{"id":1473,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1473"},"modified":"2017-04-03T14:48:09","modified_gmt":"2017-04-03T14:48:09","slug":"section-exercises-43","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/section-exercises-43\/","title":{"raw":"Section Exercises","rendered":"Section Exercises"},"content":{"raw":"

1. Explain why we cannot find inverse functions for all polynomial functions.\r\n\r\n2.\u00a0Why must we restrict the domain of a quadratic function when finding its inverse?\r\n\r\n3. When finding the inverse of a radical function, what restriction will we need to make?\r\n\r\n4.\u00a0The inverse of a quadratic function will always take what form?\r\n\r\nFor the following exercises, find the inverse of the function on the given domain.\r\n\r\n5. [latex]f\\left(x\\right)={\\left(x - 4\\right)}^{2}, \\left[4,\\infty \\right)[\/latex]\r\n\r\n6.\u00a0[latex]f\\left(x\\right)={\\left(x+2\\right)}^{2}, \\left[-2,\\infty \\right)[\/latex]\r\n\r\n7. [latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-3, \\left[-1,\\infty \\right)[\/latex]\r\n\r\n8.\u00a0[latex]f\\left(x\\right)=2-\\sqrt{3+x}[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=3{x}^{2}+5,\\left(-\\infty ,0\\right],\\left[0,\\infty \\right)[\/latex]\r\n\r\n10.\u00a0[latex]f\\left(x\\right)=12-{x}^{2}, \\left[0,\\infty \\right)[\/latex]\r\n\r\n11. [latex]f\\left(x\\right)=9-{x}^{2}, \\left[0,\\infty \\right)[\/latex]\r\n\r\n12.\u00a0[latex]f\\left(x\\right)=2{x}^{2}+4, \\left[0,\\infty \\right)[\/latex]\r\n\r\nFor the following exercises, find the inverse of the functions.\r\n\r\n13. [latex]f\\left(x\\right)={x}^{3}+5[\/latex]\r\n\r\n14.\u00a0[latex]f\\left(x\\right)=3{x}^{3}+1[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=4-{x}^{3}[\/latex]\r\n\r\n16.\u00a0[latex]f\\left(x\\right)=4 - 2{x}^{3}[\/latex]\r\n\r\nFor the following exercises, find the inverse of the functions.\r\n\r\n17. [latex]f\\left(x\\right)=\\sqrt{2x+1}[\/latex]\r\n\r\n18.\u00a0[latex]f\\left(x\\right)=\\sqrt{3 - 4x}[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=9+\\sqrt{4x - 4}[\/latex]\r\n\r\n20.\u00a0[latex]f\\left(x\\right)=\\sqrt{6x - 8}+5[\/latex]\r\n\r\n21. [latex]f\\left(x\\right)=9+2\\sqrt[3]{x}[\/latex]\r\n\r\n22.\u00a0[latex]f\\left(x\\right)=3-\\sqrt[3]{x}[\/latex]\r\n\r\n23. [latex]f\\left(x\\right)=\\frac{2}{x+8}[\/latex]\r\n\r\n24.\u00a0[latex]f\\left(x\\right)=\\frac{3}{x - 4}[\/latex]\r\n\r\n25. [latex]f\\left(x\\right)=\\frac{x+3}{x+7}[\/latex]\r\n\r\n26.\u00a0[latex]f\\left(x\\right)=\\frac{x - 2}{x+7}[\/latex]\r\n\r\n27. [latex]f\\left(x\\right)=\\frac{3x+4}{5 - 4x}[\/latex]\r\n\r\n28.\u00a0[latex]f\\left(x\\right)=\\frac{5x+1}{2 - 5x}[\/latex]\r\n\r\n29. [latex]f\\left(x\\right)={x}^{2}+2x, \\left[-1,\\infty \\right)[\/latex]\r\n\r\n30. [latex]f\\left(x\\right)={x}^{2}+4x+1, \\left[-2,\\infty \\right)[\/latex]\r\n\r\n31. [latex]f\\left(x\\right)={x}^{2}-6x+3, \\left[3,\\infty \\right)[\/latex]\r\n\r\nFor the following exercises, find the inverse of the function and graph both the function and its inverse.\r\n\r\n32. [latex]f\\left(x\\right)={x}^{2}+2,x\\ge 0[\/latex]\r\n\r\n33. [latex]f\\left(x\\right)=4-{x}^{2},x\\ge 0[\/latex]\r\n\r\n34.\u00a0[latex]f\\left(x\\right)={\\left(x+3\\right)}^{2},x\\ge -3[\/latex]\r\n\r\n35. [latex]f\\left(x\\right)={\\left(x - 4\\right)}^{2},x\\ge 4[\/latex]\r\n\r\n36.\u00a0[latex]f\\left(x\\right)={x}^{3}+3[\/latex]\r\n\r\n37. [latex]f\\left(x\\right)=1-{x}^{3}[\/latex]\r\n\r\n38. [latex]f\\left(x\\right)={x}^{2}+4x,x\\ge -2[\/latex]\r\n\r\n39. [latex]f\\left(x\\right)={x}^{2}-6x+1,x\\ge 3[\/latex]\r\n\r\n40.\u00a0[latex]f\\left(x\\right)=\\frac{2}{x}[\/latex]\r\n\r\n41. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}},x\\ge 0[\/latex]\r\n\r\nFor the following exercises, use a graph to help determine the domain of the functions.\r\n\r\n42. [latex]f\\left(x\\right)=\\sqrt{\\frac{\\left(x+1\\right)\\left(x - 1\\right)}{x}}[\/latex]\r\n\r\n43. [latex]f\\left(x\\right)=\\sqrt{\\frac{\\left(x+2\\right)\\left(x - 3\\right)}{x - 1}}[\/latex]\r\n\r\n44.