{"id":919,"date":"2015-11-12T18:37:58","date_gmt":"2015-11-12T18:37:58","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=919"},"modified":"2017-03-31T20:20:50","modified_gmt":"2017-03-31T20:20:50","slug":"section-exercises-58","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/section-exercises-58\/","title":{"raw":"Section Exercises","rendered":"Section Exercises"},"content":{"raw":"<p>1. How does one find the domain of the quotient of two functions, [latex]\\frac{f}{g}?[\/latex]\r\n\r\n2. What is the composition of two functions, [latex]f\\circ g?[\/latex]\r\n\r\n3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.\r\n\r\n4. How do you find the domain for the composition of two functions, [latex]f\\circ g?[\/latex]\r\n\r\n5. Given [latex]f\\left(x\\right)={x}^{2}+2x\\text{ }[\/latex] and [latex] g\\left(x\\right)=6-{x}^{2}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.\r\n\r\n6. Given [latex]f\\left(x\\right)=-3{x}^{2}+x\\text{ }[\/latex] and [latex]\\text{ }g\\left(x\\right)=5[\/latex], find [latex]f+g,f-g,fg[\/latex], and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.\r\n\r\n7. Given [latex]f\\left(x\\right)=2{x}^{2}+4x\\text{ }[\/latex] and [latex]\\text{ }g\\left(x\\right)=\\frac{1}{2x}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.\r\n\r\n8. Given [latex]f\\left(x\\right)=\\frac{1}{x - 4}[\/latex] and [latex]g\\left(x\\right)=\\frac{1}{6-x}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.\r\n\r\n9. Given [latex]f\\left(x\\right)=3{x}^{2}[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x - 5}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.\r\n\r\n10. Given [latex]f\\left(x\\right)=\\sqrt{x}[\/latex] and [latex]g\\left(x\\right)=|x - 3|[\/latex], find [latex]\\frac{g}{f}[\/latex]. Determine the domain of the function in interval notation.\r\n\r\n11. Given [latex]f\\left(x\\right)=2{x}^{2}+1[\/latex] and [latex]g\\left(x\\right)=3x - 5[\/latex], find the following:\r\n<\/p><p style=\"padding-left: 60px;\">[latex]f\\left(g\\left(2\\right)\\right)[\/latex]\r\n[latex]f\\left(g\\left(x\\right)\\right)[\/latex]\r\n[latex]g\\left(f\\left(x\\right)\\right)[\/latex]\r\n[latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]\r\n[latex]\\left(f\\circ f\\right)\\left(-2\\right)[\/latex]<\/p>\r\nFor the following exercises, use each pair of functions to find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex]. Simplify your answers.\r\n\r\n12. [latex]f\\left(x\\right)={x}^{2}+1,g\\left(x\\right)=\\sqrt{x+2}[\/latex]\r\n\r\n13. [latex]f\\left(x\\right)=\\sqrt{x}+2,g\\left(x\\right)={x}^{2}+3[\/latex]\r\n\r\n14.\u00a0[latex]f\\left(x\\right)=|x|,g\\left(x\\right)=5x+1[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=\\sqrt[3]{x},g\\left(x\\right)=\\frac{x+1}{{x}^{3}}[\/latex]\r\n\r\n16.\u00a0[latex]f\\left(x\\right)=\\frac{1}{x - 6},g\\left(x\\right)=\\frac{7}{x}+6[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=\\frac{1}{x - 4},g\\left(x\\right)=\\frac{2}{x}+4[\/latex]\r\n\r\nFor the following exercises, use each set of functions to find [latex]f\\left(g\\left(h\\left(x\\right)\\right)\\right)[\/latex]. Simplify your answers.\r\n\r\n18. [latex]f\\left(x\\right)={x}^{4}+6[\/latex], [latex]g\\left(x\\right)=x - 6[\/latex], and [latex]h\\left(x\\right)=\\sqrt{x}[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)={x}^{2}+1[\/latex], [latex]g\\left(x\\right)=\\frac{1}{x}[\/latex], and [latex]h\\left(x\\right)=x+3[\/latex]\r\n\r\n20. Given [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] and [latex]g\\left(x\\right)=x - 3[\/latex], find the following:\r\n<p style=\"padding-left: 60px;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex]\r\nthe domain of [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex] in interval notation\r\n[latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]\r\nthe domain of [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]\r\n[latex]\\left(\\frac{f}{g}\\right)x[\/latex]<\/p>\r\n21. Given [latex]f\\left(x\\right)=\\sqrt{2 - 4x}[\/latex] and [latex]g\\left(x\\right)=-\\frac{3}{x}[\/latex], find the following:\r\n<p style=\"padding-left: 60px;\">a. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">b. the domain of [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex] in interval notation<\/p>\r\n22. Given the functions [latex]f\\left(x\\right)=\\frac{1-x}{x}\\text{and}g\\left(x\\right)=\\frac{1}{1+{x}^{2}}[\/latex],\u00a0find the following:\r\n<p style=\"padding-left: 60px;\">a. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">b. [latex]\\left(g\\circ f\\right)\\left(\\text{2}\\right)[\/latex]<\/p>\r\n23. Given functions [latex]p\\left(x\\right)=\\frac{1}{\\sqrt{x}}[\/latex] and [latex]m\\left(x\\right)={x}^{2}-4[\/latex], state the domain of each of the following functions using interval notation:\r\n<p style=\"padding-left: 60px;\">[latex]\\frac{p\\left(x\\right)}{m\\left(x\\right)}[\/latex]\r\n[latex]p\\left(m\\left(x\\right)\\right)[\/latex]\r\n[latex]m\\left(p\\left(x\\right)\\right)[\/latex]<\/p>\r\n24. Given functions [latex]q\\left(x\\right)=\\frac{1}{\\sqrt{x}}[\/latex] and [latex]h\\left(x\\right)={x}^{2}-9[\/latex], state the domain of each of the following functions using interval notation.\r\n<p style=\"padding-left: 60px;\">[latex]\\frac{q\\left(x\\right)}{h\\left(x\\right)}[\/latex]\r\n[latex]q\\left(h\\left(x\\right)\\right)[\/latex]\r\n[latex]h\\left(q\\left(x\\right)\\right)[\/latex]<\/p>\r\n25. For [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x - 1}[\/latex], write the domain of [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex] in interval notation.\r\n\r\nFor the following exercises, find functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] so the given function can be expressed as [latex]h\\left(x\\right)=f\\left(g\\left(x\\right)\\right)[\/latex].\r\n\r\n26. [latex]h\\left(x\\right)={\\left(x+2\\right)}^{2}[\/latex]\r\n\r\n27. [latex]h\\left(x\\right)={\\left(x - 5\\right)}^{3}[\/latex]\r\n\r\n28.\u00a0[latex]h\\left(x\\right)=\\frac{3}{x - 5}[\/latex]\r\n\r\n29. [latex]h\\left(x\\right)=\\frac{4}{{\\left(x+2\\right)}^{2}}[\/latex]\r\n\r\n30.\u00a0[latex]h\\left(x\\right)=4+\\sqrt[3]{x}[\/latex]\r\n\r\n31. [latex]h\\left(x\\right)=\\sqrt[3]{\\frac{1}{2x - 3}}[\/latex]\r\n\r\n32. [latex]h\\left(x\\right)=\\frac{1}{{\\left(3{x}^{2}-4\\right)}^{-3}}[\/latex]\r\n\r\n33. [latex]h\\left(x\\right)=\\sqrt[4]{\\frac{3x - 2}{x+5}}[\/latex]\r\n\r\n34.\u00a0[latex]h\\left(x\\right)={\\left(\\frac{8+{x}^{3}}{8-{x}^{3}}\\right)}^{4}[\/latex]\r\n\r\n35. [latex]h\\left(x\\right)=\\sqrt{2x+6}[\/latex]\r\n\r\n36.\u00a0[latex]h\\left(x\\right)={\\left(5x - 1\\right)}^{3}[\/latex]\r\n\r\n37. [latex]h\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex]\r\n\r\n38.\u00a0[latex]h\\left(x\\right)=\\left|{x}^{2}+7\\right|[\/latex]\r\n\r\n39. [latex]h\\left(x\\right)=\\frac{1}{{\\left(x - 2\\right)}^{3}}[\/latex]\r\n\r\n40.