College Algebra, 2013a

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570 Systems of Equations and Matrices Our next step is to eliminate the x’s from E2 andE3. From a matrix standpoint, this means we need 0’s below the leading 1 in R1. This guarantees the leading 1 in R2 will be to the right of the leading 1 in R1 in accordance with the second requirement of Definition 8.4. ⎡ ⎣ 1 2 −1 4 3 −1 1 8 2 3 −4 10 ⎤ ⎦ Replace R2 with−3R1+R2 −−−−−−−−−−−−−−−−−→ Replace R3 with−2R1+R3 ⎡ ⎣ 1 2 −1 4 0 −7 4 −4 0 −1 −2 2 Now we repeat the above process for the variable y which means we need to get the leading entry in R2 tobe1. ⎡ ⎣ 1 2 −1 4 0 −7 4 −4 0 −1 −2 2 ⎤ ⎦ Replace R2 with− 1 7 R2 −−−−−−−−−−−−−−→ ⎡ ⎣ ⎤ 1 2 −1 4 0 1 − 4 4 ⎦ 7 7 0 −1 −2 2 To guarantee the leading 1 in R3 is to the right of the leading 1 in R2, we get a 0 in the second column of R3. ⎡ ⎤ ⎡ ⎤ 1 2 −1 4 1 2 −1 4 ⎣ 0 1 − 4 4 Replace R3 withR2+R3 ⎦ ⎢ 7 7 −−−−−−−−−−−−−−−−→ ⎣ 0 1 − 4 4 ⎥ 7 7 ⎦ 0 −1 −2 2 0 0 − 18 18 7 7 Finally, we get the leading entry in R3 tobe1. ⎡ ⎤ 1 2 −1 4 ⎢ ⎣ 0 1 − 4 4 ⎥ 7 7 ⎦ Replace R3 with− 7 18 −−−−−−−−−−−−−−−→ R3 0 0 − 18 18 7 7 Decoding from the matrix gives a system in triangular form ⎡ ⎤ ⎧ 1 2 −1 4 ⎨ ⎣ 0 1 − 4 4 ⎦ Decode from the matrix 7 7 −−−−−−−−−−−−−−→ ⎩ 0 0 1 −1 ⎡ ⎣ ⎤ ⎦ ⎤ 1 2 −1 4 0 1 − 4 4 ⎦ 7 7 0 0 1 −1 x +2y − z = 4 y − 4 7 z = 4 7 z = −1 We get z = −1, y = 4 7 z + 4 7 = 4 7 (−1) + 4 7 = 0 and x = −2y + z +4=−2(0) + (−1) + 4 = 3 for a final answer of (3, 0, −1). We leave it to the reader to check. As part of Gaussian Elimination, we used row operations to obtain 0’s beneath each leading 1 to put the matrix into row echelon form. If we also require that 0’s are the only numbers above a leading 1, we have what is known as the reduced row echelon form of the matrix. Definition 8.5. A matrix is said to be in reduced row echelon form provided both of the following conditions hold: 1. The matrix is in row echelon form. 2. The leading 1s are the only nonzero entry in their respective columns.