\u00a0[latex]f\\left(x\\right)=\\sqrt{\\frac{x\\left(x+3\\right)}{x - 4}}[\/latex]\r\n\r\n45. [latex]f\\left(x\\right)=\\sqrt{\\frac{{x}^{2}-x - 20}{x - 2}}[\/latex]\r\n\r\n46.\u00a0[latex]f\\left(x\\right)=\\sqrt{\\frac{9-{x}^{2}}{x+4}}[\/latex]\r\n\r\nFor 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.\r\n\r\n47. [latex]f\\left(x\\right)={x}^{3}-x - 2,y=1, 2, 3[\/latex]\r\n\r\n48. [latex]f\\left(x\\right)={x}^{3}+x - 2, y=0, 1, 2[\/latex]\r\n\r\n49. [latex]f\\left(x\\right)={x}^{3}+3x - 4, y=0, 1, 2[\/latex]\r\n\r\n50.\u00a0[latex]f\\left(x\\right)={x}^{3}+8x - 4, y=-1, 0, 1[\/latex]\r\n\r\n51. [latex]f\\left(x\\right)={x}^{4}+5x+1, y=-1, 0, 1[\/latex]\r\n\r\nFor the following exercises, find the inverse of the functions with a<\/em>, b<\/em>, c<\/em>\u00a0positive real numbers.\r\n\r\n52. [latex]f\\left(x\\right)=a{x}^{3}+b[\/latex]\r\n\r\n53. [latex]f\\left(x\\right)={x}^{2}+bx[\/latex]\r\n\r\n54.\u00a0[latex]f\\left(x\\right)=\\sqrt{a{x}^{2}+b}[\/latex]\r\n\r\n55. [latex]f\\left(x\\right)=\\sqrt[3]{ax+b}[\/latex]\r\n\r\n56.\u00a0[latex]f\\left(x\\right)=\\frac{ax+b}{x+c}[\/latex]\r\n\r\nFor the following exercises, determine the function described and then use it to answer the question.\r\n\r\n57. An object dropped from a height of 200 meters has a height, [latex]h\\left(t\\right)[\/latex], in meters after t<\/em>\u00a0seconds have lapsed, such that [latex]h\\left(t\\right)=200 - 4.9{t}^{2}[\/latex]. Express t<\/em>\u00a0as a function of height, h<\/em>, and find the time to reach a height of 50 meters.\r\n\r\n58.\u00a0An object dropped from a height of 600 feet has a height, [latex]h\\left(t\\right)[\/latex], in feet after t<\/em>\u00a0seconds have elapsed, such that [latex]h\\left(t\\right)=600 - 16{t}^{2}[\/latex]. Express t\u00a0<\/em>as a function of height h<\/em>, and find the time to reach a height of 400 feet.\r\n\r\n59. The volume, V<\/em>, of a sphere in terms of its radius, r<\/em>, is given by [latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]. Express r<\/em>\u00a0as a function of V<\/em>, and find the radius of a sphere with volume of 200 cubic feet.\r\n\r\n60.\u00a0The surface area, A<\/em>, of a sphere in terms of its radius, r<\/em>, is given by [latex]A\\left(r\\right)=4\\pi {r}^{2}[\/latex]. Express r<\/em>\u00a0as a function of V<\/em>, and find the radius of a sphere with a surface area of 1000 square inches.\r\n\r\n61. A container holds 100 ml of a solution that is 25 ml acid. If n<\/em>\u00a0ml 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<\/em>, as a function of the number of ml added, n<\/em>. Express n<\/em>\u00a0as a function of C<\/em>\u00a0and determine the number of mL that need to be added to have a solution that is 50% acid.\r\n\r\n62.\u00a0The period T<\/em>, in seconds, of a simple pendulum as a function of its length l<\/em>, in feet, is given by [latex]T\\left(l\\right)=2\\pi \\sqrt{\\frac{l}{32.2}}[\/latex]. Express l<\/em>\u00a0as a function of T<\/em>\u00a0and determine the length of a pendulum with period of 2 seconds.\r\n\r\n63. The volume of a cylinder, V<\/em>, in terms of radius, r<\/em>, and height, h<\/em>, 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<\/em>\u00a0and find the radius of a cylinder with volume of 300 cubic meters.\r\n\r\n64.\u00a0The surface area, A<\/em>, of a cylinder in terms of its radius, r<\/em>, and height, h<\/em>, 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<\/em>\u00a0and find the radius if the surface area is 200 square feet.\r\n\r\n65. The volume of a right circular cone, V<\/em>, in terms of its radius, r<\/em>, and its height, h<\/em>, is given by [latex]V=\\frac{1}{3}\\pi {r}^{2}h[\/latex]. Express r<\/em>\u00a0in terms of h<\/em>\u00a0if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.\r\n\r\n66.\u00a0Consider a cone with height of 30 feet. Express the radius, r<\/em>, in terms of the volume, V<\/em>, and find the radius of a cone with volume of 1000 cubic feet.<\/p>","rendered":"