\u00a0[latex]h\\left(x\\right)={\\left(\\frac{1}{2x - 3}\\right)}^{2}[\/latex]\r\n\r\n41. [latex]h\\left(x\\right)=\\sqrt{\\frac{2x - 1}{3x+4}}[\/latex]\r\n<p id=\"fs-id1165135377148\">For the following exercises, use the graphs of [latex]f[\/latex]\u00a0and [latex]g[\/latex]\u00a0to evaluate the expressions.<span id=\"fs-id1165134263075\" data-type=\"media\" data-alt=\"Graph of a function.\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200749\/CNX_Precalc_Figure_01_04_201.jpg\" alt=\"Graph of a function.\" data-media-type=\"image\/jpg\"\/><\/span><span id=\"fs-id1165135496557\" data-type=\"media\" data-alt=\"Graph of a function.\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200751\/CNX_Precalc_Figure_01_04_202.jpg\" alt=\"Graph of a function.\" data-media-type=\"image\/jpg\"\/><\/span><\/p>\r\n\r\n<div id=\"fs-id1165137534543\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137534545\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137534547\">42. [latex]f\\left(g\\left(3\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137529964\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137529966\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137529968\">43. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135650744\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165135650746\">44. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134336712\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134336714\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137741078\">45. [latex]g\\left(f\\left(0\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137759779\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137759781\">46.\u00a0[latex]f\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137434226\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135696670\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135696673\">47. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137874891\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137874893\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137874895\">48. [latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135503742\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135503744\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135503746\">49. [latex]g\\left(g\\left(0\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165132944970\">For the following exercises, use graphs of [latex]f\\left(x\\right)[\/latex]\u00a0[latex]g\\left(x\\right)[\/latex], and [latex]h\\left(x\\right)[\/latex], to evaluate the expressions.<span id=\"fs-id1165137894613\" data-type=\"media\" data-alt=\"\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200752\/CNX_Precalc_Figure_01_04_203.jpg\" alt=\"\" data-media-type=\"image\/jpg\"\/><\/span><span id=\"fs-id1165134558626\" data-type=\"media\" data-alt=\"Graph of a parabola.\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200753\/CNX_Precalc_Figure_01_04_204.jpg\" alt=\"Graph of a parabola.\" data-media-type=\"image\/jpg\"\/><\/span><span id=\"fs-id1165135349122\" data-type=\"media\" data-alt=\"Graph of a square root function.\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200754\/CNX_Precalc_Figure_01_04_205.jpg\" alt=\"Graph of a square root function.\" data-media-type=\"image\/jpg\"\/><\/span><\/p>\r\n\r\n<div id=\"fs-id1165135703261\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137932616\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137932618\">50. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137401837\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135148970\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135148972\">51. [latex]g\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135641698\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165134169487\">52.\u00a0[latex]f\\left(g\\left(4\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135541712\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135541714\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135543250\">53. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137694050\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137579588\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137579590\">54. [latex]f\\left(h\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137438320\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134031228\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134031230\">55. [latex]h\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137666936\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137408697\">56.\u00a0[latex]f\\left(g\\left(h\\left(4\\right)\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134238765\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134238767\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134238769\">57. [latex]f\\left(g\\left(f\\left(-2\\right)\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137501367\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165135587789\">For the following exercises, use the function values for [latex]f\\text{ and }g[\/latex]\u00a0to evaluate each expression.<\/p>\r\n\r\n<table id=\"Table_01_04_03\" summary=\"Eleven columns and three rows. The first row is labeled,\"><tbody><tr><td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<\/tr><tr><td>0<\/td>\r\n<td>7<\/td>\r\n<td>9<\/td>\r\n<\/tr><tr><td>1<\/td>\r\n<td>6<\/td>\r\n<td>5<\/td>\r\n<\/tr><tr><td>2<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<\/tr><tr><td>3<\/td>\r\n<td>8<\/td>\r\n<td>2<\/td>\r\n<\/tr><tr><td>4<\/td>\r\n<td>4<\/td>\r\n<td>1<\/td>\r\n<\/tr><tr><td>5<\/td>\r\n<td>0<\/td>\r\n<td>8<\/td>\r\n<\/tr><tr><td>6<\/td>\r\n<td>2<\/td>\r\n<td>7<\/td>\r\n<\/tr><tr><td>7<\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<\/tr><tr><td>8<\/td>\r\n<td>9<\/td>\r\n<td>4<\/td>\r\n<\/tr><tr><td>9<\/td>\r\n<td>3<\/td>\r\n<td>0<\/td>\r\n<\/tr><\/tbody><\/table><div id=\"fs-id1165134226798\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134226800\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134226802\">58. [latex]f\\left(g\\left(8\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135298467\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135538768\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135538770\">59. [latex]f\\left(g\\left(5\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134061990\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165134495228\">60.\u00a0[latex]g\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137761923\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137761925\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137761927\">61. [latex]g\\left(f\\left(3\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137443134\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137443136\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137443138\">62. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134262482\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134262484\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134262486\">63. [latex]f\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135417784\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165135417786\">64.