{"id":3903,"date":"2024-09-12T16:58:06","date_gmt":"2024-09-12T16:58:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=3903"},"modified":"2025-08-13T23:17:29","modified_gmt":"2025-08-13T23:17:29","slug":"algebraic-operations-on-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/algebraic-operations-on-functions-get-stronger\/","title":{"raw":"Algebraic Operations on Functions: Get Stronger","rendered":"Algebraic Operations on Functions: Get Stronger"},"content":{"raw":"<h2><span data-sheets-root=\"1\">Combinations and Compositions of Functions<\/span><\/h2>\r\nFor the following exercises, determine the domain for each function in interval notation.\r\n<ol>\r\n \t<li>Given [latex]f(x) = x^2 + 2x[\/latex] and [latex]g(x) = 6 - x^2[\/latex], find [latex]f + g[\/latex], [latex]f - g[\/latex], [latex]fg[\/latex], and [latex]\\dfrac{f}{g}[\/latex].<\/li>\r\n \t<li>Given [latex]f(x) = 2x^2 + 4x[\/latex] and [latex]g(x) = \\dfrac{1}{2x}[\/latex], find [latex]f + g[\/latex], [latex]f - g[\/latex], [latex]fg[\/latex], and [latex]\\dfrac{f}{g}[\/latex].<\/li>\r\n \t<li>Given [latex]f(x) = 3x^2[\/latex] and [latex]g(x) = \\sqrt{x - 5}[\/latex], find [latex]f + g[\/latex], [latex]f - g[\/latex], [latex]fg[\/latex], and [latex]\\dfrac{f}{g}[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercise, find the indicated function given [latex]f(x) = 2x^2 + 1[\/latex] and [latex]g(x) = 3x - 5[\/latex].\r\n<ol start=\"4\">\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>[latex]f(g(2))[\/latex]<\/li>\r\n \t<li>[latex]f(g(x))[\/latex]<\/li>\r\n \t<li>[latex]g(f(x))[\/latex]<\/li>\r\n \t<li>[latex](g \\circ g)(x)[\/latex]<\/li>\r\n \t<li>[latex](f \\circ f)(-2)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use each pair of functions to find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex]. Simplify your answers.\r\n<ol start=\"10\">\r\n \t<li>Given [latex]f(x) = \\sqrt{x} + 2[\/latex] and [latex]g(x) = x^2 + 3[\/latex], find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex].<\/li>\r\n \t<li>Given [latex]f(x) = \\sqrt[3]{x}[\/latex] and [latex]g(x) = \\dfrac{x + 1}{x^3}[\/latex], find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex].<\/li>\r\n \t<li>Given [latex]f(x) = \\dfrac{1}{x - 4}[\/latex] and [latex]g(x) = \\dfrac{2}{x} + 4[\/latex], find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercises, use each set of functions to find [latex]f(g(h(x)))[\/latex]. Simplify your answers.\r\n<ol start=\"13\">\r\n \t<li>Given [latex]f(x) = x^2 + 1[\/latex], [latex]g(x) = \\dfrac{1}{x}[\/latex], and [latex]h(x) = x + 3[\/latex], find [latex]f(g(h(x)))[\/latex].<\/li>\r\n \t<li>Given [latex]f(x) = \\sqrt{2 - 4x}[\/latex] and [latex]g(x) = -\\dfrac{3}{x}[\/latex], find the following:\r\n<ol type=\"a\">\r\n \t<li>[latex](g \\circ f)(x)[\/latex]<\/li>\r\n \t<li>The domain of [latex](g \\circ f)(x)[\/latex] in interval notation<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Given the functions [latex]p(x) = \\dfrac{1}{\\sqrt{x}}[\/latex] and [latex]m(x) = x^2 - 4 [\/latex], state the domain of each of the following functions using interval notation:\r\n<ol type=\"a\">\r\n \t<li>[latex]\\dfrac{p(x)}{m(x)}[\/latex]<\/li>\r\n \t<li>[latex]p(m(x))[\/latex]<\/li>\r\n \t<li>[latex]m(p(x))[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>For [latex]f(x) = \\dfrac{1}{x}[\/latex] and [latex]g(x) = \\sqrt{x - 1}[\/latex], write the domain of [latex](f \\circ g)(x)[\/latex] in interval notation.<\/li>\r\n<\/ol>\r\nFor the following exercises, find functions [latex]f(x)[\/latex] and [latex] g(x)[\/latex] so the given function can be expressed as [latex]h(x) = f(g(x))[\/latex].