AN34 Answers: Chapter 5 33. (a) { } ( )( ) = + ≥ − f g x x x x 2 5; 5 2 (b) g f x x x x 2 5; 0 ( )( ) { } = + ≥ (c) f f x x x x ; 0 4 ( )( ) { } = ≥ (d) ( )( ) = + g g x x4 15; all real numbers 35. (a) f g x x x x ; 7 ( )( ) { } = ≥ (b) ( )( ) = g f x x ; all real numbers (c) f f x x x 14 56; 4 2 ( )( ) = + + all real numbers (d) g g x x x x 7 7; 56 ( ) { } ( ) = − − ≥ 37. (a) f g x x x x x x 4 17 2 1 ; 3; 1 2 { } ( )( ) = − − − ≠ ≠ (b) g f x x x x x x 3 3 2 8 ; 4; 1 ( )( ) { } = − − + ≠ − ≠ − (c) f f x x x x x x 2 5 2 ; 1; 2 ( )( ) { } = − + − ≠ − ≠ (d) g g x x x x x x 3 4 2 11 ; 11 2 ; 3 { } ( )( ) = − − − ≠ ≠ 39. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = ⋅ = = = = ⋅ = f g x f g x f x x x g f x g f x g x x x 1 2 2 1 2 ; 2 1 2 2 41. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = = = f g x f g x f x x x g f x g f x g x x x ; 3 3 3 3 3 3 43. f g x f g x f x x x x g f x g f x g x x x x 1 9 6 9 1 9 6 6 6 6 ; 9 6 1 9 9 6 6 1 9 9 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = + = ⋅ + − = + − = = = − = − + = ⋅ = 45. ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = − = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ + = f g x f g x f a x b a a x b b x 1 1 ; ( ) ( ) ( ) ( ) ( ) ( ) = = + = + − = g f x g f x g ax b a ax b b x 1 47. f x x g x x ; 2 3 4 ( ) ( ) = = + (Other answers are possible.) 49. f x x g x x ; 1 2 ( ) ( ) = = + (Other answers are possible.) 51. f x x g x x ; 2 1 ( ) ( ) = = + (Other answers are possible.) 53. ( ) ( ) ( ) ( ) = = f g x g f x 11; 2 55. −3, 3 57. (a) ( )( ) = + + f g x acx ad b (b) ( )( ) = + + g f x acx bc d (c) The domains of both f g and g f are all real numbers. (d) f g g f = when ad b bc d + = + 59. π ( ) = S t t 16 9 6 61. C t t t 15,000 800,000 40,000 2 ( ) = + − 63. C p p p 2 100 25 600, 0 100 ( ) = − + ≤ ≤ 65. V r r 2 3 π ( ) = 67. (a) f x x 0.9196 ( ) = (b) g x x 170.622 ( ) = (c) g f x x 156.9039912 ( ) ( ) = (d) 156,903.9912 yen 69. (a) f p p 200 ( ) = − (b) g p p 0.8 ( ) = (c) f g p p g f p p 0.8 200; 0.8 160; ( ) ( ) ( ) ( ) = − = − The 20% discount followed by the $200 rebate is the better deal. 71. 2 2, 3, 0, 3, 2 2 − − 73. 15 75. f is an odd function, so f x f x g. ( ) ( ) − = − is an even function, so g x g x . ( ) ( ) − = Then f g x f g x f g x f g x . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − = − = = So f g is even. Also, g f x g f x g f x g f x g f x , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − = − = − = = so g f is even. 77. 5− 79. f g x x4 3; ( )( ) + = + Domain: all real numbers; f g x x2 13; ( )( ) − = + Domain: all real numbers; f g x x x 3 7 40; 2 ( )( ) ⋅ = − − Domain: all real numbers; f g x x x 3 8 5 ; ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = + − Domain: x x 5 { } ≠ 80. 1 4 , 4 81. Domain: x x 3 { } ≠ ; Vertical asymptote: x 3 = ; Oblique asymptote: y x 9 = + 82. Vertex: 3, 8 ( ); Axis of symmetry: x 3 = ; Concave down 83. x 1 7 − ≤ ≤ 84. b 3 = 85. 4, 11 , 1, 5 ( ) ( ) − − 86. 5 10 87. x c 3 1 1 ( )( ) − + + 88. 6, 0, 6 { } − 31. 5, 3, 9, 2, 2, 1, 11,0, 5,1 ( ) ( ) { } ( ) ( ) ( ) − − − − Domain: 5, 9, 2, 11, 5 { } − Range: 3, 2, 1,0,1 { } − − − 33. 1, 2, 2, 3, 0, 10, 9,1, 4,2 ( ) ( ) ( ) ( ) ( ) { } − − − Domain: 1, 2, 0, 9, 4 { } Range: 2, 3, 10, 1, 2 { } − − − 35. y 2.5 x 2.5 (2, 1) (1, 0) (22, 22) f 21 f 21 (0, 21) y 5 x 37. y 2.5 x 2.5 (1, 2) (21, 21) f 21 y 5 x 39. y 2.5 x 2.5 f 21 y 5 x 41. f g x f x x x x g f x g x x x x 1 3 4 3 1 3 4 4 4 4 3 4 1 3 3 4 4 1 3 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] = − = ⋅ − + = − + = = + = + − = ⋅ = 43. f g x f x x x x g f x g x x x x 4 2 4 4 2 8 8 8 4 8 4 8 4 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + = + − = + − = = − = − + = − + = 45. f g x f x x x x g f x g x x x x 8 8 8 8 8 8 8 8 3 3 3 3 3 3 3 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + = + − = + − = = − = − + = = 47. f g x f x x x x g f x g x x x x 1 1 1 ; 0, 1 1 1 , 0 ( ) ( ) ( ) ( ) ( ) ( ) = = = ≠ = = = ≠ 27. 49.7 43.8 4.2 61.9 12.8 Annual Rainfall (inches) Location Atlanta, Georgia Boston, Massachusetts Las Vegas, Nevada Miami, Florida Los Angeles, California Domain: 49.7, 43.8, 4.2, 61.9, 12.8 { } Range: Atlanta,Boston,LasVegas,Miami,LosAngeles { } 29. $27.50 $42.00 $105.00 30 40 50 Monthly Cost of Life Insurance Age Domain: $27.50, $42.00, $105.00 { } Range: 30, 40, 50 { } 5.2 Assess Your Understanding (page 290) 5. f x f x 1 2 ( ) ( ) ≠ 6. one-to-one 7. 3 8. y x = 9. 4, [ )∞ 10. T 11. a 12. d 13. one-to-one 15. not one-to-one 17. not one-to-one 19. one-to-one 21. one-to-one 23. not one-to-one 25. one-to-one
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