College Algebra, 2013a

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vi Table of Contents 7.5 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 7.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 7.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 8 Systems of Equations and Matrices 549 8.1 Systems of Linear Equations: Gaussian Elimination . . . . . . . . . . . . . . . . . . 549 8.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 8.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 8.2 Systems of Linear Equations: Augmented Matrices . . . . . . . . . . . . . . . . . . 567 8.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 8.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 8.3 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 8.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 8.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 8.4 Systems of Linear Equations: Matrix Inverses . . . . . . . . . . . . . . . . . . . . . 598 8.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 8.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 8.5 Determinants and Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 8.5.1 Definition and Properties of the Determinant . . . . . . . . . . . . . . . . . 614 8.5.2 Cramer’s Rule and Matrix Adjoints . . . . . . . . . . . . . . . . . . . . . . 618 8.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 8.5.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 8.6 Partial Fraction Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 8.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 8.6.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 8.7 Systems of Non-Linear Equations and Inequalities . . . . . . . . . . . . . . . . . . . 637 8.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 8.7.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 9 Sequences and the Binomial Theorem 651 9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 9.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 9.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 9.2 Summation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 9.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 9.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 9.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 9.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 9.3.2 Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 9.4 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 9.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 9.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
Preface Thank you for your interest in our book, but more importantly, thank you for taking the time to read the Preface. I always read the Prefaces of the textbooks which I use in my classes because I believe it is in the Preface where I begin to understand the authors - who they are, what their motivation for writing the book was, and what they hope the reader will get out of reading the text. Pedagogical issues such as content organization and how professors and students should best use a book can usually be gleaned out of its Table of Contents, but the reasons behind the choices authors make should be shared in the Preface. Also, I feel that the Preface of a textbook should demonstrate the authors’ love of their discipline and passion for teaching, so that I come away believing that they really want to help students and not just make money. Thus, I thank my fellow Preface-readers again for giving me the opportunity to share with you the need and vision which guided the creation of this book and passion which both Carl and I hold for Mathematics and the teaching of it. Carl and I are natives of Northeast Ohio. We met in graduate school at Kent State University in 1997. I finished my Ph.D in Pure Mathematics in August 1998 and started teaching at Lorain County Community College in Elyria, Ohio just two days after graduation. Carl earned his Ph.D in Pure Mathematics in August 2000 and started teaching at Lakeland Community College in Kirtland, Ohio that same month. Our schools are fairly similar in size and mission and each serves a similar population of students. The students range in age from about 16 (Ohio has a Post-Secondary Enrollment Option program which allows high school students to take college courses for free while still in high school.) to over 65. Many of the “non-traditional” students are returning to school in order to change careers. A majority of the students at both schools receive some sort of financial aid, be it scholarships from the schools’ foundations, state-funded grants or federal financial aid like student loans, and many of them have lives busied by family and job demands. Some will be taking their Associate degrees and entering (or re-entering) the workforce while others will be continuing on to a four-year college or university. Despite their many differences, our students share one common attribute: they do not want to spend $200 on a College Algebra book. The challenge of reducing the cost of textbooks is one that many states, including Ohio, are taking quite seriously. Indeed, state-level leaders have started to work with faculty from several of the colleges and universities in Ohio and with the major publishers as well. That process will take considerable time so Carl and I came up with a plan of our own. We decided that the best way to help our students right now was to write our own College Algebra book and give it away electronically for free. We were granted sabbaticals from our respective institutions for the Spring
Page 1 and 2: College Algebra Version ⌊π⌋ Co
Page 3 and 4: Table of Contents Preface ix 1 Rela
Page 5: Table of Contents v 4.3.3 Answers .
Page 9 and 10: xi text and we have gone to great l
Page 11 and 12: Chapter 1 Relations and Functions 1
Page 13 and 14: 1.1 Sets of Real Numbers and the Ca
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Page 35 and 36: 1.2 Relations 25 ‘connecting the
Page 37 and 38: 1.2 Relations 27 (x − 2) 2 + y 2
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Page 43 and 44: 1.2 Relations 33 1.2.3 Answers 1. y
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Page 47 and 48: 1.2 Relations 37 41. y = x 2 +1 The
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1.3 Introduction to Functions 47 Al
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1.3 Introduction to Functions 49 1.
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1.3 Introduction to Functions 51 23
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1.3 Introduction to Functions 53 1.
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1.4 Function Notation 55 1.4 Functi
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1.4 Function Notation 57 (c) To fin
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1.4 Function Notation 59 1 − 4x x
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1.4 Function Notation 61 these limi
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1.4 Function Notation 63 1.4.2 Exer
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1.4 Function Notation 65 ⎧ ⎪⎨
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1.4 Function Notation 67 (b) How mu
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1.4 Function Notation 69 1.4.3 Answ
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1.4 Function Notation 71 18. For f(
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1.4 Function Notation 73 26. For f(
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1.4 Function Notation 75 69. C(0) =
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1.5 Function Arithmetic 77 ( ) 4. F
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1.5 Function Arithmetic 79 Next, we
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1.5 Function Arithmetic 81 r(x + h)
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1.5 Function Arithmetic 83 Solution
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1.5 Function Arithmetic 85 27. f(x)
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1.5 Function Arithmetic 87 1.5.2 An
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1.5 Function Arithmetic 89 13. For
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1.5 Function Arithmetic 91 37. 39.
