Section 2.5 AN9 31. (a) All real numbers (b) ( ) 0, 1 (c) (0, 1) (1, 2) (21, 22) y x 2.5 2.5 (d) { } ( ) ( ) ≠ −∞ ∪ ∞ y y 0 ; , 0 0, 33. (a) All real numbers (b) ( ) 0, 3 (c) y x 5 5 (2, 4) (1, 1) (0, 3) (d) { } [ ) ≥ ∞ y y 1; 1, 35. (a) { } [ ) ≥ − − ∞ x x 2; 2, (b) ( ) 0, 3 , ( ) 2, 0 (c) (1, 1) (1, 5) (1, 4) (22, 1) (2, 0) y x 5 5 (0, 3) (d) { } ( ) { } < = −∞ ∪ y y y 4, 5 ; , 4 5 37. (a) All real numbers (b) ( ) ( ) −1, 0 , 0, 0 (c) (0, 1) (0, 0) (1, 1) (21, 0) y x 2.5 2.5 (d) All real numbers 39. (a) { } [ ) ( ) ≥ − ≠ − ∪ ∞ x x x 2, 0 ; 2,0 0, (b) No intercepts (c) (1, 1) (22, 2) y x 2.5 2.5 (d) { } ( ) > ∞ y y 0; 0, 41. (a) Domain ( )∞ 0, (b) No intercepts (c) (2, 4) (0, 0) (5, 7) (d) Range: ( ] 0, 7 43. f x x x x x if 1 0 if 0 2 1 2 ( ) = − − ≤ ≤ < ≤ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪⎪ 45. f x x x x x if 0 2 if 0 2 ( ) = − ≤ − + < ≤ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ 47. (a) 3 (b) 5 (c) −8 49. (a) (0, 1) (6, 22) (2, 1) (2, 6) (1, 0) y x 6 6 (b) [ ] 0, 6 (c) Absolute maximum: ( ) = f 2 6; absolute minimum: ( ) = − f 6 2 51. (a) $26.88 (b) $34.29 (c) $52.10 53. (a) $48 (b) $205 (c) C x x x x x x x 2 if 0 50 100 1.5 50 if 50 200 325 1.15 200 if 200 ( ) ( ) ( ) = ≤ ≤ + − < ≤ + − > ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (d) y x 300 100 200 100 200 Cost (dollars) Cupcakes Ordered (200, 325) (0, 0) (50, 100) 61. (a) ° 10 C (b) °4 C (c) − °3 C (d) − °4 C (e) The wind chill is equal to the air temperature. (f) At wind speed greater than 20 m s, the wind chill factor depends only on the air temperature. 63. ( ) ( ) ( ) = + ≤ ≤ + − < ≤ + − > ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ S x x x x x x x 45,000 0.04 if 0 250,000 55,000 0.06 250,000 if 250,000 500,000 70,000 0.09 500,000 if 500,000 65. ( ) ( ) + = − + ≤ − < < + − ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ f x g x x x x x x x x 2 4 if 0 3 4 if 0 2 6 7 if 2 2 67. Each graph is that of = y x ,2 but shifted horizontally. If ( ) = − > y x k k , 0, 2 the shift is right k units; if ( ) = + > y x k k , 0, 2 the shift is left k units. 69. The graph of ( ) = − y f x is the reflection about the x-axis of the graph of ( ) = y f x . 71. Yes. The graph of ( ) = − + y x 1 2 3 is the graph of = y x3 shifted right 1 unit and up 2 units. 73. They all have the same general shape. All three go through the points ( ) − − 1, 1, ( ) 0, 0 , and ( ) 1, 1 . As the exponent increases, the steepness of the curve increases (except near = x 0). 76. x y 6 10 77. ( ) ( ) = = h k r , 0, 3 ; 5 78. { } −8 79. CD: $22,000; Mutual fund: $38,000 80. Quotient: + − x x 2; 2 Remainder: −2 81. + i 3 2 2 82. − x2 7 83. + t t 5 1 25 2 3 84. { } ≥ − x x 7 or [ ) − ∞ 7, 85. ( ) ( ) − + x y x y 3 2 6 2 57. (a) (400, 170) y x 275 100 500 1000 200 Cost (dollars) Distance (miles) (800, 270) (960, 270) (0, 0) (100, 50) (b) ( ) = + C x x 10 0.4 (c) ( ) = + C x x 70 0.25 59. (a) ( ) = ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ C s s s s s s s 9000 if 659 8250 if 660 679 5250 if 680 699 3750 if 700 719 2250 if 720 739 1500 if 740 (b) $2250 (c) $8250 55. f x x x x x x x x x x x x x x x 0.1 if 0 11,600 1160 0.12 11,600 if 11,600 47,150 5426 0.22 47,150 if 47,150 100,525 17,168.50 0.24 100,525 if 100,525 191,950 39,110.50 0.32 191,950 if 191,950 243,725 55,678.50 0.35 243,725 if 243,725 609,350 183,647.25 0.37 609,350 if 609,350 ( ) ( ) ( ) ( ) ( ) ( ) ( ) = < ≤ + − < ≤ + − < ≤ + − < ≤ + − < ≤ + − < ≤ + − > ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ 2.5 Assess Your Understanding (page 121) 5. horizontal; right 6. y 7. F 8. T 9. d 10. b 11. B 13. H 15. I 17. L 19. F 21. G 23. ( ) = − y x 4 3 25. = + y x 4 3 27. = − y x3 29. = y x5 3 31. ( ) = = y x x 2 8 3 3 33. ( ) = − − + y x 2 35. = + + y x 3 5 4 37. c 39. c 41. y x 2.5 2.5 (1, 0) (21, 0) (0, 21) Domain: ( ) −∞∞, ; Range: [ ) − ∞1, 43. y x 5 5 (3, 3) (0, 0) 1 3 , 1 Domain: [ )∞ 0, ; Range: [ )∞ 0, 45. y x 5 5 (2, 2) (22, 0) (21, 1) Domain: [ ) − ∞ 2, ; Range: [ )∞ 0, 47. y x 5 5 (2, 3) (1, 2) (0, 1) Domain: ( ) −∞∞, ; Range: ( ) −∞∞, 49. y x 9 8 (4, 8) (1, 4) (0, 0) Domain: [ )∞ 0, ; Range: [ )∞ 0,
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