\r\n<ol start=\"22\">\r\n \t<li>[latex]h(x) = (x - 5)^3[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\dfrac{4}{(x+2)^2}[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\sqrt[3]{\\dfrac{1}{2x - 3}}[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\sqrt[4]{\\dfrac{3x - 2}{x + 5}}[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\sqrt{2x + 6}[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\sqrt[3]{x - 1}[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\dfrac{1}{(x - 2)^3}[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\sqrt{\\dfrac{2x-1}{3x+4}}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graphs of [latex]f[\/latex], shown in the first graph, and [latex]g[\/latex], shown in the second graph, to evaluate the expressions.\r\n<img class=\"alignnone size-medium wp-image-6111\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190149\/Algebraic_Operations_1-300x173.jpeg\" alt=\"\" width=\"300\" height=\"173\" \/>\r\n<img class=\"alignnone size-medium wp-image-6113\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190201\/Algebraic_Operations_2-300x173.jpeg\" alt=\"\" width=\"300\" height=\"173\" \/>\r\n<ol start=\"30\">\r\n \t<li>[latex]f(g(1))[\/latex]<\/li>\r\n \t<li>[latex]g(f(0))[\/latex]<\/li>\r\n \t<li>[latex]f(f(4))[\/latex]<\/li>\r\n \t<li>[latex]g(g(0))[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use graphs of [latex]f(x)[\/latex], [latex]g(x)[\/latex], and [latex]h(x)[\/latex], to evaluate the expressions.\r\n<img class=\"alignnone size-medium wp-image-6114\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190559\/Algebraic_Operations_3-294x300.jpeg\" alt=\"\" width=\"294\" height=\"300\" \/>\r\n<img class=\"alignnone size-full wp-image-6115\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190600\/Algebraic_Operations_4.jpeg\" alt=\"\" width=\"244\" height=\"250\" \/>\r\n<img class=\"alignnone size-medium wp-image-6116\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190601\/Algebraic_Operations_5-300x176.jpeg\" alt=\"\" width=\"300\" height=\"176\" \/>\r\n<ol start=\"34\">\r\n \t<li>[latex]g(f(2))[\/latex]<\/li>\r\n \t<li>[latex]f(g(1))[\/latex]<\/li>\r\n \t<li>[latex]h(f(2))[\/latex]<\/li>\r\n \t<li>[latex]f(g(f(-2)))[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the function values for [latex]f[\/latex] and [latex]g[\/latex] shown in the table below to evaluate each expression.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th>[latex]g(x)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"38\">\r\n \t<li>[latex]f(g(5))[\/latex]<\/li>\r\n \t<li>[latex]g(f(3))[\/latex]<\/li>\r\n \t<li>[latex]f(f(1))[\/latex]<\/li>\r\n \t<li>[latex]g(g(6))[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the function values for [latex]f[\/latex] and [latex]g[\/latex] shown in the table below to evaluate the expressions.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th>[latex]g(x)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<td>[latex]-8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]-8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"42\">\r\n \t<li>[latex](f \\circ g)(1)[\/latex]<\/li>\r\n \t<li>[latex](g \\circ f)(3)[\/latex]<\/li>\r\n \t<li>[latex](f \\circ f)(3)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use each pair of functions to find [latex]f(g(0))[\/latex] and [latex]g(f(0))[\/latex].\r\n<ol start=\"45\">\r\n \t<li>[latex]f(x) = 5x + 7[\/latex], [latex]g(x) = 4 - 2x^2[\/latex]<\/li>\r\n \t<li>[latex]f(x) = \\dfrac{1}{x+2}[\/latex], [latex]g(x) = 4x + 3[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the functions [latex]f(x) = 2x^2 + 1[\/latex] and [latex]g(x) = 3x + 5[\/latex] to evaluate or find the composite function as indicated.