\u00a0[latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134199326\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134199328\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134199330\">65. [latex]g\\left(g\\left(6\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134190700\">For the following exercises, use the function values for [latex]f\\text{ and }g[\/latex] to evaluate the expressions.<\/p>\r\n\r\n<table id=\"Table_01_04_04\" summary=\"Three columns and eight rows. The first column is labeled,\"><tbody><tr><td data-align=\"left\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td data-align=\"left\"><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td data-align=\"center\"><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<\/tr><tr><td data-align=\"center\">-3<\/td>\r\n<td data-align=\"center\">11<\/td>\r\n<td data-align=\"center\">-8<\/td>\r\n<\/tr><tr><td data-align=\"center\">-2<\/td>\r\n<td data-align=\"center\">9<\/td>\r\n<td data-align=\"center\">-3<\/td>\r\n<\/tr><tr><td data-align=\"center\">-1<\/td>\r\n<td data-align=\"center\">7<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<\/tr><tr><td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<\/tr><tr><td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<\/tr><tr><td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">-3<\/td>\r\n<\/tr><tr><td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">-1<\/td>\r\n<td data-align=\"center\">-8<\/td>\r\n<\/tr><\/tbody><\/table><div id=\"fs-id1165135192528\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135188435\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135188438\">66. [latex]\\left(f\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135593131\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135593133\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165133334338\">67. [latex]\\left(f\\circ g\\right)\\left(2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165133047570\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165133047572\">68.\u00a0[latex]\\left(g\\circ f\\right)\\left(2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135369160\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137441811\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137441814\">69. [latex]\\left(g\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137427576\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137427578\">70.\u00a0[latex]\\left(g\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137400455\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137400457\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137400459\">71. [latex]\\left(f\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137626894\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137939470\">For the following exercises, use each pair of functions to find [latex]f\\left(g\\left(0\\right)\\right)[\/latex] and [latex]g\\left(f\\left(0\\right)\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135191363\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135191365\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135191367\">72. [latex]f\\left(x\\right)=4x+8,g\\left(x\\right)=7-{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135149221\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135149223\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134199478\">73. [latex]f\\left(x\\right)=5x+7,g\\left(x\\right)=4 - 2{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135657107\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165135657110\">74.\u00a0[latex]f\\left(x\\right)=\\sqrt{x+4},g\\left(x\\right)=12-{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137405114\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135187526\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135187529\">75. [latex]f\\left(x\\right)=\\frac{1}{x+2},g\\left(x\\right)=4x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137422427\">For the following exercises, use the functions [latex]f\\left(x\\right)=2{x}^{2}+1[\/latex] and [latex]g\\left(x\\right)=3x+5[\/latex]\u00a0to evaluate or find the composite function as indicated.<\/p>\r\n\r\n<div id=\"fs-id1165137464556\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134558023\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134558025\">76. [latex]f\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135309916\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135309918\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135169383\">77. [latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137480928\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137480930\">78.\u00a0[latex]g\\left(f\\left(-3\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137761622\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137761624\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137761626\">79. [latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137758394\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137554206\">For the following exercises, use [latex]f\\left(x\\right)={x}^{3}+1[\/latex] and [latex]g\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165135253759\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135253761\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135253763\">80. Find [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex] and [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]. Compare the two answers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137456144\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137647741\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137647743\">81. Find [latex]\\left(f\\circ g\\right)\\left(2\\right)[\/latex] and [latex]\\left(g\\circ f\\right)\\left(2\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135198523\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165135198525\">82.\u00a0What is the domain of [latex]\\left(g\\circ f\\right)\\left(x\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165131891789\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165131891791\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137414486\">83. What is the domain of [latex]\\left(f\\circ g\\right)\\left(x\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165133190973\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137939800\">84.\u00a0Let [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135657099\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135657101\" class=\"problem\" data-type=\"problem\">\r\n<p style=\"padding-left: 60px;\">a. Find [latex]\\left(f\\circ f\\right)\\left(x\\right)[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">b. Is [latex]\\left(f\\circ f\\right)\\left(x\\right)[\/latex] for any function [latex]f[\/latex] the same result as the answer to part (a) for any function? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137432638\">For the following exercises, let [latex]F\\left(x\\right)={\\left(x+1\\right)}^{5}[\/latex], [latex]f\\left(x\\right)={x}^{5}[\/latex], and [latex]g\\left(x\\right)=x+1[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165133280659\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135194278\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165135194281\">85. True or False: [latex]\\left(g\\circ f\\right)\\left(x\\right)=F\\left(x\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137837867\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137837869\">86.\u00a0True or False: [latex]\\left(f\\circ g\\right)\\left(x\\right)=F\\left(x\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135261458\">For the following exercises, find the composition when [latex]f\\left(x\\right)={x}^{2}+2[\/latex] for all [latex]x\\ge 0[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x - 2}[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165134557320\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134481985\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134481987\">87. [latex]\\left(f\\circ g\\right)\\left(6\\right);\\left(g\\circ f\\right)\\left(6\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137939500\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137939502\">88.\u00a0[latex]\\left(g\\circ f\\right)\\left(a\\right);\\left(f\\circ g\\right)\\left(a\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137887392\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137887394\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165137887396\">89. [latex]\\left(f\\circ g\\right)\\left(11\\right);\\left(g\\circ f\\right)\\left(11\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137823068\" class=\"solution\" data-type=\"solution\">\r\n\r\n90.\u00a0The function [latex]D\\left(p\\right)[\/latex] gives the number of items that will be demanded when the price is [latex]p[\/latex]. The production cost [latex]C\\left(x\\right)[\/latex] is the cost of producing [latex]x[\/latex] items. To determine the cost of production when the price is $6, you would do which of the following?\r\n<div id=\"fs-id1165135314808\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135314810\" class=\"problem\" data-type=\"problem\">\r\n<p style=\"padding-left: 60px;\">a. Evaluate [latex]D\\left(C\\left(6\\right)\\right)[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">b. Evaluate [latex]C\\left(D\\left(6\\right)\\right)[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">c. Solve [latex]D\\left(C\\left(x\\right)\\right)=6[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">d. Solve [latex]C\\left(D\\left(p\\right)\\right)=6[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137563586\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137563588\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165133249135\">91. The function [latex]A\\left(d\\right)[\/latex] gives the pain level on a scale of 0 to 10 experienced by a patient with [latex]d[\/latex] milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient\u2019s system after [latex]t[\/latex] minutes is modeled by [latex]m\\left(t\\right)[\/latex]. Which of the following would you do in order to determine when the patient will be at a pain level of 4?<\/p>\r\n<p style=\"padding-left: 60px;\">a. Evaluate [latex]A\\left(m\\left(4\\right)\\right)[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">b. Evaluate [latex]m\\left(A\\left(4\\right)\\right)[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">c. Solve [latex]A\\left(m\\left(t\\right)\\right)=4[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">d. Solve [latex]m\\left(A\\left(d\\right)\\right)=4[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137419674\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165137419676\">92.\u00a0A store offers customers a 30% discount on the price [latex]x[\/latex] of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function [latex]P\\left(x\\right)[\/latex] that computes the final price of the item in terms of the original price [latex]x[\/latex]. (Hint: Use function composition to find your answer.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134492355\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134492358\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134492360\">93. A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to [latex]r\\left(t\\right)=25\\sqrt{t+2}[\/latex], find the area of the ripple as a function of time. Find the area of the ripple at [latex]t=2[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135530544\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165135530546\">94.\u00a0A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula [latex]r\\left(t\\right)=2t+1[\/latex], express the area burned as a function of time, [latex]t[\/latex] (minutes).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135431042\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135431044\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165134031324\">95. Use the function you found in the previous exercise to find the total area burned after 5 minutes.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134031328\" class=\"solution\" data-type=\"solution\">\r\n<p id=\"fs-id1165134031330\">96.\u00a0The radius [latex]r[\/latex], in inches, of a spherical balloon is related to the volume, [latex]V[\/latex], by [latex]r\\left(V\\right)=\\sqrt[3]{\\frac{3V}{4\\pi }}[\/latex]. Air is pumped into the balloon, so the volume after [latex]t[\/latex] seconds is given by [latex]V\\left(t\\right)=10+20t[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134480894\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134480896\" class=\"problem\" data-type=\"problem\">\r\n<p style=\"padding-left: 60px;\">a. Find the composite function [latex]r\\left(V\\left(t\\right)\\right)[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">b. Find the <em data-effect=\"italics\">exact<\/em> time when the radius reaches 10 inches.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133362937\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165133362939\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"fs-id1165133362941\">97. The number of bacteria in a refrigerated food product is given by [latex]N\\left(T\\right)=23{T}^{2}-56T+1[\/latex], [latex]3&lt;T&lt;33[\/latex], where [latex]T[\/latex] is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by [latex]T\\left(t\\right)=5t+1.5[\/latex], where [latex]t[\/latex] is the time in hours.<\/p>\r\n<p style=\"padding-left: 60px;\">a. Find the composite function [latex]N\\left(T\\left(t\\right)\\right)[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">b. Find the time (round to two decimal places) when the bacteria count reaches 6752.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<p>1. How does one find the domain of the quotient of two functions, [latex]\\frac{f}{g}?