.      SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   ; D = {all real   numbers}   ANSWER:   40.      SOLUTION:   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   41.    SOLUTION:     Substitute x = –2. ANSWER:     ANSWER:     25   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   42.  41.      SOLUTION:   SOLUTION:   Substitute x = 3. Substitute x = –2.     ANSWER:   ANSWER:   –69 25     43.  42.      SOLUTION:   SOLUTION:     Substitute x = –5. Substitute x = 3.     ANSWER:   –69     ANSWER:   43.  483     SOLUTION:   44.    SOLUTION:     Substitute x = –5.     Substitute x = 2.     ANSWER:   483   44.      ANSWER:   SOLUTION:   –1   45.    SOLUTION:     Substitute x = 2.     Substitute x = –3.     ANSWER:     –1   ANSWER:   –5 45.      46.  SOLUTION:     SOLUTION:     Substitute x = –3.     Substitute x = 9.     ANSWER:   –5     ANSWER:   46.  2303     SOLUTION:   47.    SOLUTION:     Substitute x = 9.     ANSWER:   –30a + 5     48.  ANSWER:     2303 SOLUTION:     47.    SOLUTION:     ANSWER:     5a2 + 70a + 240   ANSWER:   –30a + 5 49.      48.  SOLUTION:     SOLUTION:     ANSWER:     2 –10a + 10a + 1 ANSWER:     2 5a + 70a + 240   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   49.  a. TABULAR Make a table showing values for f   (x), g(x), ( f + g) (x), and ( f –g)(x).   SOLUTION:   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   ANSWER:   2 SOLUTION:   –10a + 10a + 1   a. 50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) b. on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   c.   b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) c. (x).   ANSWER:   a. b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. c. b. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   51. EMPLOYMENT The number of women and men c. age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is d. Sample answer: For each value of x, the vertical the function for the number of women, what does ( f distance between the graph of g(x) and the x-axis is – g)(x) represent? the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) (x). SOLUTION:     a. Add the functions. The total number of men and women employed is 51. EMPLOYMENT The number of women and men given by age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands.   women: y =1086.4x + 56,610 b. The function represents the difference men: y = 999.2x +66,450 in the number of men and women employed in the   U.S. a. Write a function that models the total number of   men and women employed in the United States during this time. ANSWER:     a. y = 2085.6x +123,060 b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f b. The function represents the difference in the – g)(x) represent? number of men and women employed in the U.S.     SOLUTION:   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x a. Add the functions. + 1, find each value. The total number of men and women employed is   given by 52.      SOLUTION:   b. The function represents the difference in the number of men and women employed in the   U.S. Substitute x = 3.     ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S. ANSWER:     –180   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value. 53.      52.    SOLUTION:   SOLUTION:     Substitute x = 3.     Substitute x = 1.   ANSWER:     –180   ANSWER:   0   53.    54.  SOLUTION:     SOLUTION:     Substitute x = 1.       ANSWER:   ANSWER:   0     54.  55.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   1   56.      55.  SOLUTION:     SOLUTION:     ANSWER:     –33   ANSWER:   1 57.      56.  SOLUTION:     SOLUTION:     ANSWER:   256     ANSWER:   58. MULTIPLE REPRESENTATIONS You will –33 explore ,  , , and   2  if f(x) = x + 1 and g(x) = x – 3. 57.      a. Tabular Make a table showing values for SOLUTION:     ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and   ANSWER:   on the same coordinate plane. 256     c. Verbal Explain the relationship between 58. MULTIPLE REPRESENTATIONS You will  and  . explore ,  , , and   2 d. Graphical Use a graphing calculator to graph [f  if f(x) = x + 1 and g(x) = x – 3. ◦ g](x), and [g ◦ f](x) on the same coordinate   plane. a. Tabular Make a table showing values for     ,  , , and e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x). .      SOLUTION:   b. Graphical Use a graphing calculator to a. graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:   a.       b.         b.

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is not a subset of the domain of . So, is undefined. Therefore: ANSWER: is undefined;. Find and. , if they exist. State. eSolutions Manual - Powered by .. your reasoning. c. How much will Ms. Smith pay for the HDTV? eSolutions Manual - Powered by Cognero. Page 17. 6-1 Operations on Functions
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Indicate any restrictions in domain or range. 1. SOLUTIO PDF

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Preview Indicate any restrictions in domain or range. 1. SOLUTIO

Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   6-1 Operations on Functions   Find (f + g)(x), (f – g)(x), (x),and 2.    (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. SOLUTION:     1.      SOLUTION:               ANSWER:     ANSWER:     For each pair of functions, find and , if they exist. State the domain and range for each composed function.     3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} 2.    SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . eSolutionsManual-PoweredbyCognero So,  is undefined. Page1       Therefore:         ANSWER:   is undefined;     ANSWER:   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. For each pair of functions, find and , if   they exist. State the domain and range for each composed function.     Therefore: 3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is undefined; So,  is undefined.     Find and , if they exist. State the domain and range for each composed   function. Therefore:   5.    SOLUTION:       ANSWER:     is undefined;   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)}   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   ANSWER:   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.   6.      Therefore: SOLUTION:     ANSWER:     is undefined;   Find and , if they exist. State the domain and range for each composed function.     5.  ANSWER:     SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan ANSWER:   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   6.  ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college SOLUTION:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      ANSWER:   SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any 9.  restrictions in domain or range.     SOLUTION:   8.    SOLUTION:                   ANSWER:     ANSWER:         10.    9.  SOLUTION:     SOLUTION:               ANSWER:       ANSWER:       11.        SOLUTION:   10.    SOLUTION:     ANSWER:         ANSWER:     12.  SOLUTION:       11.    SOLUTION:   ANSWER:       ANSWER:   13.    SOLUTION:       12.  SOLUTION:     ANSWER:     ANSWER:   14.    SOLUTION:     13.    SOLUTION:       ANSWER:     ANSWER:     15.    SOLUTION:     14.    SOLUTION:     ANSWER:     ANSWER:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined   after any number of years?  b. What is the difference in the populations of the 15.  two cities?      SOLUTION:   SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if   they exist. State the domain and range for each ANSWER:   composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10     b. (f - g)(x) = 25x + 40 ANSWER:       For each pair of functions, find and , if they exist. State the domain and range for each composed function. 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)} 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)}   g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} SOLUTION:     SOLUTION:   Therefore:   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     ANSWER:   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined. 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   ANSWER:   is The range of is not a subset of the domain of undefined; .   So,  is undefined. The range of is not a subset of the domain of 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} .   So,  is undefined. SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is undefined;  is undefined.   So,  is undefined. 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:     ANSWER:   The range of is not a subset of the domain of is undefined; .   So,  is undefined.   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} ANSWER:     ; is undefined. SOLUTION:     The range of is not a subset of the domain of 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} . g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   So,  is undefined. SOLUTION:     The range of is not a subset of the domain of ANSWER:   is undefined; . .   So,  is undefined.   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)} ANSWER:     ; is undefined. SOLUTION:     24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . So,  is undefined. . The range of is not a subset of the domain of So,  is undefined.   . So,  is undefined. ANSWER:     ; is undefined.   ANSWER:   is undefined;  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)}   g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)} SOLUTION:     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} is undefined;  is undefined. g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     ANSWER:     ANSWER:     Find and , if they exist. State the domain and range for each composed   function.   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} 27.  g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     28.  ANSWER:     SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers}   ANSWER:   29.    , D = {all real numbers}, R = {all SOLUTION:   even numbers} , D = {all real numbers}, R = {all odd numbers}   28.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   30.  , D = {all real numbers}, R = {all   real numbers} SOLUTION:   , D = {all real numbers}, R = {all real numbers}   29.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –14} real numbers} , D = {all real numbers}, R = {y | y   ≥ –10} ANSWER:     31.  , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all SOLUTION:   real numbers}   30.      SOLUTION:     , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} ANSWER:     32.  , D = {all real numbers}, R = {y | y   ≥ –14} , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –10}   31.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} ANSWER:     33.  , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –17}   32.      SOLUTION:       , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0.875} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 6.5} real numbers}     ANSWER:   34.    , D = {all real numbers}, R = {y | y ≥ 0.875} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ 6.5}   33.      SOLUTION:     , D = {all real numbers}, R = {y | y   ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     ANSWER:   35.  , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all SOLUTION:   real numbers}   34.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0}   ANSWER:   36. FINANCE A ceramics store manufactures and sells , D = {all real numbers}, R = {y | y coffee mugs. The revenue r(x) from the sale of x ≥ 1} coffee mugs is given by r(x) = 6.5x. Suppose the , D = {all real numbers}, R = {y | y function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850. ≥ 0}     a. Write the profit function.   35.  b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:     b.   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     a. P(x) = 5.75x – 1850   ANSWER:   b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900   , D = {all real numbers}, R = {y | y 37. CCSS SENSE-MAKING Ms. Smith wants to buy ≥ 0} an HDTV, which is on sale for 35% off the original , D = {all real numbers}, R = {y | y price of $2299. The sales tax is 6.25%. ≥ 0}     a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x). 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x b. Which composition of functions represents the coffee mugs is given by r(x) = 6.5x. Suppose the price of the HDTV, or ? Explain function for the cost of manufacturing x coffee mugs your reasoning. is c(x) = 0.75x + 1850.     c. How much will Ms. Smith pay for the HDTV? a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee SOLUTION:   mugs. a. Let p(x) be the price after the discount where x is   the original price. Discount = 35%(x) = 0.35x SOLUTION:   Therefore: a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. Let t(x) be the price after the sales tax. So: Sales tax = 6.25%(x) = 0.0625x Therefore:   b.   b. Since = , either function represents the price.   c.  ANSWER:   Substitute x = 2299. a. P(x) = 5.75x – 1850   b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900     Therefore, Mr. Smith will pay $1587.75 for HDTV.   37. CCSS SENSE-MAKING Ms. Smith wants to buy ANSWER:   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. a. p(x) = 0.65x; t(x) = 1.0625x     a. Write two functions representing the price after b. Since = , either function the discount p(x) and the price after sales tax t(x). represents the price.     b. Which composition of functions represents the c. $1587.75 price of the HDTV, or ? Explain   your reasoning.   Perform each operation if f(x) = x2 + x – 12 and c. How much will Ms. Smith pay for the HDTV? g(x) = x – 3. State the domain of the resulting   function.   SOLUTION:   38.  a. Let p(x) be the price after the discount where x is   the original price. Discount = 35%(x) = 0.35x SOLUTION:   Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x D = {all real numbers} Therefore:   ANSWER:   2   (f – g)(x) = x – 9; D = {all real numbers} b. Since = , either function   represents the price.   39.  c.    SOLUTION:   Substitute x = 2299. D = {all real numbers}     ANSWER:   Therefore, Mr. Smith will pay $1587.75 for HDTV. ; D = {all real   numbers} ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x   40.  b. Since = , either function represents the price.     SOLUTION:   c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function.   38.    SOLUTION:     ANSWER:   D = {all real numbers}   ANSWER:     2 (f – g)(x) = x – 9; D = {all real numbers} 2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x +   8, find each value.   39.  41.      SOLUTION:   SOLUTION:   D = {all real numbers} Substitute x = –2.   ANSWER:     ; D = {all real numbers} ANSWER:     25   40.  42.      SOLUTION:   SOLUTION:   Substitute x = 3.     ANSWER:   –69 ANSWER:     43.      SOLUTION:   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   41.      Substitute x = –5.   SOLUTION:   Substitute x = –2.   ANSWER:     483   ANSWER:   25 44.      SOLUTION:   42.    SOLUTION:     Substitute x = 2. Substitute x = 3.     ANSWER:   –69     ANSWER:   43.  –1     SOLUTION:   45.    SOLUTION:     Substitute x = –5.     Substitute x = –3.     ANSWER:     483   ANSWER:   –5 44.      46.  SOLUTION:     SOLUTION:     Substitute x = 2.     Substitute x = 9.       ANSWER:   ANSWER:   –1 2303     45.  47.      SOLUTION:   SOLUTION:       Substitute x = –3. ANSWER:     –30a + 5   48.      SOLUTION:   ANSWER:   –5   46.      SOLUTION:   ANSWER:   2 5a + 70a + 240     Substitute x = 9. 49.      SOLUTION:     ANSWER:   2303     ANSWER:   47.  2 –10a + 10a + 1     SOLUTION:   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     ANSWER:   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid. –30a + 5     c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid. 48.      d. VERBAL Describe the relationship among the SOLUTION:   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   ANSWER:   2 5a + 70a + 240     b. 49.    SOLUTION:     ANSWER:     2 –10a + 10a + 1 c.   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the d. Sample answer: For each value of x, the vertical graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). distance between the graph of g(x) and the x-axis is   the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) SOLUTION:   (x). a.   ANSWER:   a.   b. b.   c. c. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs 51. EMPLOYMENT The number of women and men of f(x) and (f + g)(x) and between f(x) and (f – g) age 16 and over employed each year in the United (x). States can be modeled by the following equations,   where x is the number of years since 1994 and y is the number of people in thousands. ANSWER:   women: y =1086.4x + 56,610 a. men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is b. the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference c. in the number of men and women employed in the U.S.   ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S.   If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x2 – 2x d. Sample answer: For each value of x, the vertical + 1, find each value. distance between the graph of g(x) and the x-axis is   the same as the vertical distance between the graphs 52.  of f(x) and (f + g)(x) and between f(x) and (f – g) (x).     SOLUTION:   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United   States can be modeled by the following equations, Substitute x = 3. where x is the number of years since 1994 and y is   the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of ANSWER:   men and women employed in the United States during this time. –180     b. If f is the function for the number of men, and g is the function for the number of women, what does ( f 53.  – g)(x) represent?     SOLUTION:   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   Substitute x = 1.     b. The function represents the difference in the number of men and women employed in the U.S.     ANSWER:   a. y = 2085.6x +123,060 ANSWER:     0 b. The function represents the difference in the   number of men and women employed in the U.S.   54.  2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value.   SOLUTION:   52.    SOLUTION:     Substitute x = 3.     ANSWER:     ANSWER:   –180 55.      SOLUTION:   53.    SOLUTION:       Substitute x = 1. ANSWER:     1   56.      