College algebra in context

651 Pages·36.68 MB·English

by  Harshbarger, Ronald J

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College Algebra in Context
Ronald J. Harshbarger Lisa Yocco
ISBN 978-1-29204-023-3
Fourth Edition
9 781292 040233
Pearson New International Edition

College Algebra in Context
Ronald J. Harshbarger Lisa Yocco
Fourth Edition
International_PCL_TP.indd 1 7/29/13 11:23 AM
ISBN 10: 1-292-04023-8
ISBN 13: 978-1-292-04023-3
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ISBN 10: 1-292-04023-8
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
1. Functions, Graphs, and Models; Linear Functions
Ronald J. Harshbarger/Lisa Yocco 1
2. Linear Models, Equations, and Inequalities
Ronald J. Harshbarger/Lisa Yocco 91
3. Quadratic, Piecewise-Defined, and Power Functions
Ronald J. Harshbarger/Lisa Yocco 171
4. Additional Topics with Functions
Ronald J. Harshbarger/Lisa Yocco 259
5. Exponential and Logarithmic Functions
Ronald J. Harshbarger/Lisa Yocco 333
6. Higher-Degree Polynomial and Rational Functions
Ronald J. Harshbarger/Lisa Yocco 439
7. Systems of Equations and Matrices
Ronald J. Harshbarger/Lisa Yocco 531
Appendix: Basic Calculator Guide
Ronald J. Harshbarger/Lisa Yocco 603
Appendix: Basic Guide to Excel 2003, Excel 2007, and Excel 2010
Ronald J. Harshbarger/Lisa Yocco 623
Index 641
I
II
Functions, Graphs,
and Models;
Linear Functions

