36 Chapter 3 59. The volume V of the wind sock described in the previous question 61. Given two odd functions f and show that f og is also odd. Verify this fact with the particular functions 8, is given by the formula V =tr'h 3 f (x) = x and g(x) = %3D 3x2 -9 where r is the radius of the wind sock and h is the height of the wind sock. If the height h increases with time t Recall that a function is odd is f(-x) = -f(x) for all x in the domain of f. according to the formula h(t)= -t 4 %3D find the volume V of the wind sock as a function of time t and radius r. 62. Given two even functions f and g,show that the product is also even. Verify this fact with the particular functions 60. A widget factory produces widgets in t hours of a single day. The number of widgets the factory produces is given by the formula n(t) = 10,000t – 25t², 0sts9. The cost c in dollars of producing n widgets is given by the formula c(n)= 2040+1.74n. Find the cost c as a function of time t. п f(x) = 2x* – x² and g(x) =- %3D 21 x? a function is even TOP Recall that f(-x)= f (x) for all x in the domain of f. %3D %3D %3D As mentioned in Topic 4, a given complex number c is said to be in the Mandelbrot set if, for the function f(z)= z' +c, the sequence of iterates f(0), f² (0), ƒ° (0), ... stays close to the origin (which is the complex number 0+ 0i). It can be shown that if any single iterate falls more than 2 units in distance (magnitude) from the origin, then the remaining iterates will grow larger and larger in magnitude. In practice, computer programs that generate the Mandelbrot set calculate the iterates up to a predecided point in the sequence, such as f"(0), and if no iterate up to this point exceeds 2 in magnitude, the number c is admitted to the set. The magnitude of a complex number a + bi is the distance between the point (a, b) and the origin, so the formula for the magnitude of a + bi is va? +b?. .2 %3D 50 In Use the above criterion to determine, without a calculator or computer, if the following complex numbers are in the Mandelbrot set or not. 63. c=0 64. c=1 65. c= i 66. c= -1 67. c = 1+i Fi 68. c=-i 69. c=1-i 70. c=-1-i 71. c=2 72. c = -2 Combining Functions Section 3.6 285 40. f(x)= /x-1 and g(x)=**1 %3D 41. f(x)= x'+4x² and g(x)=|x|-1 %3D 42. f(x)= -3x +2 and g(x)=x +2 43. f(x)= x+2 and g(x)= x² +3 Write the following functions as a composition of two functions. Answers will vary. See Example 7. 44. f(x)= V3x² – 1 45. f(x) = -1 46. f(x)=|x-2|+3 %3D 47. f(x) = x+ Vx+2 – 5 48. f(x)= |x' - 5x|+7 Vx-3 49. f(x)= %3D x* -6х+9 50. f(x)= /2x³ – 3 – 4 51. f(x)=|x² +3x|- 3 3 52. f(x)= -2 %3D In each of the following problems, use the information given to find g(x). 53. f(x)=|x+3| and (f+g)(x)=|x+ 3| + Vx + 5 54. f(x)= x and (f°g)(x)=-" x+12 %D -3 55. f(x)= x² – 3 and (f-g)(x)=x³ +x² +4 56. f(x)=x² and (g•f)(x)= V-x² + 5 + 4 Solve the following application problems. 57. The volume of a right circular cylinder is given by the formula V = Tr²h. If the height h is three times the radius r, show the volume V as a function of r. %3D 58. The surface area S of a wind sock is given by the formula S = tr\r² +h² , where r is the radius of the base of .2 the wind sock and h is the height of the wind sock. As the wind sock is being knitted by an automated knitter, the height h increases with time t according to the formula t². Find the surface area S h(t)= 4. of the wind sock as a function of time t and radius r.