ANSWER:   SOLUTION:   0   54.      SOLUTION:   ANSWER:   –33   57.      SOLUTION:   ANSWER:       55.    ANSWER:   256 SOLUTION:     58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.     a. Tabular Make a table showing values for ANSWER:     ,  , , and 1 .      56.  b. Graphical Use a graphing calculator to   graph  ,  , , and SOLUTION:   on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate ANSWER:   plane. –33     e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x). 57.      SOLUTION:   SOLUTION:   a.   ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .        b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .     b. d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:     a.       b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.      2.  SOLUTION:     SOLUTION:                 ANSWER:     ANSWER:     2.      For each pair of functions, find and , if SOLUTION:   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined.       Therefore:   ANSWER:       ANSWER:   is undefined; 6-1 Operations on Functions     For each pair of functions, find and , if 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} they exist. State the domain and range for each g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} composed function.     SOLUTION:   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined. The range of is not a subset of the domain of   . So,  is undefined.   Therefore:     Therefore:   ANSWER:   is undefined;   Find and , if they exist. State   the domain and range for each composed function.     5.  ANSWER:   is undefined;   SOLUTION:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:     Therefore:   eSolutionsManual-PoweredbyCognero Page2   6.  ANSWER:   is undefined;     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:       7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either ANSWER:   before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are   taken before or after taxes are withheld? Explain.   SOLUTION:   6.  Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college SOLUTION:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     7. CCSS MODELING Dora has 8% of her earnings 8.  deducted from her paycheck for a college savings plan. She can choose to take the deduction either   before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate SOLUTION:   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.    SOLUTION:                 ANSWER:       ANSWER:     9.      SOLUTION:     10.      SOLUTION:               ANSWER:     ANSWER:         10.      11.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:         11.  12.    SOLUTION:   SOLUTION:     ANSWER:   ANSWER:       13.  12.    SOLUTION:   SOLUTION:       ANSWER:   ANSWER:       14.  13.      SOLUTION:   SOLUTION:       ANSWER:     ANSWER:       14.    15.  SOLUTION:     SOLUTION:       ANSWER:   ANSWER:       15.  16. POPULATION In a particular county, the population of the two largest cities can be modeled   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is SOLUTION:   in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each   composed function.   16. POPULATION In a particular county, the 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} population of the two largest cities can be modeled g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is in thousands. SOLUTION:   a. What is the population of the two cities combined after any number of years?  Therefore: b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)}   g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   ANSWER:   SOLUTION:     Therefore: 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:       ANSWER:   ANSWER:       18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)}   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of .   So,  is undefined. ANSWER:     is ANSWER:   undefined; is undefined;  is undefined.     21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)} ANSWER:     is SOLUTION:   undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined. The range of is not a subset of the domain of   . 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} So,  is undefined. g = {(1, –4), (2, –3), (3, –2), (4, –1)}     ANSWER:   SOLUTION:   ; is undefined. The range of is not a subset of the domain of   . 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} So,  is undefined. g = {(4, –3), (3, –7), (9, 8), (–4, –4)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     ANSWER:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     Find and , if they exist. State the domain and range for each composed function. 28.      27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers} ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers}     29.  28.      SOLUTION:   SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     29.  30.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –14} real numbers} , D = {all real numbers}, R = {y | y   ≥ –10}   30.  31.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –10} , D = {all real numbers}, R = {y | y   ≥ –17}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –10} , D = {all real numbers}, R = {y | y   ≥ –17}   31.  32.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   32.  33.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5} , D = {all real numbers}, R = {all   real numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     34.  35.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0}   35.  36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the SOLUTION:   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:   , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y b. ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0} a. P(x) = 5.75x – 1850     b. P(500) = $1025; P(1000) = $3900; P(5000) = 36. FINANCE A ceramics store manufactures and sells $26,900 coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs 37. CCSS SENSE-MAKING Ms. Smith wants to buy is c(x) = 0.75x + 1850. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. a. Write the profit function.     a. Write two functions representing the price after b. Find the profit on 500, 1000, and 5000 coffee the discount p(x) and the price after sales tax t(x). mugs.     b. Which composition of functions represents the price of the HDTV, or ? Explain SOLUTION:   a. The profit function P(x) is given by your reasoning.    where r(x) is the revenue c. How much will Ms. Smith pay for the HDTV? function and c(x) is the cost function.   So: SOLUTION:   a. Let p(x) be the price after the discount where x is   the original price. b. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore:   ANSWER:   b. Since = , either function a. P(x) = 5.75x – 1850 represents the price.     b. P(500) = $1025; P(1000) = $3900; P(5000) = c.  $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%.   Substitute x = 2299. a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     b. Which composition of functions represents the Therefore, Mr. Smith will pay $1587.75 for HDTV.   price of the HDTV, or ? Explain your reasoning. ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x c. How much will Ms. Smith pay for the HDTV?     b. Since = , either function SOLUTION:   represents the price. a. Let p(x) be the price after the discount where x is   the original price. c. $1587.75 Discount = 35%(x) = 0.35x   Therefore: 2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting Let t(x) be the price after the sales tax. function. Sales tax = 6.25%(x) = 0.0625x   Therefore: 38.      SOLUTION:   b. Since = , either function represents the price.   c.  D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   Substitute x = 2299. 39.      Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price.   ANSWER:   c. $1587.75 ; D = {all real   numbers}   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. 40.    38.      SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     ANSWER:   39.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. D = {all real numbers}     41.  ANSWER:     ; D = {all real SOLUTION:   numbers}   40.  Substitute x = –2.   SOLUTION:     ANSWER:   25   42.    SOLUTION:     ANSWER:   Substitute x = 3.     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + ANSWER:   8, find each value. –69     41.    43.    SOLUTION:   SOLUTION:   Substitute x = –2.   Substitute x = –5.     ANSWER:   25     42.    ANSWER:   483 SOLUTION:     44.    Substitute x = 3. SOLUTION:     ANSWER:   –69     Substitute x = 2.   43.    SOLUTION:       ANSWER:   Substitute x = –5. –1     45.    SOLUTION:     ANSWER:   483     44.  Substitute x = –3.     SOLUTION:     ANSWER:   –5     Substitute x = 2.   46.    SOLUTION:       ANSWER:   Substitute x = 9. –1     45.    SOLUTION:     ANSWER:   2303     47.  Substitute x = –3.     SOLUTION:       ANSWER:   –5 ANSWER:     –30a + 5   46.    48.  SOLUTION:     SOLUTION:     Substitute x = 9.     ANSWER:   2 5a + 70a + 240     ANSWER:   49.  2303     SOLUTION:   47.    SOLUTION:       ANSWER:   2 –10a + 10a + 1 ANSWER:     –30a + 5   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x. 48.      a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x). SOLUTION:     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the ANSWER:   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). 2   5a + 70a + 240   SOLUTION:   a. 49.    SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) c. on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) b. (x).   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is ANSWER:   the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) a. (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is b. the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   c. SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is ANSWER:   the same as the vertical distance between the graphs a. y = 2085.6x +123,060 of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). b. The function represents the difference in the   number of men and women employed in the U.S.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United 2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x States can be modeled by the following equations, + 1, find each value. where x is the number of years since 1994 and y is the number of people in thousands.   women: y =1086.4x + 56,610 52.  men: y = 999.2x +66,450     a. Write a function that models the total number of SOLUTION:   men and women employed in the United States during this time.     Substitute x = 3. b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. ANSWER:   The total number of men and women employed is –180 given by   53.      b. The function represents the difference in the number of men and women employed in the SOLUTION:   U.S.   ANSWER:   a. y = 2085.6x +123,060     Substitute x = 1. b. The function represents the difference in the   number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value.     ANSWER:   52.  0     SOLUTION:   54.    Substitute x = 3.     SOLUTION:   ANSWER:   –180     53.  ANSWER:     SOLUTION:     55.      SOLUTION:   Substitute x = 1.     ANSWER:     0 ANSWER:     1   54.  56.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   –33     57.  55.      SOLUTION:   SOLUTION:     ANSWER:     256 ANSWER:     1 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and 56.  2  if f(x) = x + 1 and g(x) = x – 3.     SOLUTION:   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to   graph  ,  , , and ANSWER:   on the same coordinate plane. –33     c. Verbal Explain the relationship between 57.   and  .     SOLUTION:   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   ANSWER:   a. 256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .       d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   b. a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:         Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any ANSWER:   restrictions in domain or range.   1.    SOLUTION:       2.    SOLUTION:             ANSWER:         ANSWER:   2.    SOLUTION:     For each pair of functions, find and , if   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.       Therefore: ANSWER:         ANSWER:   For each pair of functions, find and , if is undefined; they exist. State the domain and range for each composed function.     3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)}   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.     Therefore:   Therefore:     ANSWER:   is undefined;     ANSWER:   Find and , if they exist. State is undefined; the domain and range for each composed function.     5.  4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.       Therefore:     ANSWER:   ANSWER:   is undefined; 6-1 Operations on Functions     Find and , if they exist. State the domain and range for each composed 6.  function.     SOLUTION:   5.    SOLUTION:         ANSWER:     ANSWER:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate 6.  is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are   taken before or after taxes are withheld? Explain. SOLUTION:     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college ANSWER:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only eSolutionsManual-PoweredbyCognero $62.70 will go to her college plan and $166.25 wiPlla ggeo 3 to taxes.     Find (f + g)(x), (f – g)(x), (x),and 7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   (x) for each f(x) and g(x). Indicate any plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable restrictions in domain or range. income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are 8.  taken before or after taxes are withheld? Explain.     SOLUTION:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. ANSWER:     8.    SOLUTION:       9.      SOLUTION:           ANSWER:           ANSWER:   9.    SOLUTION:         10.      SOLUTION:         ANSWER:         ANSWER:       10.    SOLUTION:       11.    SOLUTION:         ANSWER:     ANSWER:       11.    SOLUTION:     12.  SOLUTION:     ANSWER:   ANSWER:     12.    SOLUTION:   13.    SOLUTION:     ANSWER:     ANSWER:     13.      SOLUTION:   14.    SOLUTION:       ANSWER:     ANSWER:     14.    SOLUTION:     15.    SOLUTION:     ANSWER:     ANSWER:     15.    SOLUTION:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   ANSWER:     b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10     b. (f - g)(x) = 25x + 40 16. POPULATION In a particular county, the   population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is For each pair of functions, find and , if the number of years since 2000 and the population is they exist. State the domain and range for each in thousands. composed function.   a. What is the population of the two cities combined 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} after any number of years?  g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   b. What is the difference in the populations of the two cities?  SOLUTION:     SOLUTION:   Therefore: a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:     Therefore: ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   ANSWER:   SOLUTION:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   The range of is not a subset of the domain of ANSWER:   .   So,  is undefined. 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     ANSWER:   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} is g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)} undefined;     SOLUTION:   The range of is not a subset of the domain of 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} .   So,  is undefined. SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   is undefined; . is   undefined; 22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)} 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)}   g = {(3, –9), (7, 2), (8, –6), (12, 0)} SOLUTION:     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} ANSWER:     is undefined; . SOLUTION:     22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined. ANSWER:     ; is undefined. ANSWER:     ; is undefined. 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)}   g = {(1, –4), (2, –3), (3, –2), (4, –1)} 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)}   g = {(4, –3), (3, –7), (9, 8), (–4, –4)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   The range of is not a subset of the domain of 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} . g = {(–1, 5), (3, –4), (6, 4), (10, 8)} So,  is undefined.     SOLUTION:   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   ANSWER:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all   even numbers} , D = {all real numbers}, R = {all ANSWER:   odd numbers} ANSWER:     Find and , if they exist. State , D = {all real numbers}, R = {all the domain and range for each composed even numbers} function. , D = {all real numbers}, R = {all   odd numbers}   27.    28.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all even numbers} ANSWER:   , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all 28.  real numbers}     SOLUTION:   29.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all 29.  real numbers} , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   30.    SOLUTION:       , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –10} real numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers}   30.  , D = {all real numbers}, R = {y | y ≥ –14}   , D = {all real numbers}, R = {y | y ≥ –10} SOLUTION:     31.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17} , D = {all real numbers}, R = {y | y   ≥ –14} , D = {all real numbers}, R = {y | y ANSWER:   ≥ –10}   , D = {all real numbers}, R = {y | y 31.  ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} SOLUTION:     32.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 6.5} ≥ –11}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ –17}   32.  , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} SOLUTION:     33.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 6.5}   , D = {all real numbers}, R = {all real numbers} 33.  , D = {all real numbers}, R = {all real numbers}     SOLUTION:   34.    SOLUTION:         , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}   , D = {all real numbers}, R = {all real numbers} ANSWER:     , D = {all real numbers}, R = {y | y 34.  ≥ 1} , D = {all real numbers}, R = {y | y   ≥ 0} SOLUTION:     35.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     35.  , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.     a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   , D = {all real numbers}, R = {y | y a. The profit function P(x) is given by ≥ 0}  where r(x) is the revenue , D = {all real numbers}, R = {y | y function and c(x) is the cost function. ≥ 0} So:   ANSWER:     b. , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the ANSWER:   function for the cost of manufacturing x coffee mugs a. P(x) = 5.75x – 1850 is c(x) = 0.75x + 1850.     b. P(500) = $1025; P(1000) = $3900; P(5000) = a. Write the profit function. $26,900     b. Find the profit on 500, 1000, and 5000 coffee mugs. 37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. SOLUTION:     a. The profit function P(x) is given by a. Write two functions representing the price after  where r(x) is the revenue the discount p(x) and the price after sales tax t(x). function and c(x) is the cost function.   So: b. Which composition of functions represents the price of the HDTV, or ? Explain your reasoning.     b. c. How much will Ms. Smith pay for the HDTV?   SOLUTION:   a. Let p(x) be the price after the discount where x is the original price. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. ANSWER:   Sales tax = 6.25%(x) = 0.0625x a. P(x) = 5.75x – 1850 Therefore:   b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900     b. Since = , either function 37. CCSS SENSE-MAKING Ms. Smith wants to buy represents the price. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. c.    a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the price of the HDTV, or ? Explain Substitute x = 2299. your reasoning.   c. How much will Ms. Smith pay for the HDTV?     Therefore, Mr. Smith will pay $1587.75 for HDTV.   SOLUTION:   a. Let p(x) be the price after the discount where x is ANSWER:   the original price. a. p(x) = 0.65x; t(x) = 1.0625x Discount = 35%(x) = 0.35x   Therefore: b. Since = , either function represents the price.   Let t(x) be the price after the sales tax. c. $1587.75 Sales tax = 6.25%(x) = 0.0625x Therefore:   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting   function. b. Since = , either function   represents the price. 38.      c.  SOLUTION:   Substitute x = 2299. D = {all real numbers}   ANSWER:     2 Therefore, Mr. Smith will pay $1587.75 for HDTV. (f – g)(x) = x – 9; D = {all real numbers}     ANSWER:   39.  a. p(x) = 0.65x; t(x) = 1.0625x     b. Since = , either function SOLUTION:   represents the price.   c. $1587.75   D = {all real numbers} Perform each operation if f(x) = x2 + x – 12 and   g(x) = x – 3. State the domain of the resulting ANSWER:   function.   ; D = {all real 38.  numbers}     SOLUTION:   40.    SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   39.    SOLUTION:     ANSWER:   D = {all real numbers}     ANSWER:   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + ; D = {all real 8, find each value. numbers}     41.    40.  SOLUTION:     SOLUTION:   Substitute x = –2.   ANSWER:   25     42.  ANSWER:     SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + Substitute x = 3. 8, find each value.   41.      SOLUTION:   ANSWER:   –69   43.  Substitute x = –2.   SOLUTION:     ANSWER:   25     Substitute x = –5.   42.    SOLUTION:     ANSWER:   Substitute x = 3. 483   44.      ANSWER:   SOLUTION:   –69   43.    SOLUTION:     Substitute x = 2.     Substitute x = –5.     ANSWER:   –1     ANSWER:   45.  483     SOLUTION:   44.    SOLUTION:     Substitute x = –3.     Substitute x = 2.     ANSWER:   –5   46.      SOLUTION:   ANSWER:   –1   45.      Substitute x = 9.   SOLUTION:       Substitute x = –3.   ANSWER:   2303   47.      ANSWER:   SOLUTION:   –5   46.      SOLUTION:   ANSWER:   –30a + 5   48.      Substitute x = 9.   SOLUTION:       ANSWER:   2303 ANSWER:     2 5a + 70a + 240   47.    49.  SOLUTION:     SOLUTION:     ANSWER:   –30a + 5     48.  ANSWER:   2   –10a + 10a + 1   SOLUTION:   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid. ANSWER:     2 5a + 70a + 240 c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   49.  d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     SOLUTION:   SOLUTION:   a.   ANSWER:     b. 2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     c. d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a.   c. b. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). b.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time. c.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs b. The function represents the difference of f(x) and (f + g)(x) and between f(x) and (f – g) in the number of men and women employed in the (x). U.S.     51. EMPLOYMENT The number of women and men ANSWER:   age 16 and over employed each year in the United a. y = 2085.6x +123,060 States can be modeled by the following equations,   where x is the number of years since 1994 and y is b. The function represents the difference in the the number of people in thousands. number of men and women employed in the U.S. women: y =1086.4x + 56,610   men: y = 999.2x +66,450   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x a. Write a function that models the total number of + 1, find each value. men and women employed in the United States   during this time.   52.  b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f SOLUTION:   – g)(x) represent?     SOLUTION:   Substitute x = 3. a. Add the functions.   The total number of men and women employed is given by   ANSWER:   b. The function represents the difference –180 in the number of men and women employed in the   U.S.   53.  ANSWER:     a. y = 2085.6x +123,060   SOLUTION:   b. The function represents the difference in the number of men and women employed in the U.S.   If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x2 – 2x   + 1, find each value. Substitute x = 1.     52.    SOLUTION:       ANSWER:   Substitute x = 3. 0     54.    ANSWER:   SOLUTION:   –180   53.    SOLUTION:     ANSWER:     Substitute x = 1.     55.    SOLUTION:     ANSWER:   0   54.      ANSWER:   SOLUTION:   1   56.    SOLUTION:     ANSWER:       55.  ANSWER:     –33 SOLUTION:     57.    SOLUTION:     ANSWER:   1     ANSWER:   56.  256     SOLUTION:   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and   .  ANSWER:   –33     b. Graphical Use a graphing calculator to graph  ,  , , and 57.    on the same coordinate plane. SOLUTION:     c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate plane. ANSWER:     256 e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and SOLUTION:   a. 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.     e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   SOLUTION:   a.   b.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:         Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. ANSWER:     1.    SOLUTION:       2.    SOLUTION:             ANSWER:         ANSWER:   2.    SOLUTION:       For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.       Therefore: ANSWER:         For each pair of functions, find and , if ANSWER:   they exist. State the domain and range for each is undefined; composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.     Therefore:   Therefore:     ANSWER:   is undefined;     ANSWER:   Find and , if they exist. State is undefined; the domain and range for each composed function.     5.  4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.       Therefore:     ANSWER:   ANSWER:   is undefined;     Find and , if they exist. State the domain and range for each composed 6.  function.     5.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable 6.  income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain. SOLUTION:     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college ANSWER:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go 6-1 Operations on Functions to taxes.     7. CCSS MODELING Dora has 8% of her earnings Find (f + g)(x), (f – g)(x), (x),and deducted from her paycheck for a college savings plan. She can choose to take the deduction either   (x) for each f(x) and g(x). Indicate any before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate restrictions in domain or range. is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain. 8.      SOLUTION:   Either way, she will have $228.95 taken from her SOLUTION:   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range.   ANSWER:   8.    SOLUTION:       9.      SOLUTION:   eSolutionsManual-PoweredbyCognero Page4         ANSWER:           ANSWER:   9.    SOLUTION:         10.      SOLUTION:         ANSWER:         ANSWER:       10.    SOLUTION:       11.    SOLUTION:         ANSWER:     ANSWER:       11.    SOLUTION:     12.  SOLUTION:     ANSWER:   ANSWER:     12.    SOLUTION:   13.    SOLUTION:     ANSWER:     ANSWER:     13.      SOLUTION:   14.    SOLUTION:       ANSWER:     ANSWER:     14.    SOLUTION:     15.    SOLUTION:     ANSWER:     ANSWER:     15.    SOLUTION:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   ANSWER:     b. The difference in the populations of the cities is given by:   ANSWER:     a. (f + g)(x) = 375x + 10   16. POPULATION In a particular county, the b. (f - g)(x) = 25x + 40 population of the two largest cities can be modeled   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is For each pair of functions, find and , if in thousands. they exist. State the domain and range for each composed function. a. What is the population of the two cities combined   after any number of years?  17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} b. What is the difference in the populations of the   two cities?    SOLUTION:   SOLUTION:   a. The population of the cities after x years is the Therefore: sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}     ANSWER:   SOLUTION:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of .   So,  is undefined. 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     ANSWER:   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} is g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   undefined;   SOLUTION:   The range of is not a subset of the domain of 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} .   So,  is undefined. SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   is undefined; . is   undefined;   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)} 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)}   g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} ANSWER:     is undefined; . SOLUTION:     22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.   ANSWER:   ; is undefined. ANSWER:     ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)} 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)}   g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   The range of is not a subset of the domain of 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} . g = {(–1, 5), (3, –4), (6, 4), (10, 8)} So,  is undefined.     SOLUTION:   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   ANSWER:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all   even numbers} ANSWER:   , D = {all real numbers}, R = {all odd numbers} ANSWER:     Find and , if they exist. State , D = {all real numbers}, R = {all the domain and range for each composed even numbers} function. , D = {all real numbers}, R = {all   odd numbers} 27.      28.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   even numbers} ANSWER:   , D = {all real numbers}, R = {all odd numbers}   , D = {all real numbers}, R = {all real numbers} 28.  , D = {all real numbers}, R = {all real numbers}     SOLUTION:   29.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all 29.  real numbers} , D = {all real numbers}, R = {all   real numbers} SOLUTION:     30.    SOLUTION:       , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –10} real numbers}   , D = {all real numbers}, R = {all real numbers} ANSWER:     30.  , D = {all real numbers}, R = {y | y ≥ –14}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –10}   31.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ –14}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ –10}   31.  , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} SOLUTION:     32.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 6.5} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} ANSWER:     32.  , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 6.5}   33.      SOLUTION:       , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} ANSWER:     , D = {all real numbers}, R = {all real numbers} 33.  , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   34.    SOLUTION:         , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0} , D = {all real numbers}, R = {all   real numbers} ANSWER:     , D = {all real numbers}, R = {y | y 34.  ≥ 1} , D = {all real numbers}, R = {y | y   ≥ 0} SOLUTION:     35.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0} ANSWER:     35.  , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs   is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   , D = {all real numbers}, R = {y | y a. The profit function P(x) is given by ≥ 0}  where r(x) is the revenue , D = {all real numbers}, R = {y | y function and c(x) is the cost function. ≥ 0} So:   ANSWER:     b. , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the ANSWER:   function for the cost of manufacturing x coffee mugs a. P(x) = 5.75x – 1850 is c(x) = 0.75x + 1850.     b. P(500) = $1025; P(1000) = $3900; P(5000) = a. Write the profit function. $26,900     b. Find the profit on 500, 1000, and 5000 coffee mugs. 37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. SOLUTION:     a. The profit function P(x) is given by a. Write two functions representing the price after  where r(x) is the revenue the discount p(x) and the price after sales tax t(x). function and c(x) is the cost function.   So: b. Which composition of functions represents the price of the HDTV, or ? Explain your reasoning.     b. c. How much will Ms. Smith pay for the HDTV?   SOLUTION:   a. Let p(x) be the price after the discount where x is the original price. Discount = 35%(x) = 0.35x Therefore: ANSWER:   Let t(x) be the price after the sales tax. a. P(x) = 5.75x – 1850 Sales tax = 6.25%(x) = 0.0625x   Therefore: b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900     b. Since = , either function 37. CCSS SENSE-MAKING Ms. Smith wants to buy represents the price. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. c.    a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the price of the HDTV, or ? Explain Substitute x = 2299. your reasoning.   c. How much will Ms. Smith pay for the HDTV?     Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     a. Let p(x) be the price after the discount where x is ANSWER:   the original price. Discount = 35%(x) = 0.35x a. p(x) = 0.65x; t(x) = 1.0625x Therefore:   b. Since = , either function represents the price. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x c. $1587.75 Therefore:   2 Perform each operation if f(x) = x + x – 12 and   g(x) = x – 3. State the domain of the resulting b. Since = , either function function.   represents the price.   38.  c.    SOLUTION:   Substitute x = 2299. D = {all real numbers}     ANSWER:   Therefore, Mr. Smith will pay $1587.75 for HDTV. (f – g)(x) = x2 – 9; D = {all real numbers}     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x 39.      b. Since = , either function SOLUTION:   represents the price.   c. $1587.75   D = {all real numbers} 2 Perform each operation if f(x) = x + x – 12 and   g(x) = x – 3. State the domain of the resulting function. ANSWER:     ; D = {all real 38.  numbers}     SOLUTION:   40.    SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   39.    SOLUTION:     ANSWER:   D = {all real numbers}     ANSWER:   ; D = {all real If f(x) = 5x, g(x) = –2x + 1, and h(x) = x2 + 6x + numbers} 8, find each value.     41.  40.    SOLUTION:     SOLUTION:   Substitute x = –2.   ANSWER:   25     ANSWER:   42.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. Substitute x = 3.   41.      SOLUTION:   ANSWER:   –69   43.  Substitute x = –2.   SOLUTION:     ANSWER:   25     Substitute x = –5.   42.    SOLUTION:     ANSWER:   Substitute x = 3. 483   44.      ANSWER:   SOLUTION:   –69   43.    SOLUTION:     Substitute x = 2.     Substitute x = –5.     ANSWER:   –1     ANSWER:   45.  483     SOLUTION:   44.    SOLUTION:     Substitute x = –3.     Substitute x = 2.     ANSWER:   –5   46.      ANSWER:   SOLUTION:   –1   45.      Substitute x = 9. SOLUTION:       Substitute x = –3.     ANSWER:   2303   47.      ANSWER:   SOLUTION:   –5   46.      SOLUTION:   ANSWER:   –30a + 5   48.    Substitute x = 9.     SOLUTION:       ANSWER:   2303 ANSWER:     2 5a + 70a + 240   47.    49.  SOLUTION:     SOLUTION:     ANSWER:   –30a + 5     48.  ANSWER:     2 –10a + 10a + 1 SOLUTION:     50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) ANSWER:   on the same coordinate grid. 2   5a + 70a + 240 c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   49.  d. VERBAL Describe the relationship among the   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   SOLUTION:   a.   ANSWER:     –10a2 + 10a + 1 b.   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the c. graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a.   c. b. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). b.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States c. during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is   the same as the vertical distance between the graphs b. The function represents the difference of f(x) and (f + g)(x) and between f(x) and (f – g) in the number of men and women employed in the (x). U.S.     51. EMPLOYMENT The number of women and men ANSWER:   age 16 and over employed each year in the United a. y = 2085.6x +123,060 States can be modeled by the following equations,   where x is the number of years since 1994 and y is the number of people in thousands. b. The function represents the difference in the women: y =1086.4x + 56,610 number of men and women employed in the U.S. men: y = 999.2x +66,450     2 a. Write a function that models the total number of If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x men and women employed in the United States + 1, find each value. during this time.     52.  