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27. f(x) = x²¹; g(x) = x² + 4 29. f(x)=³₁: 8(x) = ²/ 31. f(x) = x1 8(x) = - 33. f(x)=√x; g(x) = 2x + 5 35. f(x) = x² + 7; g(x) = √x-7 -5 37. f(x) = = = ₁; g(x)=x+² x+1' In Problems 39-46, show that (fog)(x) = (gof)(x) = x. 39. f(x) = 2x; g(x) = x 42. f(x) = x + 5; g(x) = x - 5 45. f(x) = ax + b; g(x) = (x-b) a *0 In Problems 47-52, find functions f and g so that fog = H. 47. H(x) = (2x + 3)4 49. H(x) = √²+1 51. H(x) = 2x + 1 Applications and Extensions 53. If f(x) = 2x²-3x² + 4x 1andg(x) = 2. find (fog)(x) and (gof)(x). 40. f(x) = 4x: g(x) = x 43. f(x) = 9x - 6; g(x) = 54. If f(x)=1 find (fof)(x). x+1 55. If f(x) = 2x² + 5 and g(x) = 3x + a, find a so that the y-intercept of fog is 23. 56. If f(x) = 3x² - 7 and g(x) = 2x + a, find a so that the y-intercept of fog is 68. In Problems 57 and 58, use the functions f and g to find: (a) fog (b) gof (c) the domain of fog and of g of (d) the conditions for which fo g = gof 57. f(x) = ax + b g(x) = cx + d ax + b g(x) = mx cx + d 59. Surface Area of a Balloon The surface area S (in square meters) of a hot-air balloon is given by S(r) = 4wr² where r is the radius of the balloon (in meters). If the radius r is increasing with time t (in seconds) according to 58. f(x) = the formula r(t) = 1²,1 = 0, 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 V(r)=r³. 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 28. f(x) = x² + 1; g(x) = 2x² + 3 30. f(x) = -3; 8(x) = - 32. f(x) = 34. f(x)=√x-2; g(x) = 1 - 2x 36. f(x) = x² + 4; g(x)=√x-2 2x 1, x-21 38. f(x) = SECTION 6.1 Composite Functions 421 (x + 6) 46. f(x) = g(x) = 41. f(x) = x³; g(x) = √x 44. f(x) = 4 - 3x: g(x) =(4-x) g(x) = x + 4 48. H(x) = (1+x²)³ 50. H(x) = V1-2 52. H(x)= |2x² + 3| given by N(t) = 100r5r²,0 ≤ts 10. If the cost C (in dollars) of producing N cars is C(N) = 15,000+8000N, 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) = 200 Vt, find the area A of the oil slick as a function of the time 1. 63. Production Cost The price p, in dollars, of a certain product and the quantity x sold follow the demand equation p=- --x +100 0≤x≤ 400 Suppose that the cost C, in dollars, of producing x units is Vx 25 +600 Assuming that all items produced are sold, find the cost Cas 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+200 0≤x≤ 1000 Suppose that the cost C, in dollars, of producing x units is Vx 10 C = Assuming that all items produced are sold, find the cost C as a function of the price p. +400