[\/latex]<\/p>\n<p>2. What is the composition of two functions, [latex]f\\circ g?[\/latex]<\/p>\n<p>3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.<\/p>\n<p>4. How do you find the domain for the composition of two functions, [latex]f\\circ g?[\/latex]<\/p>\n<p>5. Given [latex]f\\left(x\\right)={x}^{2}+2x\\text{ }[\/latex] and [latex]g\\left(x\\right)=6-{x}^{2}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.<\/p>\n<p>6. Given [latex]f\\left(x\\right)=-3{x}^{2}+x\\text{ }[\/latex] and [latex]\\text{ }g\\left(x\\right)=5[\/latex], find [latex]f+g,f-g,fg[\/latex], and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.<\/p>\n<p>7. Given [latex]f\\left(x\\right)=2{x}^{2}+4x\\text{ }[\/latex] and [latex]\\text{ }g\\left(x\\right)=\\frac{1}{2x}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.<\/p>\n<p>8. Given [latex]f\\left(x\\right)=\\frac{1}{x - 4}[\/latex] and [latex]g\\left(x\\right)=\\frac{1}{6-x}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.<\/p>\n<p>9. Given [latex]f\\left(x\\right)=3{x}^{2}[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x - 5}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.<\/p>\n<p>10. Given [latex]f\\left(x\\right)=\\sqrt{x}[\/latex] and [latex]g\\left(x\\right)=|x - 3|[\/latex], find [latex]\\frac{g}{f}[\/latex]. Determine the domain of the function in interval notation.<\/p>\n<p>11. Given [latex]f\\left(x\\right)=2{x}^{2}+1[\/latex] and [latex]g\\left(x\\right)=3x - 5[\/latex], find the following:\n<\/p>\n<p style=\"padding-left: 60px;\">[latex]f\\left(g\\left(2\\right)\\right)[\/latex]<br \/>\n[latex]f\\left(g\\left(x\\right)\\right)[\/latex]<br \/>\n[latex]g\\left(f\\left(x\\right)\\right)[\/latex]<br \/>\n[latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]<br \/>\n[latex]\\left(f\\circ f\\right)\\left(-2\\right)[\/latex]<\/p>\n<p>For the following exercises, use each pair of functions to find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex]. Simplify your answers.<\/p>\n<p>12. [latex]f\\left(x\\right)={x}^{2}+1,g\\left(x\\right)=\\sqrt{x+2}[\/latex]<\/p>\n<p>13. [latex]f\\left(x\\right)=\\sqrt{x}+2,g\\left(x\\right)={x}^{2}+3[\/latex]<\/p>\n<p>14.\u00a0[latex]f\\left(x\\right)=|x|,g\\left(x\\right)=5x+1[\/latex]<\/p>\n<p>15. [latex]f\\left(x\\right)=\\sqrt[3]{x},g\\left(x\\right)=\\frac{x+1}{{x}^{3}}[\/latex]<\/p>\n<p>16.\u00a0[latex]f\\left(x\\right)=\\frac{1}{x - 6},g\\left(x\\right)=\\frac{7}{x}+6[\/latex]<\/p>\n<p>17. [latex]f\\left(x\\right)=\\frac{1}{x - 4},g\\left(x\\right)=\\frac{2}{x}+4[\/latex]<\/p>\n<p>For the following exercises, use each set of functions to find [latex]f\\left(g\\left(h\\left(x\\right)\\right)\\right)[\/latex]. Simplify your answers.<\/p>\n<p>18. [latex]f\\left(x\\right)={x}^{4}+6[\/latex], [latex]g\\left(x\\right)=x - 6[\/latex], and [latex]h\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<p>19. [latex]f\\left(x\\right)={x}^{2}+1[\/latex], [latex]g\\left(x\\right)=\\frac{1}{x}[\/latex], and [latex]h\\left(x\\right)=x+3[\/latex]<\/p>\n<p>20. Given [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] and [latex]g\\left(x\\right)=x - 3[\/latex], find the following:<\/p>\n<p style=\"padding-left: 60px;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex]<br \/>\nthe domain of [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex] in interval notation<br \/>\n[latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<br \/>\nthe domain of [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<br \/>\n[latex]\\left(\\frac{f}{g}\\right)x[\/latex]<\/p>\n<p>21. Given [latex]f\\left(x\\right)=\\sqrt{2 - 4x}[\/latex] and [latex]g\\left(x\\right)=-\\frac{3}{x}[\/latex], find the following:<\/p>\n<p style=\"padding-left: 60px;\">a. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">b. the domain of [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex] in interval notation<\/p>\n<p>22. Given the functions [latex]f\\left(x\\right)=\\frac{1-x}{x}\\text{and}g\\left(x\\right)=\\frac{1}{1+{x}^{2}}[\/latex],\u00a0find the following:<\/p>\n<p style=\"padding-left: 60px;\">a. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">b. [latex]\\left(g\\circ f\\right)\\left(\\text{2}\\right)[\/latex]<\/p>\n<p>23. Given functions [latex]p\\left(x\\right)=\\frac{1}{\\sqrt{x}}[\/latex] and [latex]m\\left(x\\right)={x}^{2}-4[\/latex], state the domain of each of the following functions using interval notation:<\/p>\n<p style=\"padding-left: 60px;\">[latex]\\frac{p\\left(x\\right)}{m\\left(x\\right)}[\/latex]<br \/>\n[latex]p\\left(m\\left(x\\right)\\right)[\/latex]<br \/>\n[latex]m\\left(p\\left(x\\right)\\right)[\/latex]<\/p>\n<p>24. Given functions [latex]q\\left(x\\right)=\\frac{1}{\\sqrt{x}}[\/latex] and [latex]h\\left(x\\right)={x}^{2}-9[\/latex], state the domain of each of the following functions using interval notation.<\/p>\n<p style=\"padding-left: 60px;\">[latex]\\frac{q\\left(x\\right)}{h\\left(x\\right)}[\/latex]<br \/>\n[latex]q\\left(h\\left(x\\right)\\right)[\/latex]<br \/>\n[latex]h\\left(q\\left(x\\right)\\right)[\/latex]<\/p>\n<p>25. For [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x - 1}[\/latex], write the domain of [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex] in interval notation.<\/p>\n<p>For the following exercises, find functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] so the given function can be expressed as [latex]h\\left(x\\right)=f\\left(g\\left(x\\right)\\right)[\/latex].<\/p>\n<p>26. [latex]h\\left(x\\right)={\\left(x+2\\right)}^{2}[\/latex]<\/p>\n<p>27. [latex]h\\left(x\\right)={\\left(x - 5\\right)}^{3}[\/latex]<\/p>\n<p>28.\u00a0[latex]h\\left(x\\right)=\\frac{3}{x - 5}[\/latex]<\/p>\n<p>29. [latex]h\\left(x\\right)=\\frac{4}{{\\left(x+2\\right)}^{2}}[\/latex]<\/p>\n<p>30.\u00a0[latex]h\\left(x\\right)=4+\\sqrt[3]{x}[\/latex]<\/p>\n<p>31. [latex]h\\left(x\\right)=\\sqrt[3]{\\frac{1}{2x - 3}}[\/latex]<\/p>\n<p>32. [latex]h\\left(x\\right)=\\frac{1}{{\\left(3{x}^{2}-4\\right)}^{-3}}[\/latex]<\/p>\n<p>33. [latex]h\\left(x\\right)=\\sqrt[4]{\\frac{3x - 2}{x+5}}[\/latex]<\/p>\n<p>34.\u00a0[latex]h\\left(x\\right)={\\left(\\frac{8+{x}^{3}}{8-{x}^{3}}\\right)}^{4}[\/latex]<\/p>\n<p>35. [latex]h\\left(x\\right)=\\sqrt{2x+6}[\/latex]<\/p>\n<p>36.\u00a0[latex]h\\left(x\\right)={\\left(5x - 1\\right)}^{3}[\/latex]<\/p>\n<p>37. [latex]h\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex]<\/p>\n<p>38.\u00a0[latex]h\\left(x\\right)=\\left|{x}^{2}+7\\right|[\/latex]<\/p>\n<p>39. [latex]h\\left(x\\right)=\\frac{1}{{\\left(x - 2\\right)}^{3}}[\/latex]<\/p>\n<p>40.\u00a0[latex]h\\left(x\\right)={\\left(\\frac{1}{2x - 3}\\right)}^{2}[\/latex]<\/p>\n<p>41. [latex]h\\left(x\\right)=\\sqrt{\\frac{2x - 1}{3x+4}}[\/latex]<\/p>\n<p id=\"fs-id1165135377148\">For the following exercises, use the graphs of [latex]f[\/latex]\u00a0and [latex]g[\/latex]\u00a0to evaluate the expressions.<span id=\"fs-id1165134263075\" data-type=\"media\" data-alt=\"Graph of a function.