With digital TV becoming more affordable by the day, the demand for
high-definition home entertainment is growing rapidly, with 115 million
Americans having digital TV in 2010. The worldwide total was 517 million
in 2010, with 40% of TV households having digital TV. More than
401 million digital TV homes are expected to be added between the
end of 2010 and the end of 2015, and by 2015 more than 1 billion
sets should be in use worldwide. Cell phone use is also on the rise,
not just in the United States but throughout the world. By the middle of
Darren Brode/Shutterstock 2006, the number of subscribers to cell phone carriers had dramati-
cally increased, and the number of total users had reached 4.6 billion
by 2010. If the numbers continue to increase at a steady rate, the
number of subscribers is expected to reach into the tens of billions
over the next few years. These projections and others are made by
collecting real-world data and creating mathematical models. The goal
of this chapter is to use real data and mathematical models to make
predictions and solve meaningful problems.
section objectives a pplications
1 Functions and Models Determine graphs, tables, and equations that Body temperature, personal computers,
represent functions; find domains and ranges; stock market, men in the workforce, public
evaluate functions and mathematical models health expenditures
2 Graphs of Functions Graph and evaluate functions with technology; Personal savings, cost-benefit, voting, U.S.
graph mathematical models; align data; graph executions, high school enrollment
data points; scale data
3 Linear Functions Identify and graph linear functions; find and inter- Hispanics in the United States, loan balances,
pret intercepts and slopes; find constant rates revenue, cost, profit, marginal cost, marginal
of change; model revenue, cost, and profit; find revenue, marginal profit
marginal revenue, marginal cost, and marginal
profit; identify special linear functions
4 Equations of Lines Write equations of lines; identify parallel and Service call charges, blood alcohol percent,
perpendicular lines; find average rates of change; inmate population, hybrid vehicle sales, high
model approximately linear data school enrollment
From Chapter 1 of College Algebra with Applications for the Managerial, Life, & Social Sciences, Fourth Edition. Ronald J. Harshbarger,
Lisa S. Yocco. Copyright © 2013 by Pearson Education, Inc. All rights reserved.
1
Algebra TOOLBOX
KEY OBJECTIVES
The Algebra Toolbox is designed to review prerequisite skills needed for success in
■ Write sets of numbers using
this chapter. In this Toolbox, we discuss sets, the real numbers, the coordinate system,
description or elements
algebraic expressions, equations, inequalities, absolute values, and subscripts.
■ Identify sets of real numbers
as being integers, rational
numbers, and/or irrational Sets
numbers
In this chapter we will use sets to write domains and ranges of functions, we will find
■ Identify the coefficients of
solution sets to equations and inequalities. A set is a well-defined collection of objects,
terms and constants in
including but not limited to numbers. In this section, we will discuss sets of real num-
algebraic expressions
bers, including natural numbers, integers, and rational numbers, and later in the text we
■ Remove parentheses and will discuss the set of complex numbers. There are two ways to define a set. One way is
simplify polynomials
by listing the elements (or members) of the set (usually between braces). For example,
■ Express inequalities as we may say that a set A contains 2, 3, 5, and 7 by writing A = {2, 3, 5, 7}. To say that 5
intervals and graph is an element of the set A , we write 5(cid:2)A. To indicate that 6 is not an element of the set,
inequalities we write 6(cid:3)A. Domains of functions and solutions to equations are sometimes given
■ Plot points on a in sets with the elements listed.
coordinate system If all the elements of the set can be listed, the set is said to be a finite set. If all
■ Use subscripts to elements of a set cannot be listed, the set is called an infinite set. To indicate that
represent fixed points a set continues with the established pattern, we use three dots. For example, B =
{1, 2, 3, 4, 5, c, 100} describes the finite set of whole numbers from 1 through 100,
and N = {1, 2, 3, 4, 5,c} describes the infinite set of all whole numbers beginning
with 1. This set is called the natural numbers.
Another way to define a set is to give its description. For example, we may write
{x(cid:2)x is a math book} to define the set of math books. This is read as “the set of all x
such that x is a math book.” N = {x(cid:2)x is a natural number} defines the set of natural
numbers, which was also defined by N = {1, 2, 3, 4, 5, c} above.
The set that contains no elements is called the empty set and is denoted by (cid:2).
EXAMPLE 1 N Write the following sets in two ways.
a. The set A containing the natural numbers less than 7.
b. The set B of natural numbers that are at least 7.
SOLUTION
a. A = {1, 2, 3, 4, 5, 6}, A = {x(cid:2)x(cid:2)N, x 6 7}
b. B = {7, 8, 9, 10,c}, B = {x(cid:2)x(cid:2)N, x Ú 7}
The relations that can exist between two sets follow.
Relations Between Sets
1. Sets X and Y are equal if they contain exactly the same elements.
2. Set A is called a subset of set B if each element of A is an element of B . This is
denoted A(cid:4)B.
3. If sets C and D have no elements in common, they are called disjoint.
2
Functions, Graphs, and Models; Linear Functions
EXAMPLE 2 N For the sets A = {x(cid:2)x … 9, x is a natural number}, B = {2, 4, 6}, C = {3, 5, 8, 10}:
a. Which of the sets A , B , and C are subsets of A ?
b. Which pairs of sets are disjoint?
c. Are any of these three sets equal?
SOLUTION
a. Every element of B is contained in A . Thus, set B is a subset of A . Because every
element of A is contained in A , A is a subset of A .
b. Sets B and C have no elements in common, so they are disjoint.
c. None of these sets have exactly the same elements, so none are equal.
The Real Numbers
B ecause most of the mathematical applications you will encounter in an applied