b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent? SOLUTION:     SOLUTION:     Substitute x = 3. a. Add the functions.   The total number of men and women employed is given by   ANSWER:   b. The function represents the difference –180 in the number of men and women employed in the   U.S.   53.  ANSWER:     a. y = 2085.6x +123,060   SOLUTION:   b. The function represents the difference in the number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value. Substitute x = 1.     52.    SOLUTION:       ANSWER:   Substitute x = 3. 0     54.    ANSWER:   –180 SOLUTION:     53.    SOLUTION:     ANSWER:     Substitute x = 1.     55.      SOLUTION:   ANSWER:   0   54.      SOLUTION:   ANSWER:   1   56.    SOLUTION:     ANSWER:       55.  ANSWER:     –33 SOLUTION:     57.    SOLUTION:     ANSWER:   1     ANSWER:   56.  256     SOLUTION:   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for     ,  , , and ANSWER:   .  –33     b. Graphical Use a graphing calculator to 57.  graph  ,  , , and   on the same coordinate plane. SOLUTION:     c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate plane. ANSWER:     256 e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x). 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and SOLUTION:   a. 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.     e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   SOLUTION:   a.   b.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range. 2.      1.  SOLUTION:     SOLUTION:                   ANSWER:   ANSWER:       For each pair of functions, find and , if they exist. State the domain and range for each composed function.   2.  3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.       Therefore:         ANSWER:     is undefined; ANSWER:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:     The range of is not a subset of the domain of For each pair of functions, find and , if . they exist. State the domain and range for each So,  is undefined. composed function.     3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     Therefore: SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:     is undefined;     Therefore: Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of ANSWER:   . So,  is undefined.       6.  Therefore:   SOLUTION:     ANSWER:   is undefined;     Find and , if they exist. State the domain and range for each composed function.   5.      ANSWER:   SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     ANSWER:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go 6.  to taxes.     ANSWER:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     ANSWER:   8.    SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan ANSWER:   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   9.  8.    SOLUTION:     SOLUTION:                     ANSWER:   ANSWER:   6-1 Operations on Functions       9.  10.      SOLUTION:   SOLUTION:               ANSWER:       ANSWER:         11.      10.  SOLUTION:     SOLUTION:   eSolutionsManual-PoweredbyCognero Page5     ANSWER:       ANSWER:     12.    SOLUTION:     11.    SOLUTION:   ANSWER:     ANSWER:     13.    SOLUTION:     12.  SOLUTION:       ANSWER:   ANSWER:     14.    SOLUTION:     13.    SOLUTION:       ANSWER:     ANSWER:     15.      SOLUTION:   14.    SOLUTION:     ANSWER:     ANSWER:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is   in thousands. a. What is the population of the two cities combined 15.  after any number of years?    b. What is the difference in the populations of the two cities?  SOLUTION:     SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10     b. (f - g)(x) = 25x + 40   ANSWER:   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}     SOLUTION:   16. POPULATION In a particular county, the population of the two largest cities can be modeled Therefore: by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40     For each pair of functions, find and , if ANSWER:   they exist. State the domain and range for each composed function.     17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:   SOLUTION:   Therefore:   ANSWER:       19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} ANSWER:   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   The range of is not a subset of the domain of 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} .   So,  is undefined. SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is So,  is undefined. undefined; The range of is not a subset of the domain of   . So,  is undefined. 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)}   g = {(3, –9), (7, 2), (8, –6), (12, 0)}   ANSWER:   SOLUTION:   is undefined;  is undefined.   The range of is not a subset of the domain of . 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} So,  is undefined. g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:     ANSWER:   is undefined;   The range of is not a subset of the domain of . 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} So,  is undefined.     SOLUTION:   ANSWER:   The range of is not a subset of the domain of ; is undefined.   . So,  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   The range of is not a subset of the domain of . 22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)} So,  is undefined.     SOLUTION:   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   The range of is not a subset of the domain of SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. .   So,  is undefined. The range of is not a subset of the domain of ANSWER:   ; is undefined. .   So,  is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} ANSWER:     is undefined;  is undefined.   SOLUTION:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined.   ANSWER:     is undefined;  is undefined.   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)} 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)}   g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   SOLUTION:       ANSWER:   ANSWER:       Find and , if they exist. State 26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} the domain and range for each composed function. g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     ANSWER:   28.      SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all odd numbers} real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all 29.  even numbers} , D = {all real numbers}, R = {all   odd numbers} SOLUTION:     28.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {all real numbers} 30.  , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   29.    SOLUTION:       , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {y | y ≥ –14} ANSWER:   , D = {all real numbers}, R = {y | y ≥ –10}   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all 31.  real numbers}     SOLUTION:   30.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}     , D = {all real numbers}, R = {y | y ANSWER:   ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   , D = {all real numbers}, R = {y | y ≥ –14} 32.  , D = {all real numbers}, R = {y | y ≥ –10}     SOLUTION:   31.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     , D = {all real numbers}, R = {y | y ≥ –11} ANSWER:   , D = {all real numbers}, R = {y | y ≥ –17}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y 33.  ≥ –17}     SOLUTION:   32.    SOLUTION:         , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {y | y ≥ 0.875} ANSWER:   , D = {all real numbers}, R = {y | y ≥ 6.5} , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all ANSWER:   real numbers}   , D = {all real numbers}, R = {y | y 34.  ≥ 0.875} , D = {all real numbers}, R = {y | y   ≥ 6.5} SOLUTION:     33.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     , D = {all real numbers}, R = {all ANSWER:   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0}   , D = {all real numbers}, R = {all real numbers} 35.  , D = {all real numbers}, R = {all real numbers}     SOLUTION:   34.    SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}     , D = {all real numbers}, R = {y | y ANSWER:   ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 1} , D = {all real numbers}, R = {y | y 36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x ≥ 0} coffee mugs is given by r(x) = 6.5x. Suppose the   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   35.  a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee SOLUTION:   mugs.   SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function.   So:   b.   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:   ANSWER:   a. P(x) = 5.75x – 1850   b. P(500) = $1025; P(1000) = $3900; P(5000) = , D = {all real numbers}, R = {y | y $26,900 ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} 37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x a. Write two functions representing the price after coffee mugs is given by r(x) = 6.5x. Suppose the the discount p(x) and the price after sales tax t(x). function for the cost of manufacturing x coffee mugs   is c(x) = 0.75x + 1850. b. Which composition of functions represents the   price of the HDTV, or ? Explain a. Write the profit function. your reasoning.     b. Find the profit on 500, 1000, and 5000 coffee c. How much will Ms. Smith pay for the HDTV? mugs.     SOLUTION:   SOLUTION:   a. Let p(x) be the price after the discount where x is a. The profit function P(x) is given by the original price.  where r(x) is the revenue Discount = 35%(x) = 0.35x function and c(x) is the cost function. Therefore: So: Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x b. Therefore:   b. Since = , either function represents the price.   c.  ANSWER:   a. P(x) = 5.75x – 1850   b. P(500) = $1025; P(1000) = $3900; P(5000) = Substitute x = 2299. $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original Therefore, Mr. Smith will pay $1587.75 for HDTV. price of $2299. The sales tax is 6.25%.     ANSWER:   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x). a. p(x) = 0.65x; t(x) = 1.0625x     b. Which composition of functions represents the b. Since = , either function price of the HDTV, or ? Explain represents the price. your reasoning.     c. $1587.75 c. How much will Ms. Smith pay for the HDTV?     2 Perform each operation if f(x) = x + x – 12 and SOLUTION:   g(x) = x – 3. State the domain of the resulting a. Let p(x) be the price after the discount where x is function. the original price.   Discount = 35%(x) = 0.35x 38.  Therefore:   SOLUTION:   Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore: D = {all real numbers}     b. Since = , either function ANSWER:   represents the price. 2 (f – g)(x) = x – 9; D = {all real numbers}     c.  39.    SOLUTION:   Substitute x = 2299.   D = {all real numbers} Therefore, Mr. Smith will pay $1587.75 for HDTV.     ANSWER:   ANSWER:   ; D = {all real a. p(x) = 0.65x; t(x) = 1.0625x   numbers}   b. Since = , either function represents the price.   40.  c. $1587.75     SOLUTION:   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function.   38.    SOLUTION:     D = {all real numbers} ANSWER:     ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     39.  If f(x) = 5x, g(x) = –2x + 1, and h(x) = x2 + 6x + 8, find each value.     SOLUTION:   41.    SOLUTION:   D = {all real numbers}   ANSWER:   Substitute x = –2. ; D = {all real numbers}     ANSWER:   40.  25     SOLUTION:   42.    SOLUTION:   Substitute x = 3.     ANSWER:   ANSWER:   –69     43.  2   If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. SOLUTION:     41.    SOLUTION:     Substitute x = –5.   Substitute x = –2.     ANSWER:   ANSWER:   25 483     44.  42.      SOLUTION:   SOLUTION:   Substitute x = 3.   Substitute x = 2.     ANSWER:   –69   43.      ANSWER:   SOLUTION:   –1   45.      Substitute x = –5. SOLUTION:       Substitute x = –3.     ANSWER:   483     44.    ANSWER:   –5 SOLUTION:     46.    SOLUTION:     Substitute x = 2.     Substitute x = 9.     ANSWER:   –1     ANSWER:   2303 45.      SOLUTION:   47.    SOLUTION:     Substitute x = –3.     ANSWER:   –30a + 5     48.  ANSWER:     –5   SOLUTION:   46.    SOLUTION:     ANSWER:     2 5a + 70a + 240 Substitute x = 9.     49.    SOLUTION:     ANSWER:   2303   47.      ANSWER:   SOLUTION:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.     a. TABULAR Make a table showing values for f ANSWER:   (x), g(x), ( f + g) (x), and ( f –g)(x). –30a + 5     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid. 48.      c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid. SOLUTION:     d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   ANSWER:   2 5a + 70a + 240   49.    b.   SOLUTION:     ANSWER:   2 –10a + 10a + 1     50. MULTIPLE REPRESENTATIONS Let f(x) = x2 c. and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     d. Sample answer: For each value of x, the vertical SOLUTION:   distance between the graph of g(x) and the x-axis is a. the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a.   b. b.   c. c. d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is d. Sample answer: For each value of x, the vertical the same as the vertical distance between the graphs distance between the graph of g(x) and the x-axis is of f(x) and (f + g)(x) and between f(x) and (f – g) the same as the vertical distance between the graphs (x). of f(x) and (f + g)(x) and between f(x) and (f – g)   (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United ANSWER:   States can be modeled by the following equations, a. where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States b. during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   a. Add the functions. The total number of men and women employed is given by c.   b. The function represents the difference in the number of men and women employed in the U.S.   ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S. d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x2 – 2x of f(x) and (f + g)(x) and between f(x) and (f – g) + 1, find each value. (x).     52.  51. EMPLOYMENT The number of women and men   age 16 and over employed each year in the United SOLUTION:   States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands.   women: y =1086.4x + 56,610 Substitute x = 3. men: y = 999.2x +66,450     a. Write a function that models the total number of men and women employed in the United States during this time.   ANSWER:   b. If f is the function for the number of men, and g is –180 the function for the number of women, what does ( f   – g)(x) represent?   53.  SOLUTION:   a. Add the functions.   The total number of men and women employed is SOLUTION:   given by   b. The function represents the difference   in the number of men and women employed in the Substitute x = 1. U.S.     ANSWER:   a. y = 2085.6x +123,060     b. The function represents the difference in the ANSWER:   number of men and women employed in the U.S.   0   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value. 54.    52.      SOLUTION:   SOLUTION:     Substitute x = 3.     ANSWER:   ANSWER:   –180     55.  53.      SOLUTION:   SOLUTION:     Substitute x = 1.     ANSWER:   1     56.  ANSWER:   0     SOLUTION:   54.    SOLUTION:     ANSWER:   –33     57.    ANSWER:   SOLUTION:     55.      SOLUTION:   ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and   2  if f(x) = x + 1 and g(x) = x – 3. ANSWER:     1 a. Tabular Make a table showing values for     ,  , , and 56.  .      SOLUTION:   b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .     ANSWER:   d. Graphical Use a graphing calculator to graph [f –33 ◦ g](x), and [g ◦ f](x) on the same coordinate   plane.   57.  e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x). SOLUTION:     SOLUTION:   a.   ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and     on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate   plane. b.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:   a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     1.    2.  SOLUTION:     SOLUTION:                 ANSWER:     ANSWER:     2.      For each pair of functions, find and , if SOLUTION:   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined.       Therefore:     ANSWER:     ANSWER:   is undefined;     For each pair of functions, find and , if 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} they exist. State the domain and range for each g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} composed function.     SOLUTION:   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   The range of is not a subset of the domain of .   So,  is undefined. Therefore:     Therefore:   ANSWER:   is undefined;   Find and , if they exist. State the domain and range for each composed   function.     5.  ANSWER:   is undefined;   SOLUTION:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:     Therefore:     6.  ANSWER:   is undefined;     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable ANSWER:   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are   taken before or after taxes are withheld? Explain.   SOLUTION:   6.  Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the SOLUTION:   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and ANSWER:     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     7. CCSS MODELING Dora has 8% of her earnings 8.  deducted from her paycheck for a college savings plan. She can choose to take the deduction either   before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate SOLUTION:   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.    SOLUTION:                 ANSWER:       ANSWER:     9.        SOLUTION:   10.      SOLUTION:               ANSWER:     ANSWER:           10.    11.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:   6-1 O  perations on Functions     11.  12.    SOLUTION:   SOLUTION:     ANSWER:   ANSWER:       13.  12.    SOLUTION:   SOLUTION:     eSolutionsManual-PoweredbyCognero Page6   ANSWER:   ANSWER:       14.  13.    SOLUTION:     SOLUTION:       ANSWER:     ANSWER:       14.    15.  