\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200749\/CNX_Precalc_Figure_01_04_201.jpg\" alt=\"Graph of a function.\" data-media-type=\"image\/jpg\" \/><\/span><span id=\"fs-id1165135496557\" data-type=\"media\" data-alt=\"Graph of a function.\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200751\/CNX_Precalc_Figure_01_04_202.jpg\" alt=\"Graph of a function.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<div id=\"fs-id1165137534543\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137534545\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137534547\">42. [latex]f\\left(g\\left(3\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137529964\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137529966\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137529968\">43. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135650744\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165135650746\">44. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134336712\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134336714\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137741078\">45. [latex]g\\left(f\\left(0\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137759779\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137759781\">46.\u00a0[latex]f\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137434226\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135696670\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135696673\">47. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874891\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137874893\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137874895\">48. [latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503742\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135503744\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135503746\">49. [latex]g\\left(g\\left(0\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165132944970\">For the following exercises, use graphs of [latex]f\\left(x\\right)[\/latex]\u00a0[latex]g\\left(x\\right)[\/latex], and [latex]h\\left(x\\right)[\/latex], to evaluate the expressions.<span id=\"fs-id1165137894613\" data-type=\"media\" data-alt=\"\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200752\/CNX_Precalc_Figure_01_04_203.jpg\" alt=\"\" data-media-type=\"image\/jpg\" \/><\/span><span id=\"fs-id1165134558626\" data-type=\"media\" data-alt=\"Graph of a parabola.\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200753\/CNX_Precalc_Figure_01_04_204.jpg\" alt=\"Graph of a parabola.\" data-media-type=\"image\/jpg\" \/><\/span><span id=\"fs-id1165135349122\" data-type=\"media\" data-alt=\"Graph of a square root function.\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200754\/CNX_Precalc_Figure_01_04_205.jpg\" alt=\"Graph of a square root function.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<div id=\"fs-id1165135703261\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137932616\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137932618\">50. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137401837\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135148970\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135148972\">51. [latex]g\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135641698\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165134169487\">52.\u00a0[latex]f\\left(g\\left(4\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541712\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135541714\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135543250\">53. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137694050\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137579588\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137579590\">54. [latex]f\\left(h\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438320\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134031228\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134031230\">55. [latex]h\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137666936\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137408697\">56.\u00a0[latex]f\\left(g\\left(h\\left(4\\right)\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134238765\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134238767\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134238769\">57. [latex]f\\left(g\\left(f\\left(-2\\right)\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137501367\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165135587789\">For the following exercises, use the function values for [latex]f\\text{ and }g[\/latex]\u00a0to evaluate each expression.<\/p>\n<table id=\"Table_01_04_03\" summary=\"Eleven columns and three rows. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>7<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>6<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>8<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>4<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>0<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>2<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>1<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>9<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>3<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165134226798\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134226800\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134226802\">58. [latex]f\\left(g\\left(8\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135298467\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135538768\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135538770\">59. [latex]f\\left(g\\left(5\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134061990\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165134495228\">60.\u00a0[latex]g\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137761923\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137761925\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137761927\">61. [latex]g\\left(f\\left(3\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137443134\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137443136\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137443138\">62. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134262482\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134262484\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134262486\">63. [latex]f\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135417784\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165135417786\">64.\u00a0[latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134199326\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134199328\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134199330\">65. [latex]g\\left(g\\left(6\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134190700\">For the following exercises, use the function values for [latex]f\\text{ and }g[\/latex] to evaluate the expressions.