College Algebra, 2013a

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136 Relations and Functions 1 is a positive number, this induces only a horizontal scaling by a factor of B . If B < 0, then we have a factor of −1 in play, and dividing by it induces a reflection about the y-axis. So we have x = a−H B as the input to g which corresponds to the input x = a to f. We now evaluate g ( ) ( a−H B = Af B · a−H B + H) + K = Af(a)+K = Ab + K. We notice that the output from f is first multiplied by A. As with the constant B, ifA>0, this induces only a vertical scaling. If A
1.7 Transformations 137 (a, f(a)) a −2x +1=a x (−4, −3) −4 −2x +1=−4 x = 5 2 (−2, 0) −2 −2x +1=−2 x = 3 2 (0, 3) 0 −2x +1=0 x = 1 2 (2, 0) 2 −2x +1=2 x = − 1 2 (4, −3) 4 −2x +1=4 x = − 3 2 Next, we take each of the x values and substitute them into g(x) =− 3 2f(−2x +1)+2togetthe corresponding y-values. Substituting x = 5 2 , and using the fact that f(−4) = −3, we get ( 5 g = − 2) 3 ( ( ) 5 2 f −2 +1 +2=− 2) 3 2 f(−4) + 2 = −3 2 (−3) + 2 = 9 13 +2= 2 2 We see that the output from f is first multiplied by − 3 2 . Thinking of this as a two step process, multiplying by 3 2 then by −1, we have a vertical stretching by a factor of 3 2 followed by a reflection across the x-axis. Adding 2 results in a vertical shift up 2 units. Continuing in this manner, we get the table below. x g(x) (x, g(x)) ( 5 13 5 2 2 2 , 13 ) 2 ( 3 2 2 3 2 ( , 1 2 , − 5 2) 1 2 − 5 2 − 1 2 2 − 3 2 13 2 ( − 1 2 ( , 2) − 3 2 , 13 ) 2 To graph g, we plot each of the points in the table above and connect them in the same order and fashion as the points to which they correspond. Plotting f and g side-by-side gives (−4, −3) 6 5 4 3 2 (0, 3) 1 (−2, 0) (2, 0) −4 −3 −2 −1 1 2 3 4 −1 −2 −3 −4 y (4, −3) x ( − 3 2 , 13 ) 2 6 5 4 ( 5 2 , 13 ) 2 , 2) 3 ( − 1 2 ( 3 2 , 2) −4 −3 −2 −1 −1 1 2 3 4 −2 −3 ( 1 2 , − 5 ) 2 −4 y x
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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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Page 107 and 108: 1.6 Graphs of Functions 97 The calc
Page 109 and 110: 1.6 Graphs of Functions 99 The calc
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Page 119 and 120: 1.6 Graphs of Functions 109 In Exer
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Page 123 and 124: 1.6 Graphs of Functions 113 y y 4 5
Page 125 and 126: 1.6 Graphs of Functions 115 6. f(x)
Page 127 and 128: 1.6 Graphs of Functions 117 17. y 1
Page 129 and 130: 1.6 Graphs of Functions 119 90. h i
Page 131 and 132: 1.7 Transformations 121 7 y 7 y (5,
Page 133 and 134: 1.7 Transformations 123 2 to all of
Page 135 and 136: 1.7 Transformations 125 2 1 (0, 0)
Page 137 and 138: 1.7 Transformations 127 Solution. 1
Page 139 and 140: 1.7 Transformations 129 y y 3 2 1 (
Page 141 and 142: 1.7 Transformations 131 A few remar
Page 143 and 144: 1.7 Transformations 133 6 y 6 y (4,
Page 145: 1.7 Transformations 135 Finally, to
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Page 151 and 152: 1.7 Transformations 141 Thecomplete
Page 153 and 154: 1.7 Transformations 143 64. The gra
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Page 161 and 162: Chapter 2 Linear and Quadratic Func
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Page 171 and 172: 2.1 Linear Functions 161 average sp
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Page 175 and 176: 2.1 Linear Functions 165 40. A rest
Page 177 and 178: 2.1 Linear Functions 167 61. y = 2
Page 179 and 180: 2.1 Linear Functions 169 2.1.2 Answ
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Page 183 and 184: 2.2 Absolute Value Functions 173 2.
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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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