SOLUTION:     SOLUTION:       ANSWER:   ANSWER:       16. POPULATION In a particular county, the 15.  population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is SOLUTION:   in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each   composed function.   16. POPULATION In a particular county, the 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} population of the two largest cities can be modeled g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is in thousands. SOLUTION:   a. What is the population of the two cities combined after any number of years?  Therefore: b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)}   g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} ANSWER:     SOLUTION:     Therefore: 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:       ANSWER:   ANSWER:       18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of   . So,  is undefined. ANSWER:     is ANSWER:   undefined; is undefined;  is undefined.     21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   ANSWER:   is SOLUTION:   undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined. The range of is not a subset of the domain of   . 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} So,  is undefined. g = {(1, –4), (2, –3), (3, –2), (4, –1)}     ANSWER:   SOLUTION:   ; is undefined. The range of is not a subset of the domain of   . So,  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     is undefined;  is undefined.   ANSWER:   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     Find and , if they exist. State the domain and range for each composed 28.  function.     27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers} ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers}     29.  28.      SOLUTION:   SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     29.  30.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ –14} real numbers} , D = {all real numbers}, R = {y | y   ≥ –10}   30.  31.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     31.  32.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0.875} ≥ –17} , D = {all real numbers}, R = {y | y   ≥ 6.5}   32.  33.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {y | y   ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5} , D = {all real numbers}, R = {all   real numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0}     34.  35.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 1} ≥ 0} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0}   35.  36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the SOLUTION:   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:   , D = {all real numbers}, R = {y | y ≥ 0}   b. , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ANSWER:   ≥ 0} a. P(x) = 5.75x – 1850     b. P(500) = $1025; P(1000) = $3900; P(5000) = 36. FINANCE A ceramics store manufactures and sells $26,900 coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs 37. CCSS SENSE-MAKING Ms. Smith wants to buy is c(x) = 0.75x + 1850. an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. a. Write the profit function.     a. Write two functions representing the price after b. Find the profit on 500, 1000, and 5000 coffee the discount p(x) and the price after sales tax t(x). mugs.     b. Which composition of functions represents the price of the HDTV, or ? Explain SOLUTION:   your reasoning. a. The profit function P(x) is given by    where r(x) is the revenue c. How much will Ms. Smith pay for the HDTV? function and c(x) is the cost function.   So: SOLUTION:   a. Let p(x) be the price after the discount where x is   the original price. b. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore:   ANSWER:   b. Since = , either function a. P(x) = 5.75x – 1850 represents the price.     b. P(500) = $1025; P(1000) = $3900; P(5000) = c.  $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. Substitute x = 2299.   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     Therefore, Mr. Smith will pay $1587.75 for HDTV. b. Which composition of functions represents the   price of the HDTV, or ? Explain your reasoning. ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x c. How much will Ms. Smith pay for the HDTV?     b. Since = , either function SOLUTION:   represents the price. a. Let p(x) be the price after the discount where x is   the original price. c. $1587.75 Discount = 35%(x) = 0.35x   Therefore: Perform each operation if f(x) = x2 + x – 12 and g(x) = x – 3. State the domain of the resulting Let t(x) be the price after the sales tax. function.   Sales tax = 6.25%(x) = 0.0625x Therefore: 38.    SOLUTION:     b. Since = , either function represents the price.   c.  D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   Substitute x = 2299. 39.      Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price.   ANSWER:   c. $1587.75 ; D = {all real   numbers}   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. 40.      38.    SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     ANSWER:   39.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value. D = {all real numbers}     41.    ANSWER:   ; D = {all real SOLUTION:   numbers}   40.  Substitute x = –2.   SOLUTION:     ANSWER:   25   42.    SOLUTION:     ANSWER:   Substitute x = 3.     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + ANSWER:   8, find each value. –69     41.  43.      SOLUTION:   SOLUTION:   Substitute x = –2.   Substitute x = –5.     ANSWER:   25     42.    ANSWER:   483 SOLUTION:     44.    Substitute x = 3. SOLUTION:     ANSWER:   –69     Substitute x = 2.   43.    SOLUTION:       ANSWER:   Substitute x = –5. –1     45.    SOLUTION:     ANSWER:   483     Substitute x = –3. 44.      SOLUTION:     ANSWER:   –5     Substitute x = 2.   46.    SOLUTION:       Substitute x = 9. ANSWER:     –1   45.    SOLUTION:     ANSWER:   2303     47.  Substitute x = –3.     SOLUTION:       ANSWER:   –5 ANSWER:     –30a + 5   46.    48.    SOLUTION:   SOLUTION:     Substitute x = 9.     ANSWER:   2 5a + 70a + 240     49.  ANSWER:   2303     SOLUTION:   47.    SOLUTION:     ANSWER:     2 –10a + 10a + 1 ANSWER:     –30a + 5   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x. 48.      a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x). SOLUTION:     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). ANSWER:     2 5a + 70a + 240   SOLUTION:   a. 49.    SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) c. on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) b. (x).   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs ANSWER:   of f(x) and (f + g)(x) and between f(x) and (f – g) a. (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is b. the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   c. SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S.   d. Sample answer: For each value of x, the vertical ANSWER:   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs a. y = 2085.6x +123,060 of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). b. The function represents the difference in the   number of men and women employed in the U.S.   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United 2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x States can be modeled by the following equations, + 1, find each value. where x is the number of years since 1994 and y is   the number of people in thousands. women: y =1086.4x + 56,610 52.  men: y = 999.2x +66,450     SOLUTION:   a. Write a function that models the total number of men and women employed in the United States during this time.     Substitute x = 3. b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   ANSWER:   a. Add the functions. The total number of men and women employed is –180 given by   53.      b. The function represents the difference SOLUTION:   in the number of men and women employed in the U.S.   ANSWER:   a. y = 2085.6x +123,060     Substitute x = 1. b. The function represents the difference in the   number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value.   ANSWER:   52.  0     SOLUTION:   54.      Substitute x = 3.   SOLUTION:   ANSWER:   –180     53.  ANSWER:     SOLUTION:     55.      SOLUTION:   Substitute x = 1.       ANSWER:   0 ANSWER:     1   54.  56.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   –33     57.  55.      SOLUTION:   SOLUTION:     ANSWER:     256   ANSWER:   1 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and 56.  2  if f(x) = x + 1 and g(x) = x – 3.     SOLUTION:   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and   on the same coordinate plane. ANSWER:   –33     c. Verbal Explain the relationship between  and  . 57.      d. Graphical Use a graphing calculator to graph [f SOLUTION:   ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   ANSWER:   a. 256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .       d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   b. a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.      SOLUTION:   2.      SOLUTION:               ANSWER:     ANSWER:     2.      SOLUTION:   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   The range of is not a subset of the domain of   . So,  is undefined.       Therefore:   ANSWER:       ANSWER:   is undefined;     For each pair of functions, find and , if they exist. State the domain and range for each 4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} composed function.   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} 3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)} SOLUTION:     The range of is not a subset of the domain of SOLUTION:   . The range of is not a subset of the domain of So,  is undefined. .   So,  is undefined.     Therefore:   Therefore:   ANSWER:   is undefined;     Find and , if they exist. State the domain and range for each composed   function.   ANSWER:   5.  is undefined;     SOLUTION:   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:     The range of is not a subset of the domain of . So,  is undefined.       Therefore: ANSWER:       ANSWER:   is undefined; 6.      Find and , if they exist. State SOLUTION:   the domain and range for each composed function.   5.      SOLUTION:       ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings ANSWER:   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   6.  SOLUTION:     Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan SOLUTION:   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range. 7. CCSS MODELING Dora has 8% of her earnings   deducted from her paycheck for a college savings plan. She can choose to take the deduction either 8.  before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are SOLUTION:   taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and ANSWER:     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.      SOLUTION:               ANSWER:       ANSWER:     9.    SOLUTION:       10.      SOLUTION:           ANSWER:         ANSWER:       10.        SOLUTION:   11.    SOLUTION:         ANSWER:     ANSWER:         11.    12.  SOLUTION:   SOLUTION:     ANSWER:   ANSWER:       12.  13.    SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   6-1 Operations on Functions     13.  14.      SOLUTION:   SOLUTION:       ANSWER:     ANSWER:     14.      SOLUTION:   15.    SOLUTION:   eSolutionsManual-PoweredbyCognero Page7   ANSWER:     ANSWER:       15.  16. POPULATION In a particular county, the   population of the two largest cities can be modeled SOLUTION:   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:     ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40     For each pair of functions, find and , if they exist. State the domain and range for each 16. POPULATION In a particular county, the composed function. population of the two largest cities can be modeled   by f(x) = 200x + 25 and g(x) = 175x - 15, where x is 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} the number of years since 2000 and the population is g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} in thousands.   a. What is the population of the two cities combined SOLUTION:   after any number of years?  b. What is the difference in the populations of the Therefore: two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}     SOLUTION:   ANSWER:   Therefore:   18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:       ANSWER:   18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)}   g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} SOLUTION:   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   is is undefined;  is undefined. undefined;     20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   is undefined; .     22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} ANSWER:   g = {(5, –4), (4, –3), (–1, 2), (2, 3)} is   undefined; SOLUTION:     21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.   ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   The range of is not a subset of the domain of ; is undefined. .   So,  is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)} ANSWER:     ; is undefined. SOLUTION:     The range of is not a subset of the domain of 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} . g = {(4, –3), (3, –7), (9, 8), (–4, –4)} So,  is undefined.   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.     25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} ANSWER:   g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} ANSWER:   g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all even numbers}   , D = {all real numbers}, R = {all odd numbers} Find and , if they exist. State   the domain and range for each composed function.   28.  27.      SOLUTION:   SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all odd numbers} real numbers}   ANSWER:   ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} odd numbers} , D = {all real numbers}, R = {all   real numbers}   28.  29.      SOLUTION:   SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all   real numbers} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all   real numbers}   29.    30.  SOLUTION:     SOLUTION:       , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14}   , D = {all real numbers}, R = {y | y ≥ –10} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14}   , D = {all real numbers}, R = {y | y ≥ –10} 30.      31.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –14}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17} ANSWER:     ANSWER:   , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17}   31.    32.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 0.875}   , D = {all real numbers}, R = {y | y ANSWER:   ≥ 6.5}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5} 32.      33.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 6.5} real numbers}   , D = {all real numbers}, R = {all ANSWER:   real numbers}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5} , D = {all real numbers}, R = {all   real numbers}   33.  34.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 1}   , D = {all real numbers}, R = {y | y ≥ 0}   34.    35.  SOLUTION:     SOLUTION:         , D = {all real numbers}, R = {y | y ≥ 1}   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y 35.  ≥ 0}     36. FINANCE A ceramics store manufactures and sells SOLUTION:   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee mugs.   SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function.   So: , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0} b.   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:     a. P(x) = 5.75x – 1850 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x b. P(500) = $1025; P(1000) = $3900; P(5000) = coffee mugs is given by r(x) = 6.5x. Suppose the $26,900 function for the cost of manufacturing x coffee mugs   is c(x) = 0.75x + 1850.   37. CCSS SENSE-MAKING Ms. Smith wants to buy a. Write the profit function. an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%.     b. Find the profit on 500, 1000, and 5000 coffee mugs. a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     SOLUTION:   b. Which composition of functions represents the a. The profit function P(x) is given by price of the HDTV, or ? Explain  where r(x) is the revenue your reasoning. function and c(x) is the cost function.   So: c. How much will Ms. Smith pay for the HDTV?   SOLUTION:     a. Let p(x) be the price after the discount where x is b. the original price. Discount = 35%(x) = 0.35x Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore: ANSWER:     a. P(x) = 5.75x – 1850   b. Since = , either function b. P(500) = $1025; P(1000) = $3900; P(5000) = represents the price. $26,900     c.  37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%.   a. Write two functions representing the price after Substitute x = 2299. the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the   price of the HDTV, or ? Explain Therefore, Mr. Smith will pay $1587.75 for HDTV. your reasoning.     c. How much will Ms. Smith pay for the HDTV? ANSWER:     a. p(x) = 0.65x; t(x) = 1.0625x   SOLUTION:   b. Since = , either function a. Let p(x) be the price after the discount where x is represents the price. the original price.   Discount = 35%(x) = 0.35x Therefore: c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and Let t(x) be the price after the sales tax. g(x) = x – 3. State the domain of the resulting Sales tax = 6.25%(x) = 0.0625x Therefore: function.   38.      b. Since = , either function SOLUTION:   represents the price.   c.  D = {all real numbers}   ANSWER:   Substitute x = 2299. (f – g)(x) = x2 – 9; D = {all real numbers}     39.  Therefore, Mr. Smith will pay $1587.75 for HDTV.     SOLUTION:   ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   b. Since = , either function represents the price. D = {all real numbers}     c. $1587.75 ANSWER:     ; D = {all real 2 Perform each operation if f(x) = x + x – 12 and numbers} g(x) = x – 3. State the domain of the resulting   function.   40.  38.      SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     39.  ANSWER:     SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + D = {all real numbers} 8, find each value.     ANSWER:   41.  ; D = {all real   numbers} SOLUTION:     40.    Substitute x = –2. SOLUTION:     ANSWER:   25   42.      