Section Exercises

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.

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