- 27 F X X 29 F X G X X 4 2 37 F X 31 F X 33 F X X G X 2x 5 35 F X X 7 G X 1 (129.79 KiB) Viewed 22 times
27. f(x) = x²; 29. f(x) = g(x) = x² + 4 2 37. f(x) == 31. f(x) = 33. f(x) = √x: g(x) = 2x + 5 35. f(x) = x² + 7; g(x) = √x-7 g(x) = =-4 =*=1: 8(x)=X+² +1' g(x) = - 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) In Problems 47-52, find functions f and g so that fog = H. 47. H(x) = (2x + 3) 4 49. H(x)=√x + 1 51. H(x)= |2x + 1] x + 1 54. If f(x)=find (fof)(x). Applications and Extensions 53. If f(x) = 2x³ 3x² + 4x 1andg(x) = 2,find (fog)(x) and (gof)(x). 57. f(x) = ax + b ax + b 58. f(x) = a 0 55. If f(x) = 2x² + 5 and g(x) = 3x + a, find a so that the y-intercept of fog is 23. (a) fog (b) gof (c) the domain of fog and of gof (d) the conditions for which fog = gof g(x) = cx + d 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: 40. f(x) = 4x; g(x)=x 43. f(x) = 9x - 6; g(x) = (x + 6) 46. f(x) = 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 the formula r(t) = 1²,1 ≥ 0, find the surface area S of the balloon as a function of the time 1. 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) = 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 28. f(x) = x² + 1; g(x) = 2x² + 3 1 30. f(x) = x +3² SECTION 6.1 Composite Functions 421 32. f(x) = ) = x + 3² 8(x) = ²/ 34. f(x)=√x - 2; g(x) = 1 - 2x 36. f(x) = x² + 4; g(x)=√x-2 38. f(x) = ²x2: 8(x) = 2x+5 8(x) = - =-²/ 41. f(x) = x³; g(x) = Vx 44. f(x) = 43x; g(x) = () = (4-x) g(x) = 48. H(x) = (1 + x²) ³ 50. H(x) = V1 -² 52. H(x) = |2x² + 3| given by N(1) 100t - 5²,0 ≤ t ≤ 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 / hours is r(t) = 200 VI, 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 1 +100 0≤x≤ 400 P: Suppose that the cost C, in dollars, of producing x units is Vx 25 C= +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 = - 3/x + 200 0≤x≤ 1000 Suppose that the cost C, in dollars, of producing x units is Vx C 10 Assuming that all items produced are sold, find the cost Cas a function of the price p. +400