SECTION 5.1 Composite Functions 279 27. f x x g x x ; 4 2 2 ( ) ( ) = = + 28. f x x g x x 1; 2 3 2 2 ( ) ( ) = + = + 29. f x x g x x 3 1 ; 2 ( ) ( ) = − = 30. ( ) ( ) = + = − f x x g x x 1 3 ; 2 31. ( ) ( ) = − = − f x x x g x x 1 ; 4 32. f x x x g x x 3 ; 2 ( ) ( ) = + = 33. f x x g x x ; 2 5 ( ) ( ) = = + 34. f x x g x x 2; 1 2 ( ) ( ) = − = − 35. f x x g x x 7; 7 2 ( ) ( ) = + = − 36. f x x g x x 4; 2 2 ( ) ( ) = + = − 37. f x x x g x x x 5 1 ; 2 3 ( ) ( ) = − + = + − 38. f x x x g x x x 2 1 2 ; 4 2 5 ( ) ( ) = − − = + − In Problems 39–46, show that f g x g f x x. ( ) ( ) ( ) ( ) = = 39. ( ) ( ) = = f x x g x x 2 ; 1 2 40. ( ) ( ) = = f x x g x x 4 ; 1 4 41. ( ) ( ) = = f x x g x x ;3 3 42. f x x g x x 5; 5 ( ) ( ) = + = − 43. ( ) ( ) ( ) = − = + f x x g x x 9 6; 1 9 6 44. ( ) ( ) ( ) = − = − f x x g x x 4 3 ; 1 3 4 45. ( ) ( ) ( ) = + = − ≠ f x ax b g x a x b a ; 1 0 46. ( ) ( ) = = f x x g x x 1 ; 1 In Problems 47–52, find functions f and g so that f g H. = 47. H x x2 3 4 ( ) ( ) = + 48. H x x 1 2 3 ( ) ( ) = + 49. H x x 1 2 ( ) = + 50. H x x 1 2 ( ) = − 51. H x x2 1 ( ) = + 52. H x x2 3 2 ( ) = + Applications and Extensions 53. If ( )= − + − f x x x x 2 3 4 1 3 2 and g x 2, ( ) = find f g x ( )( ) and g f x . ( )( ) 54. If f x x x 1 1 , ( ) = + − find f f x . ( )( ) 55. If f x x2 5 2 ( ) = + and g x x a 3 , ( ) = + find a so that the y-intercept of f g is 23. 56. If f x x3 7 2 ( ) = − and g x x a 2 , ( ) = + find a so that the y-intercept of f g is 68. In Problems 57 and 58, use the functions f and g to find: (a) f g (b) g f (c) the domain of f g and of g f (d) the conditions for which f g g f = 57. fx ax bgx cx d ( ) ( ) = + = + 58. f x ax b cx d g x mx ( ) ( ) = + + = 59. Surface Area of a Balloon The surface area S (in square meters) of a hot-air balloon is given by S r r 4 2 π ( ) = where r is the radius of the balloon (in meters). If the radius r is increasing with time t (in seconds) according to the formula ( ) = ≥ r t t t 2 3 , 0, 3 find the surface area S of the balloon as a function of the time t. 60. Volume of a Balloon The volume V (in cubic meters) of the hot-air balloon described in Problem 59 is given by π ( ) = V r r 4 3 .3 If the radius r is the same function of t as in Problem 59, find the volume V as a function of the time t. 61. Automobile Production The number N of cars produced at a certain factory in one day after t hours of operation is given by N t t t t 100 5 , 0 10. 2 ( ) = − ≤ ≤ If the cost C (in dollars) of producing N cars is C N N 15,000 8000 , ( ) = + find the cost C as a function of the time t of operation of the factory. 62. Environmental Concerns The spread of oil leaking from a tanker is in the shape of a circle. If the radius r (in feet) of the spread after t hours is r t t 200 , ( ) = find the area A of the oil slick as a function of the time t. 63. Production Cost The price p, in dollars, of a certain product and the quantity x sold follow the demand equation = − + ≤ ≤ p x x 1 4 100 0 400 Suppose that the cost C, in dollars, of producing x units is C x 25 600 = + Assuming that all items produced are sold, find the cost C as a function of the price p. [Hint: Solve for x in the demand equation and then form the composite function.] 64. Cost of a Commodity The price p, in dollars, of a certain commodity and the quantity x sold follow the demand equation = − + ≤ ≤ p x x 1 5 200 0 1000 Suppose that the cost C, in dollars, of producing x units is C x 10 400 = + Assuming that all items produced are sold, find the cost C as a function of the price p.
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