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1.6 Graphs of Functions 93 1.6 Grap
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1.6 Graphs of Functions 95 In the p
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1.6 Graphs of Functions 97 The calc
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1.6 Graphs of Functions 99 The calc
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1.6 Graphs of Functions 101 on [−
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1.6 Graphs of Functions 103 1. Find
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1.6 Graphs of Functions 105 Example
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1.6 Graphs of Functions 107 1.6.2 E
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1.6 Graphs of Functions 109 In Exer
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1.6 Graphs of Functions 111 For Exe
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1.6 Graphs of Functions 113 y y 4 5
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1.6 Graphs of Functions 115 6. f(x)
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1.6 Graphs of Functions 117 17. y 1
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1.6 Graphs of Functions 119 90. h i
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1.7 Transformations 121 7 y 7 y (5,
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1.7 Transformations 123 2 to all of
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1.7 Transformations 125 2 1 (0, 0)
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1.7 Transformations 127 Solution. 1
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1.7 Transformations 129 y y 3 2 1 (
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1.7 Transformations 131 A few remar
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1.7 Transformations 133 6 y 6 y (4,
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1.7 Transformations 135 Finally, to
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1.7 Transformations 137 (a, f(a)) a
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1.7 Transformations 139 shift right
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1.7 Transformations 141 Thecomplete
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1.7 Transformations 143 64. The gra
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1.7 Transformations 145 25. y =2−
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1.7 Transformations 147 38. g(x) =f
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1.7 Transformations 149 50. y = S 1
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Chapter 2 Linear and Quadratic Func
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2.1 Linear Functions 153 P 2 5. m =
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2.1 Linear Functions 155 Equation 2
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2.1 Linear Functions 157 y 4 3 2 y
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2.1 Linear Functions 159 4. We find
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2.1 Linear Functions 161 average sp
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2.1 Linear Functions 163 2.1.1 Exer
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2.1 Linear Functions 165 40. A rest
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2.1 Linear Functions 167 61. y = 2
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2.1 Linear Functions 169 2.1.2 Answ
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2.1 Linear Functions 171 35. (a) F
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2.2 Absolute Value Functions 173 2.
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2.2 Absolute Value Functions 177 By
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2.2 Absolute Value Functions 179 Ex
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2.2 Absolute Value Functions 181 wh
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2.2 Absolute Value Functions 185 24
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2.2 Absolute Value Functions 187 31
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2.3 Quadratic Functions 189 y y (
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2.3 Quadratic Functions 191 shows t
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2.3 Quadratic Functions 193 From g(
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2.3 Quadratic Functions 195 ( a x +
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2.3 Quadratic Functions 197 To find
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2.3 Quadratic Functions 199 Neverth
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2.3 Quadratic Functions 201 15. The
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2.3 Quadratic Functions 203 2.3.2 A
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2.3 Quadratic Functions 205 8. f(x)
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2.3 Quadratic Functions 207 25. (a)
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2.4 Inequalities with Absolute Valu
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2.5 Regression 225 2.5 Regression W
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2.5 Regression 227 2. Find the leas
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2.5 Regression 229 The coefficient
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2.5 Regression 231 4. The chart bel
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2.5 Regression 233 2.5.2 Answers 1.
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Chapter 3 Polynomial Functions 3.1
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3.1 Graphs of Polynomials 237 of th
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3.1 Graphs of Polynomials 239 In or
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3.1 Graphs of Polynomials 241 Despi
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3.1 Graphs of Polynomials 243 at th
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3.1 Graphs of Polynomials 245 Theor
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3.1 Graphs of Polynomials 247 In Ex
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3.1 Graphs of Polynomials 249 37. T
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3.1 Graphs of Polynomials 251 9. f(
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3.1 Graphs of Polynomials 253 21. g
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3.1 Graphs of Polynomials 255 33. (
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3.2 The Factor Theorem and the Rema
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3.3 Real Zeros of Polynomials 269 3
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3.3 Real Zeros of Polynomials 275 f
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3.3 Real Zeros of Polynomials 277 T
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3.3 Real Zeros of Polynomials 279 (
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3.3 Real Zeros of Polynomials 281 I
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3.4 Complex Zeros and the Fundament
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Chapter 4 Rational Functions 4.1 In
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4.1 Introduction to Rational Functi
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4.2 Graphs of Rational Functions 32
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4.3 Rational Inequalities and Appli
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Chapter 5 Further Topics in Functio
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5.1 Function Composition 361 2. As
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5.1 Function Composition 363 (+)
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5.1 Function Composition 365 9. The
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5.1 Function Composition 367 Theore
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5.1 Function Composition 369 5.1.1
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5.1 Function Composition 371 50. (g
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5.1 Function Composition 373 7. For
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5.1 Function Composition 375 20. Fo
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5.1 Function Composition 377 56. (g
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5.2 Inverse Functions 379 The main
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5.2 Inverse Functions 381 y = f −
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5.2 Inverse Functions 383 (b) We ca
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5.2 Inverse Functions 385 y = f(x)
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5.2 Inverse Functions 387 = = 2x 2x
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5.2 Inverse Functions 389 We get g
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5.2 Inverse Functions 391 Using wha
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5.2 Inverse Functions 393 1. Explai
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5.2 Inverse Functions 395 26. Show
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5.3 Other Algebraic Functions 397 5
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5.3 Other Algebraic Functions 399 I
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5.3 Other Algebraic Functions 401 b
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5.3 Other Algebraic Functions 405
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5.3 Other Algebraic Functions 415 (
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Chapter 6 Exponential and Logarithm
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6.1 Introduction to Exponential and
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6.2 Properties of Logarithms 437 6.
