College Algebra - Welcome | Department of Mathematics

506 Pages·2010·3.83 MB·English

College Algebra
√
Version 3 = 1.7320508075688772...
by
Carl Stitz, Ph.D. Jeff Zeager, Ph.D.
Lakeland Community College Lorain County Community College
Modified by
Joel Robbin and Mike Schroeder
University of Wisconsin, Madison
June 29, 2010

Table of Contents
Preface v
0 Basic Algebra 1
0.1 The Laws of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Kinds of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.3 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
0.4 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
0.5 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
0.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.7 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1 Coordinates 15
1.1 The Cartesian Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.1 Distance in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.1.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3 Graphs of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.4 Three Interesting Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4.1 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4.2 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.4.3 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1.4.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2 Functions 77
2.1 Introduction to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
iv Table of Contents
2.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.2 Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.3 Function Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.4 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.4.1 General Function Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.4.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
2.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3 Linear and Quadratic Functions 157
3.1 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.2 Defining Functions (Word Problems) . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.3 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
3.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
3.4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
3.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
3.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4 Polynomial Functions 215
4.1 Graphs of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4.2 The Factor Theorem and The Remainder Theorem . . . . . . . . . . . . . . . . . . . 235
4.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
4.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
5 Rational Functions 247
5.1 Introduction to Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
5.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
5.2 Graphs of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
5.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Table of Contents v
5.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.3 Rational Inequalities and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 284
5.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
5.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
6 Further Topics in Functions 295
6.1 Function Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
6.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
6.2 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
6.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
6.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7 Exponential and Logarithmic Functions 331
7.1 Introduction to Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 331
7.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
7.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
7.2 Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
7.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
7.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
7.3 Exponential Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 362
7.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
7.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
7.4 Logarithmic Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 373
7.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
7.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
7.5 Applications of Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 384
7.5.1 Applications of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 384
7.5.2 Applications of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
7.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
7.5.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
8 Systems of Equations 399
8.1 Systems of Linear Equations: Gaussian Elimination . . . . . . . . . . . . . . . . . . 399
8.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
8.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
8.2 Systems of Linear Equations: Augmented Matrices* . . . . . . . . . . . . . . . . . . 419
8.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
8.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
8.3 Determinants and Cramer’s Rule* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
8.3.1 Definition and Properties of the Determinant . . . . . . . . . . . . . . . . . . 432
8.3.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
8.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
vi Table of Contents
8.3.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
8.4 Systems of Non-Linear Equations and Inequalities . . . . . . . . . . . . . . . . . . . 438
8.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
8.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
9 Sequences and Series 455
9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
9.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
9.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
9.2 Series and Summation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
9.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
9.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
9.3 IRAs and Mortgages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
9.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