<\/p>\n<table id=\"Table_01_04_04\" summary=\"Three columns and eight rows. The first column is labeled,\">\n<tbody>\n<tr>\n<td data-align=\"left\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td data-align=\"left\"><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td data-align=\"center\"><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">-3<\/td>\n<td data-align=\"center\">11<\/td>\n<td data-align=\"center\">-8<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">-2<\/td>\n<td data-align=\"center\">9<\/td>\n<td data-align=\"center\">-3<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">-1<\/td>\n<td data-align=\"center\">7<\/td>\n<td data-align=\"center\">0<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">1<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">0<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">-3<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">-1<\/td>\n<td data-align=\"center\">-8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165135192528\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135188435\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135188438\">66. [latex]\\left(f\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593131\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135593133\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165133334338\">67. [latex]\\left(f\\circ g\\right)\\left(2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133047570\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165133047572\">68.\u00a0[latex]\\left(g\\circ f\\right)\\left(2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135369160\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137441811\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137441814\">69. [latex]\\left(g\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137427576\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137427578\">70.\u00a0[latex]\\left(g\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137400455\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137400457\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137400459\">71. [latex]\\left(f\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137626894\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137939470\">For the following exercises, use each pair of functions to find [latex]f\\left(g\\left(0\\right)\\right)[\/latex] and [latex]g\\left(f\\left(0\\right)\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191363\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135191365\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135191367\">72. [latex]f\\left(x\\right)=4x+8,g\\left(x\\right)=7-{x}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135149221\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135149223\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134199478\">73. [latex]f\\left(x\\right)=5x+7,g\\left(x\\right)=4 - 2{x}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135657107\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165135657110\">74.\u00a0[latex]f\\left(x\\right)=\\sqrt{x+4},g\\left(x\\right)=12-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137405114\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135187526\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135187529\">75. [latex]f\\left(x\\right)=\\frac{1}{x+2},g\\left(x\\right)=4x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137422427\">For the following exercises, use the functions [latex]f\\left(x\\right)=2{x}^{2}+1[\/latex] and [latex]g\\left(x\\right)=3x+5[\/latex]\u00a0to evaluate or find the composite function as indicated.<\/p>\n<div id=\"fs-id1165137464556\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134558023\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134558025\">76. [latex]f\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135309916\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135309918\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135169383\">77. [latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137480928\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137480930\">78.\u00a0[latex]g\\left(f\\left(-3\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137761622\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137761624\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137761626\">79. [latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137758394\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137554206\">For the following exercises, use [latex]f\\left(x\\right)={x}^{3}+1[\/latex] and [latex]g\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex].<\/p>\n<div id=\"fs-id1165135253759\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135253761\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135253763\">80. Find [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex] and [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]. Compare the two answers.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137456144\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137647741\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137647743\">81. Find [latex]\\left(f\\circ g\\right)\\left(2\\right)[\/latex] and [latex]\\left(g\\circ f\\right)\\left(2\\right)[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135198523\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165135198525\">82.\u00a0What is the domain of [latex]\\left(g\\circ f\\right)\\left(x\\right)?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131891789\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165131891791\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137414486\">83. What is the domain of [latex]\\left(f\\circ g\\right)\\left(x\\right)?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133190973\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137939800\">84.\u00a0Let [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135657099\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135657101\" class=\"problem\" data-type=\"problem\">\n<p style=\"padding-left: 60px;\">a. Find [latex]\\left(f\\circ f\\right)\\left(x\\right)[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">b. Is [latex]\\left(f\\circ f\\right)\\left(x\\right)[\/latex] for any function [latex]f[\/latex] the same result as the answer to part (a) for any function? Explain.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137432638\">For the following exercises, let [latex]F\\left(x\\right)={\\left(x+1\\right)}^{5}[\/latex], [latex]f\\left(x\\right)={x}^{5}[\/latex], and [latex]g\\left(x\\right)=x+1[\/latex].<\/p>\n<div id=\"fs-id1165133280659\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135194278\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165135194281\">85. True or False: [latex]\\left(g\\circ f\\right)\\left(x\\right)=F\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165137837867\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137837869\">86.