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. [latex]f\left(x\right)={\left(x - 4\right)}^{2}, \left[4,\infty \right)\\[/latex]

6. [latex]f\left(x\right)={\left(x+2\right)}^{2}, \left[-2,\infty \right)\\[/latex]

7. [latex]f\left(x\right)={\left(x+1\right)}^{2}-3, \left[-1,\infty \right)\\[/latex]

8. [latex]f\left(x\right)=2-\sqrt{3+x}\\[/latex]

9. [latex]f\left(x\right)=3{x}^{2}+5,\left(-\infty ,0\right],\left[0,\infty \right)\\[/latex]

10. [latex]f\left(x\right)=12-{x}^{2}, \left[0,\infty \right)\\[/latex]

11. [latex]f\left(x\right)=9-{x}^{2}, \left[0,\infty \right)\\[/latex]

12. [latex]f\left(x\right)=2{x}^{2}+4, \left[0,\infty \right)\\[/latex]

For the following exercises, find the inverse of the functions.

13. [latex]f\left(x\right)={x}^{3}+5\\[/latex]

14. [latex]f\left(x\right)=3{x}^{3}+1\\[/latex]

15. [latex]f\left(x\right)=4-{x}^{3}\\[/latex]

16. [latex]f\left(x\right)=4 - 2{x}^{3}\\[/latex]

For the following exercises, find the inverse of the functions.

17. [latex]f\left(x\right)=\sqrt{2x+1}\\[/latex]

18. [latex]f\left(x\right)=\sqrt{3 - 4x}\\[/latex]

19. [latex]f\left(x\right)=9+\sqrt{4x - 4}\\[/latex]

20. [latex]f\left(x\right)=\sqrt{6x - 8}+5\\[/latex]

21. [latex]f\left(x\right)=9+2\sqrt[3]{x}\\[/latex]

22. [latex]f\left(x\right)=3-\sqrt[3]{x}\\[/latex]

23. [latex]f\left(x\right)=\frac{2}{x+8}\\[/latex]

24. [latex]f\left(x\right)=\frac{3}{x - 4}\\[/latex]

25. [latex]f\left(x\right)=\frac{x+3}{x+7}\\[/latex]

26. [latex]f\left(x\right)=\frac{x - 2}{x+7}\\[/latex]

27. [latex]f\left(x\right)=\frac{3x+4}{5 - 4x}\\[/latex]

28. [latex]f\left(x\right)=\frac{5x+1}{2 - 5x}\\[/latex]

29. [latex]f\left(x\right)={x}^{2}+2x, \left[-1,\infty \right)\\[/latex]

30. [latex]f\left(x\right)={x}^{2}+4x+1, \left[-2,\infty \right)\\[/latex]

31. [latex]f\left(x\right)={x}^{2}-6x+3, \left[3,\infty \right)\\[/latex]

For the following exercises, find the inverse of the function and graph both the function and its inverse.

32. [latex]f\left(x\right)={x}^{2}+2,x\ge 0\\[/latex]

33. [latex]f\left(x\right)=4-{x}^{2},x\ge 0\\[/latex]

34. [latex]f\left(x\right)={\left(x+3\right)}^{2},x\ge -3\\[/latex]

35. [latex]f\left(x\right)={\left(x - 4\right)}^{2},x\ge 4\\[/latex]

36. [latex]f\left(x\right)={x}^{3}+3\\[/latex]

37. [latex]f\left(x\right)=1-{x}^{3}\\[/latex]

38. [latex]f\left(x\right)={x}^{2}+4x,x\ge -2\\[/latex]

39. [latex]f\left(x\right)={x}^{2}-6x+1,x\ge 3\\[/latex]

40. [latex]f\left(x\right)=\frac{2}{x}\\[/latex]

41. [latex]f\left(x\right)=\frac{1}{{x}^{2}},x\ge 0\\[/latex]

For the following exercises, use a graph to help determine the domain of the functions.

42. [latex]f\left(x\right)=\sqrt{\frac{\left(x+1\right)\left(x - 1\right)}{x}}\\[/latex]

43. [latex]f\left(x\right)=\sqrt{\frac{\left(x+2\right)\left(x - 3\right)}{x - 1}}\\[/latex]

44. [latex]f\left(x\right)=\sqrt{\frac{x\left(x+3\right)}{x - 4}}\\[/latex]

45. [latex]f\left(x\right)=\sqrt{\frac{{x}^{2}-x - 20}{x - 2}}\\[/latex]

46. [latex]f\left(x\right)=\sqrt{\frac{9-{x}^{2}}{x+4}}\\[/latex]

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. [latex]f\left(x\right)={x}^{3}-x - 2,y=1, 2, 3\\[/latex]

48. [latex]f\left(x\right)={x}^{3}+x - 2, y=0, 1, 2\\[/latex]

49. [latex]f\left(x\right)={x}^{3}+3x - 4, y=0, 1, 2\\[/latex]

50. [latex]f\left(x\right)={x}^{3}+8x - 4, y=-1, 0, 1\\[/latex]

51. [latex]f\left(x\right)={x}^{4}+5x+1, y=-1, 0, 1\\[/latex]

For the following exercises, find the inverse of the functions with a, b, c positive real numbers.

52. [latex]f\left(x\right)=a{x}^{3}+b\\[/latex]

53. [latex]f\left(x\right)={x}^{2}+bx\\[/latex]

54. [latex]f\left(x\right)=\sqrt{a{x}^{2}+b}\\[/latex]

55. [latex]f\left(x\right)=\sqrt[3]{ax+b}\\[/latex]

56. [latex]f\left(x\right)=\frac{ax+b}{x+c}\\[/latex]

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, [latex]h\left(t\right)\\[/latex], in meters after t seconds have lapsed, such that [latex]h\left(t\right)=200 - 4.9{t}^{2}\\[/latex]. 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, [latex]h\left(t\right)\\[/latex], in feet after t seconds have elapsed, such that [latex]h\left(t\right)=600 - 16{t}^{2}\\[/latex]. 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 [latex]V\left(r\right)=\frac{4}{3}\pi {r}^{3}\\[/latex]. 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 [latex]A\left(r\right)=4\pi {r}^{2}\\[/latex]. 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 [latex]C\left(n\right)=\frac{25+.6n}{100+n}\\[/latex] 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 [latex]T\left(l\right)=2\pi \sqrt{\frac{l}{32.2}}\\[/latex]. 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 [latex]V=\pi {r}^{2}h\\[/latex]. 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 [latex]A=2\pi {r}^{2}+2\pi rh\\[/latex]. 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 [latex]V=\frac{1}{3}\pi {r}^{2}h\\[/latex]. 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.