nontechnical setting use real numbers, the emphasis in this text is the real number
system. Real numbers can be rational or irrational. Rational numbers include integers,
fractions containing only integers (with no 0 in a denominator), and decimals that either
terminate or repeat. Some examples of rational numbers are
1 4
-9, , 0, 12, - , 6.58, -7.3
2 7
Irrational numbers are real numbers that are not rational. Some examples of
irrational numbers are p (a number familiar to us from the study of circles), 12, 135,
and 13 -10.
The types of real numbers are described in Table 1 .
Table 1
Types of Real Numbers Descriptions
Natural numbers 1, 2, 3, 4, c
Integers Natural numbers, zero, and the negatives of the natural num-
bers: . . . , (cid:2)3, (cid:2)2, (cid:2)1, 0, 1, 2, 3, . . .
p
Rational numbers All numbers that can be written in the form , where p and q
q
are both integers with q (cid:2) 0. Rational numbers can be written
as terminating or repeating decimals.
Irrational numbers All real numbers that are not rational numbers. Irrational num-
bers cannot be written as terminating or repeating decimals.
W e can represent real numbers on a real number line. Exactly one real number is
associated with each point on the line, and we say there is a one-to-one correspondence
between the real numbers and the points on the line. That is, the real number line is a
graph of the real numbers (see Figure 1 ).
(cid:2)9 (cid:2)7.3 (cid:2)3(cid:2)(cid:3)5(cid:3)0 (cid:2) 13 0 (cid:2)(cid:3)3 (cid:3) 9 6.568 12
6 2
Figure 1
3
Functions, Graphs, and Models; Linear Functions
Notice the number p on the real number line in F igure 1 . This special number, which
can be approximated by 3.14, results when the circumference of (distance around) any
circle is divided by the diameter of the circle. Another special real number is e ; it is
denoted by
e (cid:5) 2.71828
We will discuss this number, which is important in financial and biological applications,
later in the text.
Inequalities and Intervals on the Number Line
In this chapter, we will sometimes use inequalities and interval notation to describe
domains and ranges of functions. An inequality is a statement that one quantity is greater
(or less) than another quantity. We say that a is less than b (written a 6 b ) if the point
representing a is to the left of the point representing b on the real number line. We may
indicate that the number a is greater than or equal to b by writing a Ú b. The subset
of real numbers x that lie between a and b (excluding a and b ) can be denoted by the
double inequality a 6 x 6 b or by the open interval ( a, b ). This is called an open inter-
val because neither of the endpoints is included in the interval. The closed interval [ a, b ]
represents the set of all real numbers satisfying a … x … b. Intervals containing one
endpoint, such as [ a, b ) or (a , b ], are called half-open intervals. We can represent the
inequality x Ú a by the interval [a, q), and we can represent the inequality x 6 a by
the interval (-q, a). Note that q and -q are not numbers, but q is used in [a, q)
to represent the fact that x increases without bound and -q is used in (-q, a) to
indicate that x decreases without bound. Table 2 shows the graphs of different types of
intervals.
Table 2
Interval Inequality Verbal
Notation Notation Description Number Line Graph
( a , q ) x 7 a x is greater than a x
a
[ a , q ) x Ú a x is greater than or equal to a x
a
( -q , b ) x 6 b x is less than b x
b
( -q , b ] x … b x is less than or equal to b x
b
( a, b ) a 6 x 6 b x is between a and b , not including x
either a or b a b
[ a, b ) a … x 6 b x is between a and b , including a x
but not including b a b
( a, b ] a 6 x … b x is between a and b , not including x
a but including b a b
[ a, b ] a … x … b x is between a and b , including x
both a and b a b
Note that open circles may be used instead of parentheses and solid circles may be used instead of brackets in the
number line graphs.
4
Functions, Graphs, and Models; Linear Functions
EXAMPLE 3 N Intervals
Write the interval corresponding to each of the inequalities in parts (a)–(e), and then
graph the inequality.
a. -1 … x … 2 b. 2 6 x 6 4 c. -2 6 x … 3 d. x Ú 3 e. x 6 5
SOLUTION
a. [-1, 2] x
(cid:3)1 2
b. (2, 4) x
2 4
c. (-2, 3] x
(cid:3)2 3
d. [3, q) x
3
e. (-q, 5) x
5
Algebraic Expressions
In algebra we deal with a combination of real numbers and letters. Generally, the letters
are symbols used to represent unknown quantities or fixed but unspecified constants.
Letters representing unknown quantities are usually called variables, and letters rep-
resenting fixed but unspecified numbers are called literal constants. An expression
created by performing additions, subtractions, or other arithmetic operations with one
or more real numbers and variables is called an algebraic expression. Unless otherwise
specified, the variables represent real numbers for which the algebraic expression is a
real number. Examples of algebraic expressions include
3x - 5
5x - 2y, , and 7z + 2
12 + 5y
A term of an algebraic expression is the product of one or more variables and a real
number; the real number is called a numerical coefficient or simply a coefficient. A
constant is also considered a term of an algebraic expression and is called a constant
term. For instance, the term 5y z is the product of the factors 5, y , and z ; this term has
coefficient 5.
Polynomials
An algebraic expression containing a finite number of additions, subtractions, and mul-
tiplications of constants and nonnegative integer powers of variables is called a poly-
nomial. When simplified, a polynomial cannot contain negative powers of variables,
fractional powers of variables, variables in a denominator, or variables inside a radical.
3x - 5
The expressions 5x - 2y and 7z3 + 2y are polynomials, but and 3x2 - 61x
12 + 5y
are not polynomials. If the only variable in the polynomial is x , then the polynomial is
called a polynomial in x . The general form of a polynomial in x is
an xn + an-1xn-1 + g+ a1x + a0
where a0 and each coefficient an, an-1,c are real numbers and each exponent
n, n - 1,c is a positive integer.
5