1. Explain why we cannot find inverse functions for all polynomial functions.
2. Why must we restrict the domain of a quadratic function when finding its inverse?
3. When finding the inverse of a radical function, what restriction will we need to make?
4. The inverse of a quadratic function will always take what form?
For the following exercises, find the inverse of the function on the given domain.
5. f(x)=(x−4)2,[4,∞)
6. f(x)=(x+2)2,[−2,∞)
7. f(x)=(x+1)2−3,[−1,∞)
8. f(x)=2−3+x
9. f(x)=3x2+5,(−∞,0],[0,∞)
10. f(x)=12−x2,[0,∞)
11. f(x)=9−x2,[0,∞)
12. f(x)=2x2+4,[0,∞)
For the following exercises, find the inverse of the functions.
13. f(x)=x3+5
14. f(x)=3x3+1
15. f(x)=4−x3
16. f(x)=4−2x3
For the following exercises, find the inverse of the functions.
17. f(x)=2x+1
18. f(x)=3−4x
19. f(x)=9+4x−4
20. f(x)=6x−8+5
21. f(x)=9+23x
22. f(x)=3−3x
23. f(x)=x+82
24. f(x)=x−43
25. f(x)=x+7x+3
26. f(x)=x+7x−2
27. f(x)=5−4x3x+4
28. f(x)=2−5x5x+1
29. f(x)=x2+2x,[−1,∞)
30. f(x)=x2+4x+1,[−2,∞)
31. f(x)=x2−6x+3,[3,∞)
For the following exercises, find the inverse of the function and graph both the function and its inverse.
32. f(x)=x2+2,x≥0
33. f(x)=4−x2,x≥0
34. f(x)=(x+3)2,x≥−3
35. f(x)=(x−4)2,x≥4
36. f(x)=x3+3
37. f(x)=1−x3
38. f(x)=x2+4x,x≥−2
39. f(x)=x2−6x+1,x≥3
40. f(x)=x2
41. f(x)=x21,x≥0
For the following exercises, use a graph to help determine the domain of the functions.
42. f(x)=x(x+1)(x−1)
43. f(x)=x−1(x+2)(x−3)
44. f(x)=x−4x(x+3)
45. f(x)=x−2x2−x−20
46. f(x)=x+49−x2
For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.
47. f(x)=x3−x−2,y=1,2,3
48. f(x)=x3+x−2,y=0,1,2
49. f(x)=x3+3x−4,y=0,1,2
50. f(x)=x3+8x−4,y=−1,0,1
51. f(x)=x4+5x+1,y=−1,0,1
For the following exercises, find the inverse of the functions with a, b, c positive real numbers.
52. f(x)=ax3+b
53. f(x)=x2+bx
54. f(x)=ax2+b
55. f(x)=3ax+b
56. f(x)=x+cax+b
For the following exercises, determine the function described and then use it to answer the question.
57. An object dropped from a height of 200 meters has a height, h(t), in meters after t seconds have lapsed, such that h(t)=200−4.9t2. Express t as a function of height, h, and find the time to reach a height of 50 meters.
58. An object dropped from a height of 600 feet has a height, h(t), in feet after t seconds have elapsed, such that h(t)=600−16t2. Express t as a function of height h, and find the time to reach a height of 400 feet.
59. The volume, V, of a sphere in terms of its radius, r, is given by V(r)=34πr3. Express r as a function of V, and find the radius of a sphere with volume of 200 cubic feet.
60. The surface area, A, of a sphere in terms of its radius, r, is given by A(r)=4πr2. Express r as a function of V, and find the radius of a sphere with a surface area of 1000 square inches.
61. A container holds 100 ml of a solution that is 25 ml acid. If n ml of a solution that is 60% acid is added, the function C(n)=100+n25+.6n gives the concentration, C, as a function of the number of ml added, n. Express n as a function of C and determine the number of mL that need to be added to have a solution that is 50% acid.
62. The period T, in seconds, of a simple pendulum as a function of its length l, in feet, is given by T(l)=2π32.2l. Express l as a function of T and determine the length of a pendulum with period of 2 seconds.
63. The volume of a cylinder, V, in terms of radius, r, and height, h, is given by V=πr2h. If a cylinder has a height of 6 meters, express the radius as a function of V and find the radius of a cylinder with volume of 300 cubic meters.
64. The surface area, A, of a cylinder in terms of its radius, r, and height, h, is given by A=2πr2+2πrh. If the height of the cylinder is 4 feet, express the radius as a function of V and find the radius if the surface area is 200 square feet.
65. The volume of a right circular cone, V, in terms of its radius, r, and its height, h, is given by V=31πr2h. Express r in terms of h if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.
66. Consider a cone with height of 30 feet. Express the radius, r, in terms of the volume, V, and find the radius of a cone with volume of 1000 cubic feet.
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Precalculus.Provided by: OpenStaxAuthored by: Jay Abramson, et al..Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions.License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..