C H A P T E R 2 Polynomial and Rational Functions

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208 Chapter 2 Polynomial and Rational Functions 19. 23. 25. f x 1 3 x 3 1 3 0 x 3 x 3 3 f x Domain: all real numbers x except x 1 and x 3 x2 1 x2 x 1x 1 x 1 , x 1 24. 2x 3 x 1x 3 x 3 Horizontal asymptote: y 1 Degree of Nx degree of Dx Vertical asymptote: x 3 Since x 1 is a common factor of Nx and Dx, x 1 is not a vertical asymptote of f x. f x x2 3x 4 2x 2 x 1 1 3 x 3 x 6 is a zero of f x. x 1x 4 2x 1x 1 x 4 , x 1 2x 1 Domain: all real numbers except and Horizontal asymptote: Vertical asymptote: x Since x 1 is a common factor of Nx and Dx, x 1 is not a vertical asymptote of f x. 1 y Degree of Nx degree of Dx 2 1 x x 1 2 1 x 2 20. 26. gx x3 8 x 2 1 x3 8 x2 0 1 x 3 8 0 x 3 8 x 2 is a real zero of gx. 21. x 4 f x x Domain: all real numbers x except x ±4 Horizontal asymptote: y 0 Degree of Nx < degree of Dx Vertical asymptote: x 4 Since x 4 is a common factor of Nx and Dx, x 4 is not a vertical asymptote of f x. 2 1 , x 4 16 x 4 22. x 3 f x x Domain: all real numbers x except x ±3 The degree of the numerator is less than the degree of the denominator, so the graph has the line y 0 as a horizontal asymptote. Vertical asymptote: x 3 Since x 3 is a common factor of Nx and Dx, x 3 is not a vertical asymptote of f x. 2 9 x 3 1 , x 3 x 3x 3 x 3 f x x2 4 x 2 3x 2 x 2x 2 x 2x 1 Domain: all real numbers x except x 1 and x 2 Horizontal asymptote: x 2 , x 2 x 1 y 1 Degree of Nx degree of Dx Vertical asymptote: x 1 Since x 2 is a common factor of Nx and Dx, x 2 is not a vertical asymptote of f x. f x 6x2 11x 3 6x 2 7x 3 x 2 2x 33x 1 2x 33x 1 Domain: all real numbers x except x or Horizontal asymptote: y 1 3 2 Degree of Nx degree of Dx 3x 1 3 , x 3x 1 2 x 1 3 Vertical asymptote: Since is a common factor of and x is not a vertical asymptote of f x. 3 x 2x 3 Nx Dx, 2 1 3
27. fx (a) Domain: all real numbers x except x 2 1 x 2 (b) y-intercept: (c) Vertical asymptote: x 2 Horizontal asymptote: y 0 (d) 29. hx (a) Domain: all real numbers x except x 2 (b) y-intercept: 0, 1 2 (c) Vertical asymptote: x 2 Horizontal asymptote: y 0 1 x 2 (d) x 0 y 1 1 4 3 1 1 1 2 1 0, 2 x 4 3 1 0 1 y 1 1 1 2 −3 −1 −4 −3 −1 (( 0, − 1 2 2 1 −1 −2 2 1 −1 −2 y y (( 0, 1 Note: This is the graph of fx (Exercise 27) reflected about the x-axis. 1 x 2 1 2 2 x x 2 1 3 Section 2.6 Rational Functions 209 28. f x (a) Domain: all real numbers x except x 3 (b) 1 y- intercept: 0, 3 (c) Vertical asymptote: x 3 Horizontal asymptote: y 0 1 x 3 (d) 30. gx (a) Domain: all real numbers x except x 3 1 1 3 x x 3 (b) y-intercept: (c) Vertical asymptote: x 3 Horizontal asymptote: y 0 (d) 3 2 1 −1 −2 −3 3 2 1 −1 −2 −3 x 0 1 2 4 5 6 y y (( 0, (( 0, 1 1 3 y 1 1 1 − 3 3 1 0, 3 x 0 1 2 4 5 6 y 1 1 1 1 1 3 2 4 5 6 2 1 2 1 2 4 Note: This is the graph of fx (Exercise 28) reflected about the x-axis. 1 x 3 x x 1 2 1 2 1 3 3
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CHAPTER 2 Polynomial and Rational F
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9. (a) (c) 10. (a) (c) y 1 2 x2 (b
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13. f x x 2 5 14. Vertex: Axis of
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25. 27. 29. hx 4x 2 4x 21 Vertex
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41. 2, 2 is the vertex. 42. 2, 0 is
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61. fx 2x 10 2 7x 30 62. x-inter
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75. —CONTINUED— (d) A 8 x50 x
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86. (a) and (c) (b) (d) 1996 (e) Ve
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Section 2.2 Polynomial Functions of
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11. (a) (b) fx x4 fx x 3 3 4 y
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Section 2.2 Polynomial Functions of
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Page 25 and 26: 49. 51. 53. 55. 57. 59. 61. fx x
Page 27 and 28: 70. 72. 74. gx x 2 10x 16 x 2x
Page 29 and 30: 81. 83. 85. Zeros: 0, 2, 2 all of m
Page 31 and 32: 92. (a) 93. (c) 200 7 140 V 4 3r 3
Page 33 and 34: 111. 113. 115. 117. 12x 2 11x 5
Page 35 and 36: 2x 4 4. and y2 x 3 x (a) and (b
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Page 39 and 40: 45. fx 4x (a) 1 4 0 13 4 4 4 4 9 3
Page 41 and 42: 58. f x 3x Factors: x 3, x 2 3
Page 43 and 44: 67. 69. 71. ht t 3 2t 2 7t 2 (a
Page 45 and 46: 79. 83. x n 3 ) x 3n 9x 2n 27x n
Page 47 and 48: 1. 4. 7. a 10 b 6 Section 2.4 Comp
Page 49 and 50: 52. 53. 55. 57. 59. 8 16i 2i 3i 4
Page 51 and 52: 74. 4.5x2 3x 12 0; a 4.5, b 3,
Page 53 and 54: Section 2.5 Zeros of Polynomial Fun
Page 55 and 56: 18. 19. f x 3x 3 19x 2 33x 9 Po
Page 57 and 58: −24 (c) Real zeros: 2, 1 145 ± 8
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Page 61 and 62: 54. f x x Since 1 3i is a zero, s
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Page 95 and 96: 39. −2 −1 0 1 2 3 4 5 x Section
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3. V l w h x 2 x 3 x 2 x 3 2
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Problem Solving for Chapter 2 261 1
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19. Use synthetic division to show
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C H A P T E R 2 Polynomial and Rational Functions

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