9.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
9.4 Infinite sums and Repeating Decimals* . . . . . . . . . . . . . . . . . . . . . . . . . . 473
9.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
10 Complex Numbers and the Fundamental Theorem of Algebra 477
10.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
10.2 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 480
10.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
10.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
A The Laws of Algebra Proved 491
A.1 The Laws of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
A.2 The Analogy between Addition and Multiplication . . . . . . . . . . . . . . . . . . . 491
A.3 Consequences of the Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . . . 496
Preface
This book is a modified version of the Open Source Precalculus Project initiated by Carl Stitz and
Jeff Seager. The original version is available at
http://www.stitz-zeager.com/Free_College_Algebra_Book_Download.html.
As indicated on that website you may go to
http://www.lulu.com/product/paperback/college-algebra/11396948
to order a low-cost, royalty free printed version of the book from lulu.com. Neither author receives
royalties from lulu.com, and, in most cases, it is far cheaper to purchase the printed version from
lulu than to print out the entire book at home.
The version you are viewing was modified by Joel Robbin and Mike Schroeder for use in
Math 112 at the University of Wisconsin Madison. A companion workbook for the course is being
publishedbyKendallHuntPublishingCo. 4050WestmarkDrive,Dubuque,IA52002. NeitherJoel
Robbin nor Mike Schroeder nor anyone else at the University of Wisconsin receives any royalties
from sales of the workbook to UW students.
The original version of this book contains the following acknowledgements:
The authors are indebted to the many people who support this project. From Lake-
land Community College, we wish to thank the following people: Bill Previts, who
not only class tested the book but added an extraordinary amount of exercises to it;
Rich Basich and Ivana Gorgievska, who class tested and promoted the book; Don An-
than and Ken White, who designed the electric circuit applications used in the text;
Gwen Sevits, Assistant Bookstore Manager, for her patience and her efforts to get the
book to the students in an efficient and economical fashion; Jessica Novak, Marketing
and Communication Specialist, for her efforts to promote the book; Corrie Bergeron,
Instructional Designer, for his enthusiasm and support of the text and accompanying
YouTube videos; Dr. Fred Law, Provost, and the Board of Trustees of Lakeland Com-
munity College for their strong support and deep commitment to the project. From
LorainCountyCommunityCollege, wewishtothank: IrinaLomonosovforclasstesting
thebookandgeneratingaccompanyingPowerPointslides;JorgeGerszonowicz,Kathryn
Arocho, Heather Bubnick, and Florin Muscutariu for their unwavering support of the
project; Drs. Wendy Marley and Marcia Ballinger, Lorain CCC, for the Lorain CCC
vii
viii Table of Contents
enrollment data used in the text. We would also like to extend a special thanks to
Chancellor Eric Fingerhut and the Ohio Board of Regents for their support and promo-
tion of the project. Last, but certainly not least, we wish to thank Dimitri Moonen, our
dear friend from across the Atlantic, who took the time each week to e-mail us typos
and other corrections.
Chapter 0
Basic Algebra
0.1 The Laws of Algebra
Terminology and Notation. In this section we review the notations used in algebra. Some
are peculiar to this book. For example the notation A := B indicates that the equality holds
by definition of the notations involved. Two other notations which will become important when
we solve equations are =⇒ and ⇐⇒ . The notation P =⇒ Q means that P implies Q i.e.
“If P, then Q”. For example, x = 2 =⇒ x2 = 4. (Note however that the converse statement
x2 = 4 =⇒ x = 2 is not always true since it might be that x = −2.) The notation P ⇐⇒ Q
means P =⇒ Q and Q =⇒ P, i.e. “P if and only if Q”. For example 3x−6 = 0 ⇐⇒ x = 2.
The notations =⇒ and ⇐⇒ are explained more carefully in Section 0.5 below.
Implicit Multiplication. In mathematics the absence of an operation symbol usually indicates
multiplication: ab mean a×b. Sometimes a dot is used to indicate multiplication and in computer
languages an asterisk is often used.
ab := a·b := a∗b := a×b
Order of operations. Parentheses are used to indicate the order of doing the operations: in
evaluating an expression with parentheses the innermost matching pairs are evaluated first as in
((1+2)2+5)2 = (32+5)2 = (9+5)2 = 142 = 196.
There are conventions which allow us not to write the parentheses. For example, multiplication is
done before addition
ab+c means (ab)+c and not a(b+c),
and powers are done before multiplication:
ab2c means a(b2)c and not (ab)2c.
In the absence of other rules and parentheses, the left most operations are done first.
a−b−c means (a−b)−c and not a−(b−c).
2 Basic Algebra
The long fraction line indicates that the division is done last:
a+b
means (a+b)/c and not a+(b/c).
c
In writing fractions the length of the fraction line indicates which fraction is evaluated first:
a
means a/(b/c) and not (a/b)/c,
b
c
a
b means (a/b)/c and not a/(b/c).
c
The length of the horizontal line in the radical sign indicates the order of evaluation:
√ (cid:112) √
a+b means (a+b) and not ( a)+b.
√ √ (cid:112)
a+b means ( a)+b and not (a+b).
The Laws of Algebra. There are four fundamental operations which can be performed on
numbers.
1. Addition. The sum of a and b is denoted a+b.
2. Multiplication. The product of a and b is denoted ab.
3. Reversing the sign. The negative of a is denoted −a.
1
4. Inverting. The reciprocal of a (for a (cid:54)= 0) is denoted by a−1 or by .
a
These operations satisfy the following laws.
Associative a+(b+c) = (a+b)+c a(bc) = (ab)c
Commutative a+(b+c) = (a+b)+c a(bc) = (ab)c
Identity a+0 = 0+a = a a·1 = 1·a = a
Inverse a+(−a) = (−a)+a = 0 a·a−1 = a−1·a = 1
Distributive a(b+c) = ab+ac (a+b)c = ac+bc
The operations of subtraction and division are then defined by
a 1
a−b := a+(−b) a÷b := := a·b−1 = a· .
b b