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6.2 Properties of Logarithms 439 lo
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6.2 Properties of Logarithms 441 ex
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6.2 Properties of Logarithms 443 as
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6.3 Exponential Equations and Inequ
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6.4 Logarithmic Equations and Inequ
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6.5 Applications of Exponential and
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Chapter 7 Hooked on Conics 7.1 Intr
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7.1 Introduction to Conics 497 If t
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7.2 Circles 499 4 y 3 2 1 −4 −3
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7.2 Circles 501 We close this secti
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7.2 Circles 503 7.2.2 Answers 1. (x
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7.3 Parabolas 505 7.3 Parabolas We
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7.3 Parabolas 507 The distance from
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7.3 Parabolas 509 Example 7.3.3. Gr
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7.3 Parabolas 511 Every cross secti
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7.3 Parabolas 513 7.3.2 Answers 1.
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7.3 Parabolas 515 8. ( y + 2) 3 2 (
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7.4 Ellipses 517 Minor Axis Major A
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7.4 Ellipses 519 This equation is f
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7.4 Ellipses 521 As with circles an
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7.4 Ellipses 523 From this sketch,
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7.4 Ellipses 525 7.4.1 Exercises In
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7.4 Ellipses 527 7.4.2 Answers x 2
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7.4 Ellipses 529 (x − 4) 2 (y −
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7.5 Hyperbolas 531 7.5 Hyperbolas I
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7.5 Hyperbolas 533 endpoints of the
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7.5 Hyperbolas 535 from the center
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7.5 Hyperbolas 537 As with the othe
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7.5 Hyperbolas 539 y 6 5 4 3 2 Jeff
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7.5 Hyperbolas 541 7.5.1 Exercises
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7.5 Hyperbolas 543 is a hyperbola.
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7.5 Hyperbolas 545 (y − 3) 2 (x
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7.5 Hyperbolas 547 19. (x − 1) 2
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Chapter 8 Systems of Equations and
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8.1 Systems of Linear Equations: Ga
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8.2 Systems of Linear Equations: Au
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8.3 Matrix Arithmetic 579 ⎡ ⎤
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8.3 Matrix Arithmetic 581 As did ma
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8.3 Matrix Arithmetic 583 While the
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8.3 Matrix Arithmetic 585 Using Ri
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8.3 Matrix Arithmetic 587 C 2 − 5
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8.3 Matrix Arithmetic 589 ( √2 2
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8.3 Matrix Arithmetic 591 8.3.1 Exe
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8.3 Matrix Arithmetic 593 28. Now l
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8.3 Matrix Arithmetic 595 8.3.2 Ans
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8.3 Matrix Arithmetic 597 [ ] −9
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8.4 Systems of Linear Equations: Ma
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8.5 Determinants and Cramer’s Rul
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8.6 Partial Fraction Decomposition
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8.7 Systems of Non-Linear Equations
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20. Solve the following system ⎧
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Chapter 9 Sequences and the Binomia
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9.1 Sequences 653 3. From {2n − 1
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9.1 Sequences 655 Solution. A good
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9.1 Sequences 657 Looking at the de
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9.1 Sequences 659 28. 0.9, 0.99, 0.
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9.2 Summation Notation 661 9.2 Summ
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9.2 Summation Notation 663 5∑ (
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9.2 Summation Notation 665 S = n (2
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9.2 Summation Notation 667 payment
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9.2 Summation Notation 669 n∑ k=1
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9.2 Summation Notation 671 In Exerc
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9.3 Mathematical Induction 673 9.3
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9.3 Mathematical Induction 675 This
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9.3 Mathematical Induction 677 it i
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9.3 Mathematical Induction 679 9.3.
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9.4 The Binomial Theorem 681 9.4 Th
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9.4 The Binomial Theorem 683 songs
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9.4 The Binomial Theorem 685 (a + b
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9.4 The Binomial Theorem 687 ∑k
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9.4 The Binomial Theorem 689 ( 0 0)
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9.4 The Binomial Theorem 691 9.4.1
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Index n th root of a complex number
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Index 1071 central angle, 701 chang
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Index 1073 reflective property, 523
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Index 1075 matrix, multiplicative,
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Index 1077 midpoint definition of,
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Index 1079 instantaneous, 161, 472
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Index 1081 inconsistent, 553 indepe
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College Algebra, 2013a

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