1. Explain why we cannot find inverse functions for all polynomial functions.<\/p>\n

2.\u00a0Why must we restrict the domain of a quadratic function when finding its inverse?<\/p>\n

3. When finding the inverse of a radical function, what restriction will we need to make?<\/p>\n

4.\u00a0The inverse of a quadratic function will always take what form?<\/p>\n

For the following exercises, find the inverse of the function on the given domain.<\/p>\n

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

6.\u00a0[latex]f\\left(x\\right)={\\left(x+2\\right)}^{2}, \\left[-2,\\infty \\right)[\/latex]<\/p>\n

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

8.\u00a0[latex]f\\left(x\\right)=2-\\sqrt{3+x}[\/latex]<\/p>\n

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

10.\u00a0[latex]f\\left(x\\right)=12-{x}^{2}, \\left[0,\\infty \\right)[\/latex]<\/p>\n

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

12.\u00a0[latex]f\\left(x\\right)=2{x}^{2}+4, \\left[0,\\infty \\right)[\/latex]<\/p>\n

For the following exercises, find the inverse of the functions.<\/p>\n

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

14.\u00a0[latex]f\\left(x\\right)=3{x}^{3}+1[\/latex]<\/p>\n

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

16.\u00a0[latex]f\\left(x\\right)=4 - 2{x}^{3}[\/latex]<\/p>\n

For the following exercises, find the inverse of the functions.<\/p>\n

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

18.\u00a0[latex]f\\left(x\\right)=\\sqrt{3 - 4x}[\/latex]<\/p>\n

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

20.\u00a0[latex]f\\left(x\\right)=\\sqrt{6x - 8}+5[\/latex]<\/p>\n

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

22.\u00a0[latex]f\\left(x\\right)=3-\\sqrt[3]{x}[\/latex]<\/p>\n

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

24.\u00a0[latex]f\\left(x\\right)=\\frac{3}{x - 4}[\/latex]<\/p>\n

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

26.\u00a0[latex]f\\left(x\\right)=\\frac{x - 2}{x+7}[\/latex]<\/p>\n

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

28.\u00a0[latex]f\\left(x\\right)=\\frac{5x+1}{2 - 5x}[\/latex]<\/p>\n

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

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

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

For the following exercises, find the inverse of the function and graph both the function and its inverse.<\/p>\n

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

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

34.\u00a0[latex]f\\left(x\\right)={\\left(x+3\\right)}^{2},x\\ge -3[\/latex]<\/p>\n