SOLUTION:   ANSWER:   Substitute x = 3.   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x +   8, find each value.   ANSWER:   –69 41.      SOLUTION:   43.    SOLUTION:   Substitute x = –2.     Substitute x = –5. ANSWER:     25   42.      SOLUTION:   ANSWER:   483   44.  Substitute x = 3.   SOLUTION:     ANSWER:   –69     43.  Substitute x = 2.     SOLUTION:       Substitute x = –5.   ANSWER:   –1   45.      SOLUTION:   ANSWER:   483   44.      Substitute x = –3.   SOLUTION:     ANSWER:     Substitute x = 2. –5     46.    SOLUTION:     ANSWER:     –1 Substitute x = 9.     45.    SOLUTION:     ANSWER:   2303     Substitute x = –3.   47.    SOLUTION:     ANSWER:   –5     ANSWER:   46.  –30a + 5     SOLUTION:   48.    SOLUTION:     Substitute x = 9.     ANSWER:     5a2 + 70a + 240   ANSWER:   2303 49.      47.  SOLUTION:     SOLUTION:       ANSWER:   ANSWER:   –30a + 5 –10a2 + 10a + 1     48.  50. MULTIPLE REPRESENTATIONS Let f(x) = x2   and g(x) = x.   SOLUTION:   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   ANSWER:   d. VERBAL Describe the relationship among the 2 5a + 70a + 240 graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     SOLUTION:   49.  a.   SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is b. the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   d. Sample answer: For each value of x, the vertical ANSWER:   distance between the graph of g(x) and the x-axis is a. the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United b. States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent? c.   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the d. Sample answer: For each value of x, the vertical U.S.   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs ANSWER:   of f(x) and (f + g)(x) and between f(x) and (f – g) (x). a. y = 2085.6x +123,060     b. The function represents the difference in the 51. EMPLOYMENT The number of women and men number of men and women employed in the U.S. age 16 and over employed each year in the United   States can be modeled by the following equations, where x is the number of years since 1994 and y is 2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x the number of people in thousands. + 1, find each value. women: y =1086.4x + 56,610   men: y = 999.2x +66,450 52.    a. Write a function that models the total number of   men and women employed in the United States SOLUTION:   during this time.   b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f Substitute x = 3. – g)(x) represent?     SOLUTION:   a. Add the functions. The total number of men and women employed is ANSWER:   given by –180     b. The function represents the difference 53.  in the number of men and women employed in the   U.S. SOLUTION:     ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the   number of men and women employed in the U.S. Substitute x = 1.     2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value.     52.    ANSWER:   0 SOLUTION:       54.  Substitute x = 3.     SOLUTION:   ANSWER:   –180   53.      ANSWER:   SOLUTION:     55.      Substitute x = 1.   SOLUTION:     ANSWER:   0     ANSWER:   54.  1     56.  SOLUTION:     SOLUTION:     ANSWER:     ANSWER:   –33     55.  57.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   256 1     58. MULTIPLE REPRESENTATIONS You will 56.  explore ,  , , and   2  if f(x) = x + 1 and g(x) = x – 3. SOLUTION:     a. Tabular Make a table showing values for   ,  , , and .      b. Graphical Use a graphing calculator to graph  ,  , , and ANSWER:   –33 on the same coordinate plane.     57.  c. Verbal Explain the relationship between    and  . SOLUTION:     d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g]   (x), and [g ◦ f](x).   ANSWER:   SOLUTION:   256   a. 58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .     d. Graphical Use a graphing calculator to graph [f   ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:     a. b.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     1.    2.  SOLUTION:     SOLUTION:                 ANSWER:     ANSWER:     2.      For each pair of functions, find and , if they exist. State the domain and range for each SOLUTION:   composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined.     Therefore:       ANSWER:     ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} For each pair of functions, find and , if g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)} they exist. State the domain and range for each   composed function.   SOLUTION:   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} The range of is not a subset of the domain of g = {(9, 11), (6, 15), (10, 13), (5, 8)} .   So,  is undefined. SOLUTION:     The range of is not a subset of the domain of .   So,  is undefined. Therefore:       Therefore: ANSWER:   is undefined;   Find and , if they exist. State the domain and range for each composed   function.     5.  ANSWER:     is undefined; SOLUTION:     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     ANSWER:     Therefore:     6.  ANSWER:     is undefined; SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   5.      SOLUTION:     ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable ANSWER:   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her 6.  paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college   plan and $152.95 will go to taxes. If she takes the SOLUTION:   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.   ANSWER:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     Find (f + g)(x), (f – g)(x), (x),and ANSWER:     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     8.  7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings   plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable SOLUTION:   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   ANSWER:   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   8.      SOLUTION:   9.    SOLUTION:                 ANSWER:       ANSWER:     9.        SOLUTION:   10.      SOLUTION:                 ANSWER:   ANSWER:           10.    11.  SOLUTION:     SOLUTION:         ANSWER:     ANSWER:         12.  11.    SOLUTION:   SOLUTION:   ANSWER:     ANSWER:       13.    12.  SOLUTION:   SOLUTION:       ANSWER:   ANSWER:       14.    13.  SOLUTION:     SOLUTION:         ANSWER:   ANSWER:       14.  15.    SOLUTION:     SOLUTION:       ANSWER:   ANSWER:   6-1 Operations on Functions     16. POPULATION In a particular county, the 15.  population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is in thousands. SOLUTION:   a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.     16. POPULATION In a particular county, the 17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} population of the two largest cities can be modeled g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} by f(x) = 200x + 25 and g(x) = 175x - 15, where x is   the number of years since 2000 and the population is SOLUTION:   in thousands. a. What is the population of the two cities combined Therefore: after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations. eSolutionsManual-PoweredbyCognero Page8   b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.     17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)} ANSWER:     SOLUTION:     Therefore: 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:       ANSWER:   ANSWER:       18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} 19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is   undefined; ANSWER:     is undefined;  is undefined.   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)}   g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)}   g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   ANSWER:   SOLUTION:   is undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   The range of is not a subset of the domain of SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. .   So,  is undefined. ANSWER:   ; is undefined.   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:     ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   The range of is not a subset of the domain of SOLUTION:   . So,  is undefined.   ANSWER:   ; is undefined.   The range of is not a subset of the domain of . 24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} So,  is undefined. g = {(1, –4), (2, –3), (3, –2), (4, –1)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of ; is undefined. .   So,  is undefined. 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} The range of is not a subset of the domain of g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   . So,  is undefined. SOLUTION:     ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     is undefined;  is undefined. ANSWER:     25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:       ANSWER:   26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}     SOLUTION:   Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:     , D = {all real numbers}, R = {all ANSWER:   even numbers} , D = {all real numbers}, R = {all odd numbers}     Find and , if they exist. State the domain and range for each composed 28.  function.     27.  SOLUTION:     SOLUTION:     , D = {all real numbers}, R = {all   real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers} ANSWER:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all even numbers} real numbers} , D = {all real numbers}, R = {all   odd numbers}   29.  28.    SOLUTION:     SOLUTION:       , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     30.  29.      SOLUTION:   SOLUTION:         , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     30.  31.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     31.  32.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –11} ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 6.5}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –11} ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 6.5}     32.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0.875} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 6.5} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  33.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 1} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0} real numbers}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 1} real numbers} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {all ≥ 0} real numbers}     35.  34.      SOLUTION:   SOLUTION:           , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     ANSWER:   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0} , D = {all real numbers}, R = {y | y   ≥ 0}   35.  36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs SOLUTION:   is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.     SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue function and c(x) is the cost function. So:   , D = {all real numbers}, R = {y | y   ≥ 0} b. , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:   , D = {all real numbers}, R = {y | y a. P(x) = 5.75x – 1850 ≥ 0}     b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900 36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the 37. CCSS SENSE-MAKING Ms. Smith wants to buy function for the cost of manufacturing x coffee mugs an HDTV, which is on sale for 35% off the original is c(x) = 0.75x + 1850. price of $2299. The sales tax is 6.25%.     a. Write the profit function. a. Write two functions representing the price after   the discount p(x) and the price after sales tax t(x). b. Find the profit on 500, 1000, and 5000 coffee   mugs. b. Which composition of functions represents the   price of the HDTV, or ? Explain SOLUTION:   your reasoning. a. The profit function P(x) is given by    where r(x) is the revenue c. How much will Ms. Smith pay for the HDTV? function and c(x) is the cost function.   So: SOLUTION:   a. Let p(x) be the price after the discount where x is the original price.   Discount = 35%(x) = 0.35x b. Therefore: Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x Therefore:   b. Since = , either function ANSWER:   represents the price. a. P(x) = 5.75x – 1850     c.  b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. Substitute x = 2299.   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).     Therefore, Mr. Smith will pay $1587.75 for HDTV. b. Which composition of functions represents the   price of the HDTV, or ? Explain ANSWER:   your reasoning.   a. p(x) = 0.65x; t(x) = 1.0625x   c. How much will Ms. Smith pay for the HDTV?   b. Since = , either function represents the price. SOLUTION:     a. Let p(x) be the price after the discount where x is c. $1587.75 the original price.   Discount = 35%(x) = 0.35x Therefore: 2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x 38.  Therefore:   SOLUTION:     b. Since = , either function represents the price.   c.  D = {all real numbers}   ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}   Substitute x = 2299. 39.      SOLUTION:   Therefore, Mr. Smith will pay $1587.75 for HDTV.   ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price. ANSWER:     ; D = {all real c. $1587.75 numbers}     2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting 40.  function.     38.  SOLUTION:     SOLUTION:   D = {all real numbers}   ANSWER:   (f – g)(x) = x2 – 9; D = {all real numbers}     ANSWER:   39.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   D = {all real numbers} 41.      ANSWER:   SOLUTION:   ; D = {all real numbers}   Substitute x = –2. 40.      SOLUTION:   ANSWER:   25   42.    SOLUTION:     ANSWER:   Substitute x = 3.     2 ANSWER:   If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + –69 8, find each value.     41.  43.      SOLUTION:   SOLUTION:   Substitute x = –2.   Substitute x = –5.     ANSWER:   25     42.  ANSWER:     483 SOLUTION:     44.    SOLUTION:   Substitute x = 3.   ANSWER:     –69 Substitute x = 2.     43.    SOLUTION:     ANSWER:     Substitute x = –5. –1     45.    SOLUTION:     ANSWER:   483     Substitute x = –3. 44.      SOLUTION:     ANSWER:   –5     Substitute x = 2. 46.      SOLUTION:       Substitute x = 9. ANSWER:     –1   45.      SOLUTION:   ANSWER:   2303   47.    Substitute x = –3.     SOLUTION:       ANSWER:   ANSWER:   –5 –30a + 5     46.  48.      SOLUTION:   SOLUTION:     Substitute x = 9.     ANSWER:   2 5a + 70a + 240     49.  ANSWER:     2303   SOLUTION:   47.    SOLUTION:     ANSWER:     2 –10a + 10a + 1 ANSWER:     –30a + 5   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   48.  a. TABULAR Make a table showing values for f   (x), g(x), ( f + g) (x), and ( f –g)(x). SOLUTION:     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.     d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). ANSWER:     2 5a + 70a + 240 SOLUTION:     a. 49.    SOLUTION:     b.   ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     c. b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). b.   ANSWER:   a. b.   c. c.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x). d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs ANSWER:   of f(x) and (f + g)(x) and between f(x) and (f – g) a. (x).   51. EMPLOYMENT The number of women and men age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. b. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   c. a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S.   d. Sample answer: For each value of x, the vertical ANSWER:   distance between the graph of g(x) and the x-axis is a. y = 2085.6x +123,060 the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) b. The function represents the difference in the (x). number of men and women employed in the U.S.     51. EMPLOYMENT The number of women and men 2 age 16 and over employed each year in the United If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x States can be modeled by the following equations, + 1, find each value. where x is the number of years since 1994 and y is   the number of people in thousands. 52.  women: y =1086.4x + 56,610   men: y = 999.2x +66,450   SOLUTION:   a. Write a function that models the total number of men and women employed in the United States   during this time. Substitute x = 3.     b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent?   SOLUTION:   ANSWER:   a. Add the functions. –180 The total number of men and women employed is   given by 53.      b. The function represents the difference SOLUTION:   in the number of men and women employed in the U.S.   ANSWER:     a. y = 2085.6x +123,060 Substitute x = 1.     b. The function represents the difference in the number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x   + 1, find each value.   ANSWER:   52.  0     SOLUTION:   54.      Substitute x = 3. SOLUTION:     ANSWER:   –180     ANSWER:   53.    SOLUTION:     55.      SOLUTION:   Substitute x = 1.       ANSWER:   0 ANSWER:     1   54.  56.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   –33     57.    55.    SOLUTION:   SOLUTION:     ANSWER:     256   ANSWER:   1 58. MULTIPLE REPRESENTATIONS You will   explore ,  , , and 56.   if f(x) = x2 + 1 and g(x) = x – 3.     SOLUTION:   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and   on the same coordinate plane. ANSWER:   –33     c. Verbal Explain the relationship between  and  . 57.      d. Graphical Use a graphing calculator to graph [f SOLUTION:   ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   a. ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .       d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     b. SOLUTION:   a.         b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:           ANSWER:     2.  Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any SOLUTION:   restrictions in domain or range.   1.      SOLUTION:               ANSWER:     ANSWER:     For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)}   g = {(9, 11), (6, 15), (10, 13), (5, 8)}   SOLUTION:   2.  The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined.     Therefore:           ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} ANSWER:   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.     For each pair of functions, find and , if   they exist. State the domain and range for each Therefore: composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of is undefined;   . So,  is undefined. Find and , if they exist. State   the domain and range for each composed function.     