A-20 Answers to Selected Exercises 61. x y 4 2 0 4 6 –2 –4 f(x) = x + 1 x – 4 63. 4 1 2 4 –2 0 f(x) = x + 2 x – 3 x y 3 65. 4 12 0 f(x) = 4 – 2x 8 – x y x 8 67. x y 0 1 2 –1 1 2 3 –3 f(x) = 3x x2 – x – 2 (j) x y –3 2 3 –12 1 3 f(x) = –3x4 + 22x3 – 55x2 + 52x – 12 0 20. For the function in Exercise 12: {1.732; for the function in Exercise 13: {0.707; for the function in Exercise 14: -2.303, 1.303; for the function in Exercise 17: {2.236 3.5 Exercises 1. 1-∞, 02 ´10, ∞2; 1-∞, 02 ´10, ∞2 3. none; 1-∞, 02 and 10, ∞2; none 5. x = 3; y = 2 7. even; symmetry with respect to the y-axis 9. A, B, C 11. A 13. A 15. A, C, D 17. To obtain the graph of ƒ, stretch the graph of y = 1 x vertically by a factor of 2. (a) 1-∞, 02 ´10, ∞2 (b) 1-∞, 02 ´10, ∞2 (c) none (d) 1-∞, 02 and 10, ∞2 19. To obtain the graph of ƒ, shift the graph of y = 1 x to the left 2 units. (a) 1-∞, -22 ´1-2, ∞2 (b) 1-∞, 02 ´10, ∞2 (c) none (d) 1-∞, -22 and 1-2, ∞2 21. To obtain the graph of ƒ, shift the graph of y = 1 x up 1 unit. (a) 1-∞, 02 ´10, ∞2 (b) 1-∞, 12 ´11, ∞2 (c) none (d) 1-∞, 02 and 10, ∞2 23. To obtain the graph of ƒ, stretch the graph of y = 1 x2 vertically by a factor of 2 and reflect across the x-axis. (a) 1-∞, 02 ´10, ∞2 (b) 1-∞, 02 (c) 10, ∞2 (d) 1-∞, 02 25. To obtain the graph of ƒ, shift the graph of y = 1 x2 to the right 3 units. (a) 1-∞, 32 ´13, ∞2 (b) 10, ∞2 (c) 1-∞, 32 (d) 13, ∞2 27. To obtain the graph of ƒ, shift the graph of y = 1 x2 to the left 2 units, reflect across the x-axis, and shift down 3 units. (a) 1-∞, -22 ´1-2, ∞2 (b) 1-∞, -32 (c) 1-2, ∞2 (d) 1-∞, -22 29. D 31. G 33. E 35. F In selected Exercises 37–59, V.A. represents vertical asymptote, H.A. represents horizontal asymptote, and O.A. represents oblique asymptote. 37. V.A.: x = 5; H.A.: y = 0 39. V.A.: x = - 1 2 ; H.A.: y = - 3 2 41. V.A.: x = -3; O.A.: y = x - 3 43. V.A.: x = -2, x = 5 2; H.A.: y = 1 2 45. V.A.: none; H.A.: y = 1 47. (a) ƒ1x2 = 2x - 5 x - 3 (b) 5 2 (c) H.A.: y = 2; V.A.: x = 3 49. (a) y = x + 1 (b) at x = 0 and x = 1 (c) above 51. A 53. V.A.: x = 2; H.A.: y = 4; 1-∞, 22 ´12, ∞2 55. V.A.: x = {2; H.A.: y = -4; 1-∞, -22 ´1-2, 22 ´ 12, ∞2 57. V.A.: none; H.A.: y = 0; 1-∞, ∞2 59. V.A.: x = -1; O.A.: y = x - 1; 1-∞, -12 ´1-1, ∞2 y –4 –4 4 4 0 x f(x) = 2x y –3 x = –2 2 2 0 x f(x) = 1 x + 2 y –2 –3 y = 1 2 3 x f(x) = 1 + 1 x 0 y –2 1 2 0 x 2 x2 f(x) = – y 0 2 4 x x = 3 f(x) = 1 (x – 3)2 2 1 2 x y y = –3 x = –2 f(x) = –1 – 3 (x + 2)2 0 69. x y 0 2 –1 1 f(x) = x2 – 1 5x 71. –6 6 0 f(x) = (x + 6)(x – 2) (x + 3)(x – 4) x y 73. 8 8 0 f(x) = 3x 2 + 3x – 6 x2 – x – 12 y x 75. 5 4 0 f(x) = 9x2 – 1 x2 – 4 x y 77. 4 1 1 4 0 f(x) = (x – 3)(x + 1) (x – 1)2 y x 79. x y 0 f(x) = –3 3 x x2 – 9
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