\u00a0True or False: [latex]\\left(f\\circ g\\right)\\left(x\\right)=F\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135261458\">For the following exercises, find the composition when [latex]f\\left(x\\right)={x}^{2}+2[\/latex] for all [latex]x\\ge 0[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x - 2}[\/latex].<\/p>\n<div id=\"fs-id1165134557320\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134481985\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134481987\">87. [latex]\\left(f\\circ g\\right)\\left(6\\right);\\left(g\\circ f\\right)\\left(6\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137939500\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137939502\">88.\u00a0[latex]\\left(g\\circ f\\right)\\left(a\\right);\\left(f\\circ g\\right)\\left(a\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137887392\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137887394\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165137887396\">89. [latex]\\left(f\\circ g\\right)\\left(11\\right);\\left(g\\circ f\\right)\\left(11\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137823068\" class=\"solution\" data-type=\"solution\">\n<p>90.\u00a0The function [latex]D\\left(p\\right)[\/latex] gives the number of items that will be demanded when the price is [latex]p[\/latex]. The production cost [latex]C\\left(x\\right)[\/latex] is the cost of producing [latex]x[\/latex] items. To determine the cost of production when the price is $6, you would do which of the following?<\/p>\n<div id=\"fs-id1165135314808\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135314810\" class=\"problem\" data-type=\"problem\">\n<p style=\"padding-left: 60px;\">a. Evaluate [latex]D\\left(C\\left(6\\right)\\right)[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">b. Evaluate [latex]C\\left(D\\left(6\\right)\\right)[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">c. Solve [latex]D\\left(C\\left(x\\right)\\right)=6[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">d. Solve [latex]C\\left(D\\left(p\\right)\\right)=6[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137563586\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137563588\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165133249135\">91. The function [latex]A\\left(d\\right)[\/latex] gives the pain level on a scale of 0 to 10 experienced by a patient with [latex]d[\/latex] milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient\u2019s system after [latex]t[\/latex] minutes is modeled by [latex]m\\left(t\\right)[\/latex]. Which of the following would you do in order to determine when the patient will be at a pain level of 4?<\/p>\n<p style=\"padding-left: 60px;\">a. Evaluate [latex]A\\left(m\\left(4\\right)\\right)[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">b. Evaluate [latex]m\\left(A\\left(4\\right)\\right)[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">c. Solve [latex]A\\left(m\\left(t\\right)\\right)=4[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">d. Solve [latex]m\\left(A\\left(d\\right)\\right)=4[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165137419674\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165137419676\">92.\u00a0A store offers customers a 30% discount on the price [latex]x[\/latex] of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function [latex]P\\left(x\\right)[\/latex] that computes the final price of the item in terms of the original price [latex]x[\/latex]. (Hint: Use function composition to find your answer.)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134492355\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134492358\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134492360\">93. A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to [latex]r\\left(t\\right)=25\\sqrt{t+2}[\/latex], find the area of the ripple as a function of time. Find the area of the ripple at [latex]t=2[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135530544\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165135530546\">94.\u00a0A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula [latex]r\\left(t\\right)=2t+1[\/latex], express the area burned as a function of time, [latex]t[\/latex] (minutes).<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135431042\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135431044\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165134031324\">95. Use the function you found in the previous exercise to find the total area burned after 5 minutes.<\/p>\n<\/div>\n<div id=\"fs-id1165134031328\" class=\"solution\" data-type=\"solution\">\n<p id=\"fs-id1165134031330\">96.\u00a0The radius [latex]r[\/latex], in inches, of a spherical balloon is related to the volume, [latex]V[\/latex], by [latex]r\\left(V\\right)=\\sqrt[3]{\\frac{3V}{4\\pi }}[\/latex]. Air is pumped into the balloon, so the volume after [latex]t[\/latex] seconds is given by [latex]V\\left(t\\right)=10+20t[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134480894\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134480896\" class=\"problem\" data-type=\"problem\">\n<p style=\"padding-left: 60px;\">a. Find the composite function [latex]r\\left(V\\left(t\\right)\\right)[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">b. Find the <em data-effect=\"italics\">exact<\/em> time when the radius reaches 10 inches.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133362937\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165133362939\" class=\"problem\" data-type=\"problem\">\n<p id=\"fs-id1165133362941\">97. The number of bacteria in a refrigerated food product is given by [latex]N\\left(T\\right)=23{T}^{2}-56T+1[\/latex], [latex]3<T<33[\/latex], where [latex]T[\/latex] is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by [latex]T\\left(t\\right)=5t+1.5[\/latex], where [latex]t[\/latex] is the time in hours.<\/p>\n<p style=\"padding-left: 60px;\">a. Find the composite function [latex]N\\left(T\\left(t\\right)\\right)[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">b. Find the time (round to two decimal places) when the bacteria count reaches 6752.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-919\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-919","chapter","type-chapter","status-publish","hentry"],"part":901,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/919","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/919\/revisions"}],"predecessor-version":[{"id":2806,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/919\/revisions\/2806"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/901"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/919\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=919"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=919"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=919"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}