Therefore: 5.    SOLUTION:         ANSWER:   is undefined;     4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} ANSWER:   g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:     The range of is not a subset of the domain of . 6.  So,  is undefined.     SOLUTION:     Therefore:     ANSWER:   is undefined;   Find and , if they exist. State the domain and range for each composed   function.   ANSWER:   5.      SOLUTION:   7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is   $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the ANSWER:   college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   Either way, she will have $228.95 taken from her 6.  paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only SOLUTION:   $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any   restrictions in domain or range.   8.      SOLUTION:   ANSWER:       7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable   income, or after taxes are withheld. Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.     SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes. ANSWER:     ANSWER:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go   to taxes.   Find (f + g)(x), (f – g)(x), (x),and 9.    (x) for each f(x) and g(x). Indicate any   SOLUTION:   restrictions in domain or range.   8.      SOLUTION:                 ANSWER:     ANSWER:       10.      SOLUTION:   9.    SOLUTION:               ANSWER:       ANSWER:       11.    SOLUTION:       10.    SOLUTION:     ANSWER:         ANSWER:     12.  SOLUTION:       11.    SOLUTION:   ANSWER:       13.  ANSWER:     SOLUTION:       12.  SOLUTION:     ANSWER:     14.  ANSWER:     SOLUTION:     13.    SOLUTION:       ANSWER:     ANSWER:     15.    SOLUTION:     14.    SOLUTION:     ANSWER:       ANSWER:   16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?    b. What is the difference in the populations of the two cities?    15.  SOLUTION:     a. The population of the cities after x years is the sum of the individual populations. SOLUTION:     b. The difference in the populations of the cities is given by:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each   composed function.   ANSWER:   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years?  b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.   b. The difference in the populations of the cities is given by:     ANSWER:   a. (f + g)(x) = 375x + 10 ANSWER:     6-1 Ob.p (efr -a tgi)o(nxs) =o n2 5Fxu +n c4t0i ons     For each pair of functions, find and , if 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} they exist. State the domain and range for each g = {(6, 8), (–12, –5), (0, 5), (5, 1)} composed function.     SOLUTION:   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of .   So,  is undefined. The range of is not a subset of the domain of 18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} . g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   So,  is undefined.   SOLUTION:   eSolutionsManual-PoweredbyCog nero ANSWER:   Page9 is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)}   g = {(–8, 2), (–4, 1), (3, –3), (5, 7)} ANSWER:     is SOLUTION:   undefined; The range of is not a subset of the domain of   . 21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} So,  is undefined. g = {(3, –9), (7, 2), (8, –6), (12, 0)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; .   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is So,  is undefined. undefined;     ANSWER:   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} ; is undefined. g = {(3, –9), (7, 2), (8, –6), (12, 0)}     SOLUTION:   23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} The range of is not a subset of the domain of   . SOLUTION:   So,  is undefined. The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is undefined; .     ANSWER:   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} ; is undefined. g = {(5, –4), (4, –3), (–1, 2), (2, 3)}     SOLUTION:   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.     ANSWER:   ANSWER:   ; is undefined. is undefined;  is undefined.     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(4, –3), (3, –7), (9, 8), (–4, –4)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. The range of is not a subset of the domain of .   So,  is undefined. 26. f = {(12, –3), (9, –2), (8, –1), (6, 3)}   g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)} ANSWER:     is undefined;  is undefined. SOLUTION:     25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     ANSWER:       ANSWER:   Find and , if they exist. State the domain and range for each composed function.     27.  26. f = {(12, –3), (9, –2), (8, –1), (6, 3)}   g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:   SOLUTION:     , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   28.      ANSWER:   SOLUTION:     Find and , if they exist. State the domain and range for each composed function.   27.      , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:   , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} even numbers}   , D = {all real numbers}, R = {all odd numbers} 29.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   28.    SOLUTION:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {all real numbers} real numbers}     30.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   29.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –14} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ –10}     ANSWER:   31.    , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   30.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ –11}   , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –14} ≥ –11} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –10} ≥ –17}     ANSWER:   32.    , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   31.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ 0.875}   , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –11} ≥ 0.875} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ –17} ≥ 6.5}     ANSWER:   33.    , D = {all real numbers}, R = {y | y SOLUTION:   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   32.      SOLUTION:     , D = {all real numbers}, R = {all real numbers}   , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0.875} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 6.5}     34.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     33.    SOLUTION:     , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y   ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y   ≥ 1} , D = {all real numbers}, R = {all , D = {all real numbers}, R = {y | y real numbers} ≥ 0} , D = {all real numbers}, R = {all   real numbers}   35.  ANSWER:     , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   34.      SOLUTION:     , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 1} , D = {all real numbers}, R = {y | y , D = {all real numbers}, R = {y | y ≥ 0} ≥ 0}     36. FINANCE A ceramics store manufactures and sells ANSWER:   coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the , D = {all real numbers}, R = {y | y function for the cost of manufacturing x coffee mugs ≥ 1} is c(x) = 0.75x + 1850. , D = {all real numbers}, R = {y | y   ≥ 0} a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee mugs. 35.      SOLUTION:   a. The profit function P(x) is given by SOLUTION:    where r(x) is the revenue function and c(x) is the cost function. So:   b.     , D = {all real numbers}, R = {y | y ≥ 0} ANSWER:   , D = {all real numbers}, R = {y | y a. P(x) = 5.75x – 1850 ≥ 0}     b. P(500) = $1025; P(1000) = $3900; P(5000) = $26,900 ANSWER:     37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original , D = {all real numbers}, R = {y | y price of $2299. The sales tax is 6.25%. ≥ 0}   , D = {all real numbers}, R = {y | y a. Write two functions representing the price after ≥ 0} the discount p(x) and the price after sales tax t(x).     b. Which composition of functions represents the 36. FINANCE A ceramics store manufactures and sells price of the HDTV, or ? Explain coffee mugs. The revenue r(x) from the sale of x your reasoning. coffee mugs is given by r(x) = 6.5x. Suppose the   function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850. c. How much will Ms. Smith pay for the HDTV?     a. Write the profit function. SOLUTION:     a. Let p(x) be the price after the discount where x is b. Find the profit on 500, 1000, and 5000 coffee the original price. mugs. Discount = 35%(x) = 0.35x   Therefore: SOLUTION:   a. The profit function P(x) is given by  where r(x) is the revenue Let t(x) be the price after the sales tax. Sales tax = 6.25%(x) = 0.0625x function and c(x) is the cost function. Therefore: So:     b. Since = , either function b. represents the price.   c.  Substitute x = 2299. ANSWER:   a. P(x) = 5.75x – 1850     b. P(500) = $1025; P(1000) = $3900; P(5000) = Therefore, Mr. Smith will pay $1587.75 for HDTV. $26,900     ANSWER:   37. CCSS SENSE-MAKING Ms. Smith wants to buy a. p(x) = 0.65x; t(x) = 1.0625x an HDTV, which is on sale for 35% off the original   price of $2299. The sales tax is 6.25%. b. Since = , either function   represents the price. a. Write two functions representing the price after   the discount p(x) and the price after sales tax t(x). c. $1587.75     b. Which composition of functions represents the price of the HDTV, or ? Explain 2 Perform each operation if f(x) = x + x – 12 and your reasoning. g(x) = x – 3. State the domain of the resulting   function. c. How much will Ms. Smith pay for the HDTV?     38.  SOLUTION:     a. Let p(x) be the price after the discount where x is SOLUTION:   the original price. Discount = 35%(x) = 0.35x Therefore: D = {all real numbers} Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x Therefore: ANSWER:   2 (f – g)(x) = x – 9; D = {all real numbers}     b. Since = , either function 39.  represents the price.     c.  SOLUTION:   D = {all real numbers} Substitute x = 2299.   ANSWER:     ; D = {all real Therefore, Mr. Smith will pay $1587.75 for HDTV. numbers}     ANSWER:   40.  a. p(x) = 0.65x; t(x) = 1.0625x     b. Since = , either function represents the price. SOLUTION:     c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function.   38.      SOLUTION:   ANSWER:   D = {all real numbers}     ANSWER:   If f(x) = 5x, g(x) = –2x + 1, and h(x) = x2 + 6x + 2 8, find each value. (f – g)(x) = x – 9; D = {all real numbers}     41.  39.      SOLUTION:   SOLUTION:   Substitute x = –2. D = {all real numbers}     ANSWER:   ANSWER:   ; D = {all real 25 numbers}     42.  40.      SOLUTION:   SOLUTION:   Substitute x = 3.   ANSWER:   –69     ANSWER:   43.    SOLUTION:     2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.     Substitute x = –5. 41.      SOLUTION:     Substitute x = –2. ANSWER:   483     44.  ANSWER:     25   SOLUTION:   42.    SOLUTION:     Substitute x = 2.   Substitute x = 3.   ANSWER:     –69 ANSWER:     –1   43.    45.  SOLUTION:     SOLUTION:     Substitute x = –5.     Substitute x = –3.       ANSWER:   ANSWER:   483   –5   44.  46.      SOLUTION:   SOLUTION:       Substitute x = 9. Substitute x = 2.         ANSWER:   2303 ANSWER:     –1   47.    45.  SOLUTION:     SOLUTION:     ANSWER:     –30a + 5 Substitute x = –3.     48.    SOLUTION:     ANSWER:   –5   46.      ANSWER:   SOLUTION:   2 5a + 70a + 240   49.      Substitute x = 9.   SOLUTION:       ANSWER:   2303 ANSWER:     2 –10a + 10a + 1   47.    50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x. SOLUTION:     a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x)   on the same coordinate grid. ANSWER:     –30a + 5 c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid.   48.  d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).     SOLUTION:   SOLUTION:   a.   ANSWER:     2 b. 5a + 70a + 240   49.    SOLUTION:       c. ANSWER:   2 –10a + 10a + 1   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x)   on the same coordinate grid. d. Sample answer: For each value of x, the vertical   distance between the graph of g(x) and the x-axis is d. VERBAL Describe the relationship among the the same as the vertical distance between the graphs graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x). of f(x) and (f + g)(x) and between f(x) and (f – g)   (x). SOLUTION:     a. ANSWER:   a.   b. b. c.   c. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).     d. Sample answer: For each value of x, the vertical 51. EMPLOYMENT The number of women and men distance between the graph of g(x) and the x-axis is age 16 and over employed each year in the United the same as the vertical distance between the graphs States can be modeled by the following equations, of f(x) and (f + g)(x) and between f(x) and (f – g) where x is the number of years since 1994 and y is (x). the number of people in thousands.   women: y =1086.4x + 56,610 men: y = 999.2x +66,450 ANSWER:     a. a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is the function for the number of women, what does ( f – g)(x) represent? b.   SOLUTION:   a. Add the functions. The total number of men and women employed is given by   b. The function represents the difference in the number of men and women employed in the U.S. c.   ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S.   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value.   d. Sample answer: For each value of x, the vertical 52.  distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) SOLUTION:   (x).     51. EMPLOYMENT The number of women and men Substitute x = 3. age 16 and over employed each year in the United   States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   ANSWER:   a. Write a function that models the total number of –180 men and women employed in the United States   during this time.   53.  b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f – g)(x) represent? SOLUTION:     SOLUTION:   a. Add the functions. The total number of men and women employed is given by   Substitute x = 1.     b. The function represents the difference in the number of men and women employed in the U.S.     ANSWER:   ANSWER:   0 a. y = 2085.6x +123,060     b. The function represents the difference in the 54.  number of men and women employed in the U.S.     2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x SOLUTION:   + 1, find each value.   52.    SOLUTION:       Substitute x = 3. ANSWER:       55.  ANSWER:   –180     SOLUTION:   53.    SOLUTION:     ANSWER:     1 Substitute x = 1.     56.    SOLUTION:     ANSWER:   0   54.      ANSWER:   SOLUTION:   –33   57.    SOLUTION:     ANSWER:       ANSWER:   55.  256     SOLUTION:   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and   ANSWER:   .  1     b. Graphical Use a graphing calculator to graph  ,  , , and 56.    on the same coordinate plane. SOLUTION:     c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate   plane.   ANSWER:   e.  Verbal Explain the relationship between [f ◦ g] –33 (x), and [g ◦ f](x).     SOLUTION:   57.  a.   SOLUTION:     ANSWER:   256   58. MULTIPLE REPRESENTATIONS You will explore ,  , , and 2  if f(x) = x + 1 and g(x) = x – 3.   a. Tabular Make a table showing values for   ,  , , and   .      b. Graphical Use a graphing calculator to graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between    and  . b.   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).     SOLUTION:   a.       b. Find (f + g)(x), (f – g)(x), (x),and   (x) for each f(x) and g(x). Indicate any restrictions in domain or range.   1.    SOLUTION:     Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     1.    SOLUTION:     ANSWER:           2.    SOLUTION:     ANSWER:         2.      SOLUTION:     ANSWER:         For each pair of functions, find and , if   they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:   ANSWER:   The range of is not a subset of the domain of . So,  is undefined.     Therefore:   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   3. f = {(2, 5), (6, 10), (12, 9), (7, 6)} g = {(9, 11), (6, 15), (10, 13), (5, 8)}     SOLUTION:     The range of is not a subset of the domain of ANSWER:   . is undefined; So,  is undefined.       4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} Therefore: g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.         Therefore: ANSWER:   is undefined;     ANSWER:   4. f = {(–5, 4), (14, 8), (12, 1), (0, –3)} is undefined; g = {(–2, –4), (–3, 2), (–1, 4), (5, –6)}     SOLUTION:   Find and , if they exist. State The range of is not a subset of the domain of the domain and range for each composed function. .   So,  is undefined. 5.        SOLUTION:   Therefore:   ANSWER:     is undefined;   Find and , if they exist. State the domain and range for each composed function.     ANSWER:   5.      SOLUTION:   6.    SOLUTION:         ANSWER:       ANSWER:   6.    SOLUTION:     7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings plan. She can choose to take the deduction either before taxes are withheld, which reduces her taxable income, or after taxes are withheld. Dora’s tax rate   is 17.5%. If her pay before taxes and deductions is $950, will she save more money if the deductions are taken before or after taxes are withheld? Explain.   SOLUTION:   Either way, she will have $228.95 taken from her   paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college ANSWER:   plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only $62.70 will go to her college plan and $166.25 will go to taxes.     ANSWER:   7. CCSS MODELING Dora has 8% of her earnings deducted from her paycheck for a college savings Either way, she will have $228.95 taken from her plan. She can choose to take the deduction either paycheck. If she takes the college savings plan before taxes are withheld, which reduces her taxable deduction before taxes, $76 will go to her college income, or after taxes are withheld. Dora’s tax rate plan and $152.95 will go to taxes. If she takes the is 17.5%. If her pay before taxes and deductions is college savings plan deduction after taxes, only $950, will she save more money if the deductions are $62.70 will go to her college plan and $166.25 will go taken before or after taxes are withheld? Explain. to taxes.     