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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. $f\left(x\right)={\left(x - 4\right)}^{2}, \left[4,\infty \right)$ 6. $f\left(x\right)={\left(x+2\right)}^{2}, \left[-2,\infty \right)$ 7. $f\left(x\right)={\left(x+1\right)}^{2}-3, \left[-1,\infty \right)$ 8. $f\left(x\right)=2-\sqrt{3+x}$ 9. $f\left(x\right)=3{x}^{2}+5,\left(-\infty ,0\right],\left[0,\infty \right)$ 10. $f\left(x\right)=12-{x}^{2}, \left[0,\infty \right)$ 11. $f\left(x\right)=9-{x}^{2}, \left[0,\infty \right)$ 12. $f\left(x\right)=2{x}^{2}+4, \left[0,\infty \right)$ For the following exercises, find the inverse of the functions. 13. $f\left(x\right)={x}^{3}+5$ 14. $f\left(x\right)=3{x}^{3}+1$ 15. $f\left(x\right)=4-{x}^{3}$ 16. $f\left(x\right)=4 - 2{x}^{3}$ For the following exercises, find the inverse of the functions. 17. $f\left(x\right)=\sqrt{2x+1}$ 18. $f\left(x\right)=\sqrt{3 - 4x}$ 19. $f\left(x\right)=9+\sqrt{4x - 4}$ 20. $f\left(x\right)=\sqrt{6x - 8}+5$ 21. $f\left(x\right)=9+2\sqrt[3]{x}$ 22. $f\left(x\right)=3-\sqrt[3]{x}$ 23. $f\left(x\right)=\frac{2}{x+8}$ 24. $f\left(x\right)=\frac{3}{x - 4}$ 25. $f\left(x\right)=\frac{x+3}{x+7}$ 26. $f\left(x\right)=\frac{x - 2}{x+7}$ 27. $f\left(x\right)=\frac{3x+4}{5 - 4x}$ 28. $f\left(x\right)=\frac{5x+1}{2 - 5x}$ 29. $f\left(x\right)={x}^{2}+2x, \left[-1,\infty \right)$ 30. $f\left(x\right)={x}^{2}+4x+1, \left[-2,\infty \right)$ 31. $f\left(x\right)={x}^{2}-6x+3, \left[3,\infty \right)$ For the following exercises, find the inverse of the function and graph both the function and its inverse. 32. $f\left(x\right)={x}^{2}+2,x\ge 0$ 33. $f\left(x\right)=4-{x}^{2},x\ge 0$ 34. $f\left(x\right)={\left(x+3\right)}^{2},x\ge -3$ 35. $f\left(x\right)={\left(x - 4\right)}^{2},x\ge 4$ 36. $f\left(x\right)={x}^{3}+3$ 37. $f\left(x\right)=1-{x}^{3}$ 38. $f\left(x\right)={x}^{2}+4x,x\ge -2$ 39. $f\left(x\right)={x}^{2}-6x+1,x\ge 3$ 40. $f\left(x\right)=\frac{2}{x}$ 41. $f\left(x\right)=\frac{1}{{x}^{2}},x\ge 0$ For the following exercises, use a graph to help determine the domain of the functions. 42. $f\left(x\right)=\sqrt{\frac{\left(x+1\right)\left(x - 1\right)}{x}}$ 43. $f\left(x\right)=\sqrt{\frac{\left(x+2\right)\left(x - 3\right)}{x - 1}}$ 44. $f\left(x\right)=\sqrt{\frac{x\left(x+3\right)}{x - 4}}$ 45. $f\left(x\right)=\sqrt{\frac{{x}^{2}-x - 20}{x - 2}}$ 46. $f\left(x\right)=\sqrt{\frac{9-{x}^{2}}{x+4}}$ 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. $f\left(x\right)={x}^{3}-x - 2,y=1, 2, 3$ 48. $f\left(x\right)={x}^{3}+x - 2, y=0, 1, 2$ 49. $f\left(x\right)={x}^{3}+3x - 4, y=0, 1, 2$ 50. $f\left(x\right)={x}^{3}+8x - 4, y=-1, 0, 1$ 51. $f\left(x\right)={x}^{4}+5x+1, y=-1, 0, 1$ For the following exercises, find the inverse of the functions with a, b, c positive real numbers. 52. $f\left(x\right)=a{x}^{3}+b$ 53. $f\left(x\right)={x}^{2}+bx$ 54. $f\left(x\right)=\sqrt{a{x}^{2}+b}$ 55. $f\left(x\right)=\sqrt[3]{ax+b}$ 56. $f\left(x\right)=\frac{ax+b}{x+c}$ 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, $h\left(t\right)$, in meters after t seconds have lapsed, such that $h\left(t\right)=200 - 4.9{t}^{2}$. 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, $h\left(t\right)$, in feet after t seconds have elapsed, such that $h\left(t\right)=600 - 16{t}^{2}$. Express 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 $V\left(r\right)=\frac{4}{3}\pi {r}^{3}$. 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 $A\left(r\right)=4\pi {r}^{2}$. 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 $C\left(n\right)=\frac{25+.6n}{100+n}$ 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 $T\left(l\right)=2\pi \sqrt{\frac{l}{32.2}}$. 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 $V=\pi {r}^{2}h$. 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 $A=2\pi {r}^{2}+2\pi rh$. 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 $V=\frac{1}{3}\pi {r}^{2}h$. 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.