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Study Guides > MATH 1314: College Algebra

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. f(x)=(x4)2,[4,)f\left(x\right)={\left(x - 4\right)}^{2}, \left[4,\infty \right)\\ 6. f(x)=(x+2)2,[2,)f\left(x\right)={\left(x+2\right)}^{2}, \left[-2,\infty \right)\\ 7. f(x)=(x+1)23,[1,)f\left(x\right)={\left(x+1\right)}^{2}-3, \left[-1,\infty \right)\\ 8. f(x)=23+xf\left(x\right)=2-\sqrt{3+x}\\ 9. f(x)=3x2+5,(,0],[0,)f\left(x\right)=3{x}^{2}+5,\left(-\infty ,0\right],\left[0,\infty \right)\\ 10. f(x)=12x2,[0,)f\left(x\right)=12-{x}^{2}, \left[0,\infty \right)\\ 11. f(x)=9x2,[0,)f\left(x\right)=9-{x}^{2}, \left[0,\infty \right)\\ 12. f(x)=2x2+4,[0,)f\left(x\right)=2{x}^{2}+4, \left[0,\infty \right)\\ For the following exercises, find the inverse of the functions. 13. f(x)=x3+5f\left(x\right)={x}^{3}+5\\ 14. f(x)=3x3+1f\left(x\right)=3{x}^{3}+1\\ 15. f(x)=4x3f\left(x\right)=4-{x}^{3}\\ 16. f(x)=42x3f\left(x\right)=4 - 2{x}^{3}\\ For the following exercises, find the inverse of the functions. 17. f(x)=2x+1f\left(x\right)=\sqrt{2x+1}\\ 18. f(x)=34xf\left(x\right)=\sqrt{3 - 4x}\\ 19. f(x)=9+4x4f\left(x\right)=9+\sqrt{4x - 4}\\ 20. f(x)=6x8+5f\left(x\right)=\sqrt{6x - 8}+5\\ 21. f(x)=9+2x3f\left(x\right)=9+2\sqrt[3]{x}\\ 22. f(x)=3x3f\left(x\right)=3-\sqrt[3]{x}\\ 23. f(x)=2x+8f\left(x\right)=\frac{2}{x+8}\\ 24. f(x)=3x4f\left(x\right)=\frac{3}{x - 4}\\ 25. f(x)=x+3x+7f\left(x\right)=\frac{x+3}{x+7}\\ 26. f(x)=x2x+7f\left(x\right)=\frac{x - 2}{x+7}\\ 27. f(x)=3x+454xf\left(x\right)=\frac{3x+4}{5 - 4x}\\ 28. f(x)=5x+125xf\left(x\right)=\frac{5x+1}{2 - 5x}\\ 29. f(x)=x2+2x,[1,)f\left(x\right)={x}^{2}+2x, \left[-1,\infty \right)\\ 30. f(x)=x2+4x+1,[2,)f\left(x\right)={x}^{2}+4x+1, \left[-2,\infty \right)\\ 31. f(x)=x26x+3,[3,)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(x)=x2+2,x0f\left(x\right)={x}^{2}+2,x\ge 0\\ 33. f(x)=4x2,x0f\left(x\right)=4-{x}^{2},x\ge 0\\ 34. f(x)=(x+3)2,x3f\left(x\right)={\left(x+3\right)}^{2},x\ge -3\\ 35. f(x)=(x4)2,x4f\left(x\right)={\left(x - 4\right)}^{2},x\ge 4\\ 36. f(x)=x3+3f\left(x\right)={x}^{3}+3\\ 37. f(x)=1x3f\left(x\right)=1-{x}^{3}\\ 38. f(x)=x2+4x,x2f\left(x\right)={x}^{2}+4x,x\ge -2\\ 39. f(x)=x26x+1,x3f\left(x\right)={x}^{2}-6x+1,x\ge 3\\ 40. f(x)=2xf\left(x\right)=\frac{2}{x}\\ 41. f(x)=1x2,x0f\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(x)=(x+1)(x1)xf\left(x\right)=\sqrt{\frac{\left(x+1\right)\left(x - 1\right)}{x}}\\ 43. f(x)=(x+2)(x3)x1f\left(x\right)=\sqrt{\frac{\left(x+2\right)\left(x - 3\right)}{x - 1}}\\ 44. f(x)=x(x+3)x4f\left(x\right)=\sqrt{\frac{x\left(x+3\right)}{x - 4}}\\ 45. f(x)=x2x20x2f\left(x\right)=\sqrt{\frac{{x}^{2}-x - 20}{x - 2}}\\ 46. f(x)=9x2x+4f\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(x)=x3x2,y=1,2,3f\left(x\right)={x}^{3}-x - 2,y=1, 2, 3\\ 48. f(x)=x3+x2,y=0,1,2f\left(x\right)={x}^{3}+x - 2, y=0, 1, 2\\ 49. f(x)=x3+3x4,y=0,1,2f\left(x\right)={x}^{3}+3x - 4, y=0, 1, 2\\ 50. f(x)=x3+8x4,y=1,0,1f\left(x\right)={x}^{3}+8x - 4, y=-1, 0, 1\\ 51. f(x)=x4+5x+1,y=1,0,1f\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(x)=ax3+bf\left(x\right)=a{x}^{3}+b\\ 53. f(x)=x2+bxf\left(x\right)={x}^{2}+bx\\ 54. f(x)=ax2+bf\left(x\right)=\sqrt{a{x}^{2}+b}\\ 55. f(x)=ax+b3f\left(x\right)=\sqrt[3]{ax+b}\\ 56. f(x)=ax+bx+cf\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(t)h\left(t\right)\\, in meters after t seconds have lapsed, such that h(t)=2004.9t2h\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(t)h\left(t\right)\\, in feet after t seconds have elapsed, such that h(t)=60016t2h\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(r)=43πr3V\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(r)=4πr2A\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(n)=25+.6n100+nC\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(l)=2πl32.2T\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=πr2hV=\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=2πr2+2πrhA=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=13πr2hV=\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.

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