8.2 Systems of Linear Equations: Augmented Matrices 571 Of what significance is the reduced row echelon form of a matrix? To illustrate, let’s take the row echelon form from Example 8.2.1 and perform the necessary steps to put into reduced row echelon form. We start by using the leading 1 in R3 tozerooutthenumbersintherowsaboveit. ⎡ ⎣ ⎤ 1 2 −1 4 0 1 − 4 4 ⎦ 7 7 0 0 1 −1 Replace R1 withR3+R1 −−−−−−−−−−−−−−−−−→ Replace R2 with 4 7 R3+R2 ⎡ ⎣ 1 2 0 3 0 1 0 0 0 0 1 −1 Finally, we take care of the 2 in R1 above the leading 1 in R2. ⎡ 1 2 0 ⎤ 3 ⎡ 1 0 0 3 ⎣ 0 1 0 0 ⎦ Replace R1 with−2R2+R1 −−−−−−−−−−−−−−−−−→ ⎣ 0 1 0 0 0 0 1 −1 0 0 1 −1 To our surprise and delight, when we decode this matrix, we obtain the solution instantly without having to deal with any back-substitution at all. ⎡ ⎤ ⎧ 1 0 0 3 ⎨ x = 3 ⎣ 0 1 0 0 ⎦ Decode from the matrix −−−−−−−−−−−−−−→ y = 0 ⎩ 0 0 1 −1 z = −1 Note that in the previous discussion, we could have started with R2 and used it to get a zero above its leading 1 and then done the same for the leading 1 in R3. By starting with R3, however, we get more zeros first, and the more zeros there are, the faster the remaining calculations will be. 2 It is also worth noting that while a matrix has several 3 row echelon forms, it has only one reduced row echelon form. The process by which we have put a matrix into reduced row echelon form is called Gauss-Jordan Elimination. Example 8.2.2. Solve the following system using an augmented matrix. Use Gauss-Jordan Elimination to put the augmented matrix into reduced row echelon form. ⎧ ⎨ ⎩ x 2 − 3x 1 + x 4 = 2 2x 1 +4x 3 = 5 4x 2 − x 4 = 3 Solution. We first encode the system into a matrix. (Pay attention to the subscripts!) ⎧ ⎡ ⎤ ⎨ ⎩ x 2 − 3x 1 + x 4 = 2 2x 1 +4x 3 = 5 4x 2 − x 4 = 3 Encode into the matrix −−−−−−−−−−−−−−→ Next, we get a leading 1 in the first column of R1. ⎡ ⎤ −3 1 0 1 2 ⎣ 2 0 4 0 5 ⎦ Replace R1 with− 1 3 −−−−−−−−−−−−−−→ R1 0 4 0 −1 3 2 Carl also finds starting with R3 to be more symmetric, in a purely poetic way. 3 infinite, in fact ⎣ ⎡ ⎣ ⎤ ⎦ ⎤ ⎦ −3 1 0 1 2 2 0 4 0 5 0 4 0 −1 3 1 − 1 3 0 − 1 3 − 2 3 2 0 4 0 5 0 4 0 −1 3 ⎦ ⎤ ⎦
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College Algebra Version ⌊π⌋ Co
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Table of Contents Preface ix 1 Rela
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Table of Contents v 4.3.3 Answers .
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Preface Thank you for your interest
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xi text and we have gone to great l
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Chapter 1 Relations and Functions 1
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1.1 Sets of Real Numbers and the Ca
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1.2 Relations 21 Solution. 1. To gr
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1.2 Relations 23 lines y = 1 and y
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1.2 Relations 25 ‘connecting the
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1.2 Relations 27 (x − 2) 2 + y 2
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1.2 Relations 29 1.2.2 Exercises In
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1.2 Relations 31 29. 5 4 3 y 30. 2
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1.2 Relations 33 1.2.3 Answers 1. y
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1.2 Relations 35 13. y 14. y 3 2
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1.2 Relations 37 41. y = x 2 +1 The
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1.2 Relations 39 45. y = √ x −
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1.2 Relations 41 49. (x +2) 2 + y 2
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1.3 Introduction to Functions 43 1.
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1.3 Introduction to Functions 45 So
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1.3 Introduction to Functions 47 Al
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1.3 Introduction to Functions 51 23
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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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Page 539 and 540: 7.4 Ellipses 529 (x − 4) 2 (y −
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Page 545 and 546: 7.5 Hyperbolas 535 from the center
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Page 553 and 554: 7.5 Hyperbolas 543 is a hyperbola.
Page 555 and 556: 7.5 Hyperbolas 545 (y − 3) 2 (x
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Page 591 and 592: 8.3 Matrix Arithmetic 581 As did 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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