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

36.\u00a0[latex]f\\left(x\\right)={x}^{3}+3[\/latex]<\/p>\n

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

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

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

40.\u00a0[latex]f\\left(x\\right)=\\frac{2}{x}[\/latex]<\/p>\n

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

For the following exercises, use a graph to help determine the domain of the functions.<\/p>\n

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

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

44.\u00a0[latex]f\\left(x\\right)=\\sqrt{\\frac{x\\left(x+3\\right)}{x - 4}}[\/latex]<\/p>\n

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

46.\u00a0[latex]f\\left(x\\right)=\\sqrt{\\frac{9-{x}^{2}}{x+4}}[\/latex]<\/p>\n

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.<\/p>\n

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

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

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

50.\u00a0[latex]f\\left(x\\right)={x}^{3}+8x - 4, y=-1, 0, 1[\/latex]<\/p>\n

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

For the following exercises, find the inverse of the functions with a<\/em>, b<\/em>, c<\/em>\u00a0positive real numbers.<\/p>\n

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

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

54.\u00a0[latex]f\\left(x\\right)=\\sqrt{a{x}^{2}+b}[\/latex]<\/p>\n

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

56.\u00a0[latex]f\\left(x\\right)=\\frac{ax+b}{x+c}[\/latex]<\/p>\n

For the following exercises, determine the function described and then use it to answer the question.<\/p>\n

57. An object dropped from a height of 200 meters has a height, [latex]h\\left(t\\right)[\/latex], in meters after t<\/em>\u00a0seconds have lapsed, such that [latex]h\\left(t\\right)=200 - 4.9{t}^{2}[\/latex]. Express t<\/em>\u00a0as a function of height, h<\/em>, and find the time to reach a height of 50 meters.<\/p>\n

58.\u00a0An object dropped from a height of 600 feet has a height, [latex]h\\left(t\\right)[\/latex], in feet after t<\/em>\u00a0seconds have elapsed, such that [latex]h\\left(t\\right)=600 - 16{t}^{2}[\/latex]. Express t\u00a0<\/em>as a function of height h<\/em>, and find the time to reach a height of 400 feet.<\/p>\n

59. The volume, V<\/em>, of a sphere in terms of its radius, r<\/em>, is given by [latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]. Express r<\/em>\u00a0as a function of V<\/em>, and find the radius of a sphere with volume of 200 cubic feet.<\/p>\n

60.\u00a0The surface area, A<\/em>, of a sphere in terms of its radius, r<\/em>, is given by [latex]A\\left(r\\right)=4\\pi {r}^{2}[\/latex]. Express r<\/em>\u00a0as a function of V<\/em>, and find the radius of a sphere with a surface area of 1000 square inches.<\/p>\n

61. A container holds 100 ml of a solution that is 25 ml acid. If n<\/em>\u00a0ml 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<\/em>, as a function of the number of ml added, n<\/em>. Express n<\/em>\u00a0as a function of C<\/em>\u00a0and determine the number of mL that need to be added to have a solution that is 50% acid.<\/p>\n

62.\u00a0The period T<\/em>, in seconds, of a simple pendulum as a function of its length l<\/em>, in feet, is given by [latex]T\\left(l\\right)=2\\pi \\sqrt{\\frac{l}{32.2}}[\/latex]. Express l<\/em>\u00a0as a function of T<\/em>\u00a0and determine the length of a pendulum with period of 2 seconds.<\/p>\n

63. The volume of a cylinder, V<\/em>, in terms of radius, r<\/em>, and height, h<\/em>, 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<\/em>\u00a0and find the radius of a cylinder with volume of 300 cubic meters.<\/p>\n

64.\u00a0The surface area, A<\/em>, of a cylinder in terms of its radius, r<\/em>, and height, h<\/em>, 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<\/em>\u00a0and find the radius if the surface area is 200 square feet.<\/p>\n

65. The volume of a right circular cone, V<\/em>, in terms of its radius, r<\/em>, and its height, h<\/em>, is given by [latex]V=\\frac{1}{3}\\pi {r}^{2}h[\/latex]. Express r<\/em>\u00a0in terms of h<\/em>\u00a0if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.<\/p>\n

66.\u00a0Consider a cone with height of 30 feet. Express the radius, r<\/em>, in terms of the volume, V<\/em>, and find the radius of a cone with volume of 1000 cubic feet.<\/p>\n\n\t\t\t

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