SOLUTION:   Find (f + g)(x), (f – g)(x), (x),and Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan   (x) for each f(x) and g(x). Indicate any deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the restrictions in domain or range. college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes. 8.      ANSWER:   SOLUTION:   Either way, she will have $228.95 taken from her paycheck. If she takes the college savings plan deduction before taxes, $76 will go to her college plan and $152.95 will go to taxes. If she takes the college savings plan deduction after taxes, only   $62.70 will go to her college plan and $166.25 will go to taxes.   Find (f + g)(x), (f – g)(x), (x),and     (x) for each f(x) and g(x). Indicate any restrictions in domain or range.     8.    SOLUTION:     ANSWER:           9.    SOLUTION:     ANSWER:         9.      SOLUTION:         ANSWER:           10.      SOLUTION:     ANSWER:       10.          SOLUTION:   ANSWER:       11.        SOLUTION:     ANSWER:       11.      SOLUTION:   ANSWER:     12.  SOLUTION:     ANSWER:     12.  SOLUTION:   ANSWER:     13.    SOLUTION:   ANSWER:       13.      ANSWER:   SOLUTION:       14.    SOLUTION:     ANSWER:     14.    SOLUTION:     ANSWER:     15.    SOLUTION:     ANSWER:     15.      SOLUTION:   ANSWER:     16. POPULATION In a particular county, the population of the two largest cities can be modeled by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined   after any number of years?  ANSWER:   b. What is the difference in the populations of the two cities?    SOLUTION:   a. The population of the cities after x years is the sum of the individual populations.     16. POPULATION In a particular county, the b. The difference in the populations of the cities is population of the two largest cities can be modeled given by: by f(x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands.   a. What is the population of the two cities combined ANSWER:   after any number of years?  a. (f + g)(x) = 375x + 10   b. What is the difference in the populations of the b. (f - g)(x) = 25x + 40 two cities?      SOLUTION:   For each pair of functions, find and , if a. The population of the cities after x years is the they exist. State the domain and range for each sum of the individual populations. composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}     b. The difference in the populations of the cities is SOLUTION:   given by: Therefore:   ANSWER:   a. (f + g)(x) = 375x + 10   b. (f - g)(x) = 25x + 40   For each pair of functions, find and , if they exist. State the domain and range for each composed function.   17. f = {(–8, –4), (0, 4), (2, 6), (–6, –2)} g = {(4, –4), (–2, –1), (–4, 0), (6, –5)}   SOLUTION:   Therefore:   ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     18. f = {(–7, 0), (4, 5), (8, 12), (–3, 6)} g = {(6, 8), (–12, –5), (0, 5), (5, 1)}   SOLUTION:     ANSWER:     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.     ANSWER:   ANSWER:   6-1 Operations on Functions is undefined;  is undefined.     19. f = {(5, 13), (–4, –2), (–8, –11), (3, 1)} 20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–8, 2), (–4, 1), (3, –3), (5, 7)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   20. f = {(–4, –14), (0, –6), (–6, –18), (2, –2)} g = {(–6, 1), (–18, 13), (–14, 9), (–2, –3)}   SOLUTION:   The range of is not a subset of the domain of .   So,  is undefined. ANSWER:   is undefined;   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined; eSolutionsManual-PoweredbyCognero Page10   21. f = {(–15, –5), (–4, 12), (1, 7), (3, 9)} g = {(3, –9), (7, 2), (8, –6), (12, 0)}     ANSWER:   SOLUTION:   is undefined; . The range of is not a subset of the domain of   . 22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} So,  is undefined. g = {(5, –4), (4, –3), (–1, 2), (2, 3)}   SOLUTION:   The range of is not a subset of the domain of   . ANSWER:   So,  is undefined. is undefined; .     ANSWER:   22. f = {(–1, 11), (2, –2), (5, –7), (4, –4)} ; is undefined. g = {(5, –4), (4, –3), (–1, 2), (2, 3)}     23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} SOLUTION:   g = {(4, –3), (3, –7), (9, 8), (–4, –4)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   ; is undefined. The range of is not a subset of the domain of   . 23. f = {(7, –3), (–10, –3), (–7, -8), (–3, 6)} So,  is undefined. g = {(4, –3), (3, –7), (9, 8), (–4, –4)}     ANSWER:   SOLUTION:   ; is undefined.   24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} g = {(1, –4), (2, –3), (3, –2), (4, –1)}   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of The range of is not a subset of the domain of . . So,  is undefined. So,  is undefined.     ANSWER:   ANSWER:   ; is undefined. is undefined;  is undefined.     24. f = {(1, –1), (2, –2), (3, –3), (4, –4)} 25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(1, –4), (2, –3), (3, –2), (4, –1)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}     SOLUTION:   SOLUTION:   The range of is not a subset of the domain of . So,  is undefined. The range of is not a subset of the domain of . So,  is undefined.   ANSWER:   is undefined;  is undefined.   25. f = {(–4, –1), (–2, 6), (–1, 10), (4, 11)} g = {(–1, 5), (3, –4), (6, 4), (10, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     26. f = {(12, –3), (9, –2), (8, –1), (6, 3)} g = {(–1, 5), (–2, 6), (–3, –1), (–4, 8)}   SOLUTION:     ANSWER:     Find and , if they exist. State the domain and range for each composed function.   27.    SOLUTION:     ANSWER:     Find and , if they exist. State   the domain and range for each composed , D = {all real numbers}, R = {all function. even numbers}   , D = {all real numbers}, R = {all odd numbers} 27.  ANSWER:     SOLUTION:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}   28.      , D = {all real numbers}, R = {all SOLUTION:   even numbers} , D = {all real numbers}, R = {all odd numbers} ANSWER:   , D = {all real numbers}, R = {all even numbers} , D = {all real numbers}, R = {all odd numbers}     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all 28.  real numbers}     SOLUTION:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   29.      , D = {all real numbers}, R = {all real numbers} SOLUTION:   , D = {all real numbers}, R = {all real numbers}   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {all 29.  real numbers} , D = {all real numbers}, R = {all   real numbers}   SOLUTION:   ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}   30.      , D = {all real numbers}, R = {all real numbers} SOLUTION:   , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     , D = {all real numbers}, R = {y | y 30.  ≥ –14} , D = {all real numbers}, R = {y | y   ≥ –10} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {y | y   ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}   31.    , D = {all real numbers}, R = {y | y   ≥ –14} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ –10}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –14} , D = {all real numbers}, R = {y | y ≥ –10}     , D = {all real numbers}, R = {y | y 31.  ≥ –11} , D = {all real numbers}, R = {y | y   ≥ –17} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {y | y   ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}   32.    , D = {all real numbers}, R = {y | y   ≥ –11} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ –17}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ –11} , D = {all real numbers}, R = {y | y ≥ –17}     , D = {all real numbers}, R = {y | y 32.  ≥ 0.875} , D = {all real numbers}, R = {y | y   ≥ 6.5} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {y | y   ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}   33.    , D = {all real numbers}, R = {y | y   ≥ 0.875} SOLUTION:   , D = {all real numbers}, R = {y | y ≥ 6.5}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 0.875} , D = {all real numbers}, R = {y | y ≥ 6.5}     , D = {all real numbers}, R = {all 33.  real numbers} , D = {all real numbers}, R = {all   real numbers} SOLUTION:     ANSWER:   , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all   real numbers}   34.      , D = {all real numbers}, R = {all SOLUTION:   real numbers} , D = {all real numbers}, R = {all real numbers}   ANSWER:     , D = {all real numbers}, R = {all real numbers} , D = {all real numbers}, R = {all real numbers}     34.  , D = {all real numbers}, R = {y | y ≥ 1}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y ≥ 1}   , D = {all real numbers}, R = {y | y ≥ 0}   35.      , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}   ANSWER:     , D = {all real numbers}, R = {y | y ≥ 1} , D = {all real numbers}, R = {y | y ≥ 0}     35.  , D = {all real numbers}, R = {y | y ≥ 0}   , D = {all real numbers}, R = {y | y SOLUTION:   ≥ 0}   ANSWER:   , D = {all real numbers}, R = {y | y   ≥ 0} , D = {all real numbers}, R = {y | y ≥ 0}   36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x   coffee mugs is given by r(x) = 6.5x. Suppose the , D = {all real numbers}, R = {y | y function for the cost of manufacturing x coffee mugs ≥ 0} is c(x) = 0.75x + 1850. , D = {all real numbers}, R = {y | y   ≥ 0} a. Write the profit function.     b. Find the profit on 500, 1000, and 5000 coffee ANSWER:   mugs.   SOLUTION:   , D = {all real numbers}, R = {y | y a. The profit function P(x) is given by ≥ 0}  where r(x) is the revenue , D = {all real numbers}, R = {y | y function and c(x) is the cost function. ≥ 0} So:   36. FINANCE A ceramics store manufactures and sells   coffee mugs. The revenue r(x) from the sale of x b. coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs is c(x) = 0.75x + 1850.   a. Write the profit function.   b. Find the profit on 500, 1000, and 5000 coffee mugs.   SOLUTION:   ANSWER:   a. The profit function P(x) is given by a. P(x) = 5.75x – 1850  where r(x) is the revenue   function and c(x) is the cost function. b. P(500) = $1025; P(1000) = $3900; P(5000) = So: $26,900   37. CCSS SENSE-MAKING Ms. Smith wants to buy   an HDTV, which is on sale for 35% off the original b. price of $2299. The sales tax is 6.25%.   a. Write two functions representing the price after the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the price of the HDTV, or ? Explain your reasoning.   c. How much will Ms. Smith pay for the HDTV? ANSWER:     a. P(x) = 5.75x – 1850   SOLUTION:   b. P(500) = $1025; P(1000) = $3900; P(5000) = a. Let p(x) be the price after the discount where x is $26,900 the original price.   Discount = 35%(x) = 0.35x Therefore: 37. CCSS SENSE-MAKING Ms. Smith wants to buy an HDTV, which is on sale for 35% off the original price of $2299. The sales tax is 6.25%. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x a. Write two functions representing the price after Therefore: the discount p(x) and the price after sales tax t(x).   b. Which composition of functions represents the   price of the HDTV, or ? Explain b. Since = , either function your reasoning. represents the price.     c. How much will Ms. Smith pay for the HDTV? c.    SOLUTION:   a. Let p(x) be the price after the discount where x is the original price. Discount = 35%(x) = 0.35x Therefore: Substitute x = 2299. Let t(x) be the price after the sales tax.   Sales tax = 6.25%(x) = 0.0625x Therefore, Mr. Smith will pay $1587.75 for HDTV. Therefore:   ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x     b. Since = , either function b. Since = , either function represents the price.   represents the price. c.    c. $1587.75   2 Perform each operation if f(x) = x + x – 12 and g(x) = x – 3. State the domain of the resulting function. Substitute x = 2299.   38.      Therefore, Mr. Smith will pay $1587.75 for HDTV. SOLUTION:     ANSWER:   a. p(x) = 0.65x; t(x) = 1.0625x   D = {all real numbers} b. Since = , either function   represents the price.   ANSWER:   c. $1587.75 (f – g)(x) = x2 – 9; D = {all real numbers}     2 Perform each operation if f(x) = x + x – 12 and 39.  g(x) = x – 3. State the domain of the resulting function.     SOLUTION:   38.    SOLUTION:   D = {all real numbers}   ANSWER:   D = {all real numbers} ; D = {all real   numbers} ANSWER:     2 (f – g)(x) = x – 9; D = {all real numbers}   40.  39.      SOLUTION:   SOLUTION:   D = {all real numbers}   ANSWER:   ; D = {all real   numbers}   ANSWER:   40.      SOLUTION:   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   41.    SOLUTION:     Substitute x = –2. ANSWER:     ANSWER:     25   2 If f(x) = 5x, g(x) = –2x + 1, and h(x) = x + 6x + 8, find each value.   42.  41.      SOLUTION:   SOLUTION:   Substitute x = 3. Substitute x = –2.     ANSWER:   ANSWER:   –69 25     43.  42.      SOLUTION:   SOLUTION:     Substitute x = –5. Substitute x = 3.     ANSWER:   –69     ANSWER:   43.  483     SOLUTION:   44.    SOLUTION:     Substitute x = –5.     Substitute x = 2.     ANSWER:   483   44.      ANSWER:   SOLUTION:   –1   45.    SOLUTION:     Substitute x = 2.     Substitute x = –3.     ANSWER:     –1   ANSWER:   –5 45.      46.  SOLUTION:     SOLUTION:     Substitute x = –3.     Substitute x = 9.     ANSWER:   –5     ANSWER:   46.  2303     SOLUTION:   47.    SOLUTION:     Substitute x = 9.     ANSWER:   –30a + 5     48.  ANSWER:     2303 SOLUTION:     47.    SOLUTION:     ANSWER:     5a2 + 70a + 240   ANSWER:   –30a + 5 49.      48.  SOLUTION:     SOLUTION:     ANSWER:     2 –10a + 10a + 1 ANSWER:     2 5a + 70a + 240   50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   49.  a. TABULAR Make a table showing values for f   (x), g(x), ( f + g) (x), and ( f –g)(x).   SOLUTION:   b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the   graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   ANSWER:   2 SOLUTION:   –10a + 10a + 1   a. 50. MULTIPLE REPRESENTATIONS Let f(x) = x2 and g(x) = x.   a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f –g)(x).     b. GRAPHICAL Graph f(x), g(x), and ( f + g)(x) b. on the same coordinate grid.   c. GRAPHICAL Graph f(x), g(x), and ( f – g)(x) on the same coordinate grid.   d. VERBAL Describe the relationship among the graphs of f(x), g(x), ( f + g)(x), and ( f – g)(x).   SOLUTION:   a.   c.   b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) c. (x).   ANSWER:   a. b.   d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   ANSWER:   a. c. b. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f(x) and (f + g)(x) and between f(x) and (f – g) (x).   51. EMPLOYMENT The number of women and men c. age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands. women: y =1086.4x + 56,610 men: y = 999.2x +66,450   a. Write a function that models the total number of men and women employed in the United States during this time.   b. If f is the function for the number of men, and g is d. Sample answer: For each value of x, the vertical the function for the number of women, what does ( f distance between the graph of g(x) and the x-axis is – g)(x) represent? the same as the vertical distance between the graphs   of f(x) and (f + g)(x) and between f(x) and (f – g) (x). SOLUTION:     a. Add the functions. The total number of men and women employed is 51. EMPLOYMENT The number of women and men given by age 16 and over employed each year in the United States can be modeled by the following equations, where x is the number of years since 1994 and y is the number of people in thousands.   women: y =1086.4x + 56,610 b. The function represents the difference men: y = 999.2x +66,450 in the number of men and women employed in the   U.S. a. Write a function that models the total number of   men and women employed in the United States during this time. ANSWER:     a. y = 2085.6x +123,060 b. If f is the function for the number of men, and g is   the function for the number of women, what does ( f b. The function represents the difference in the – g)(x) represent? number of men and women employed in the U.S.     SOLUTION:   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x a. Add the functions. + 1, find each value. The total number of men and women employed is   given by 52.      SOLUTION:   b. The function represents the difference in the number of men and women employed in the   U.S. Substitute x = 3.     ANSWER:   a. y = 2085.6x +123,060   b. The function represents the difference in the number of men and women employed in the U.S. ANSWER:     –180   2 If f(x) = x + 2, g(x) = –4x + 3, and h(x) = x – 2x + 1, find each value. 53.      52.    SOLUTION:   SOLUTION:     Substitute x = 3.     Substitute x = 1.   ANSWER:     –180   ANSWER:   0   53.    54.  SOLUTION:     SOLUTION:     Substitute x = 1.       ANSWER:   ANSWER:   0     54.  55.      SOLUTION:   SOLUTION:       ANSWER:   ANSWER:   1   56.      55.  SOLUTION:     SOLUTION:     ANSWER:     –33   ANSWER:   1 57.      56.  SOLUTION:     SOLUTION:     ANSWER:   256     ANSWER:   58. MULTIPLE REPRESENTATIONS You will –33 explore ,  , , and   2  if f(x) = x + 1 and g(x) = x – 3. 57.      a. Tabular Make a table showing values for SOLUTION:     ,  , , and .    b. Graphical Use a graphing calculator to graph  ,  , , and   ANSWER:   on the same coordinate plane. 256     c. Verbal Explain the relationship between 58. MULTIPLE REPRESENTATIONS You will  and  . explore ,  , , and   2 d. Graphical Use a graphing calculator to graph [f  if f(x) = x + 1 and g(x) = x – 3. ◦ g](x), and [g ◦ f](x) on the same coordinate   plane. a. Tabular Make a table showing values for     ,  , , and e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x). .      SOLUTION:   b. Graphical Use a graphing calculator to a. graph  ,  , , and on the same coordinate plane.   c. Verbal Explain the relationship between  and  .   d. Graphical Use a graphing calculator to graph [f ◦ g](x), and [g ◦ f](x) on the same coordinate plane.   e.  Verbal Explain the relationship between [f ◦ g] (x), and [g ◦ f](x).   SOLUTION:   a.       b.         b.

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is not a subset of the domain of . So, is undefined. Therefore: ANSWER: is undefined;. Find and. , if they exist. State. eSolutions Manual - Powered by .. your reasoning. c. How much will Ms. Smith pay for the HDTV? eSolutions Manual - Powered by Cognero. Page 17. 6-1 Operations on Functions
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