College Algebra - Department of Mathematics
[Pages:506]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
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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
.
As indicated on that website you may go to
to order a low-cost, royalty free printed version of the book from . Neither author receives royalties from , 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 published by Kendall Hunt Publishing Co. 4050 Westmark Drive, Dubuque, IA 52002. Neither Joel 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 Lakeland 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 Anthan 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 Community College for their strong support and deep commitment to the project. From Lorain County Community College, we wish to thank: Irina Lomonosov for class testing the book and generating accompanying PowerPoint slides; Jorge Gerszonowicz, 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
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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 promotion 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.
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