\r\n<ol start=\"47\">\r\n \t<li>[latex]f(g(x))[\/latex]<\/li>\r\n \t<li>[latex](g \\circ g)(x)[\/latex]<\/li>\r\n<\/ol>\r\nReal-World Applications.\r\n<ol start=\"49\">\r\n \t<li>The function [latex]A(d)[\/latex] gives the pain level on a scale of [latex]0[\/latex] to [latex]10[\/latex] 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(t)[\/latex]. Which of the following would you do in order to determine when the patient will be at a pain level of [latex]4[\/latex]?\r\n<ol type=\"a\">\r\n \t<li>Evaluate [latex]A(m(4))[\/latex].<\/li>\r\n \t<li>Evaluate [latex]m(A(4))[\/latex].<\/li>\r\n \t<li>Solve [latex]A(m(t)) = 4[\/latex].<\/li>\r\n \t<li>Solve [latex]m(A(d)) = 4[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>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(t) = 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].<\/li>\r\n \t<li>Use the function you found in the previous exercise to find the total area burned after [latex]5[\/latex] minutes.<\/li>\r\n \t<li>The number of bacteria in a refrigerated food product is given by [latex]N(T) = 23T^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(t) = 5t + 1.5[\/latex], where [latex]t[\/latex] is the time in hours.\r\n<ol type=\"a\">\r\n \t<li>Find the composite function [latex]N(T(t))[\/latex].<\/li>\r\n \t<li>Find the time (round to two decimal places) when the bacteria count reaches [latex]6752[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<h2><span data-sheets-root=\"1\">Transformations of Functions<\/span><\/h2>\r\nFor the following exercises, write a formula for the function obtained when the graph is shifted as described.\r\n<ol>\r\n \t<li>[latex]f(x) = |x|[\/latex] is shifted down 3 units and to the right 1 unit.<\/li>\r\n \t<li>[latex]f(x) = \\dfrac{1}{x^2}[\/latex] is shifted up 2 units and to the left 4 units.<\/li>\r\n<\/ol>\r\nFor the following exercises, describe how the graph of the function is a transformation of the graph of the original function [latex]f[\/latex].\r\n<ol start=\"3\">\r\n \t<li>[latex]y = f(x + 43)[\/latex]<\/li>\r\n \t<li>[latex]y = f(x - 4)[\/latex]<\/li>\r\n \t<li>[latex]y = f(x) + 8[\/latex]<\/li>\r\n \t<li>[latex]y = f(x) - 7[\/latex]<\/li>\r\n \t<li>[latex]y = f(x + 4) - 1[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine the interval(s) on which the function is increasing and decreasing.\r\n<ol start=\"8\">\r\n \t<li>[latex]g(x) = 5(x + 3)^2 - 2[\/latex]<\/li>\r\n \t<li>[latex]k(x) = -3\\sqrt{x - 1}[\/latex]<\/li>\r\n<\/ol>\r\n<ol start=\"10\">\r\n \t<li>Use the graph of [latex]f(x) = 2^x[\/latex] to sketch a graph of the transformation of [latex]f(x)[\/latex], [latex]h(x) = 2^x - 3[\/latex].\r\n<img class=\"alignnone size-medium wp-image-6118\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31192452\/Algebraic_Operations_6-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/><\/li>\r\n<\/ol>\r\nFor the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.\r\n<ol start=\"11\">\r\n \t<li>[latex]f(t) = (t + 1)^2 - 3[\/latex]<\/li>\r\n \t<li>[latex]k(x) = (x - 2)^3 - 1[\/latex]<\/li>\r\n<\/ol>\r\n<ol start=\"13\">\r\n \t<li>Tabular representations for the functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are given below. Write [latex]g(x)[\/latex] and [latex]h(x)[\/latex] as transformations of [latex]f(x)[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]-2[\/latex]<\/th>\r\n<th>[latex]-1[\/latex]<\/th>\r\n<th>[latex]0[\/latex]<\/th>\r\n<th>[latex]1[\/latex]<\/th>\r\n<th>[latex]2[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f(x)[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]-1[\/latex]<\/th>\r\n<th>[latex]0[\/latex]<\/th>\r\n<th>[latex]1[\/latex]<\/th>\r\n<th>[latex]2[\/latex]<\/th>\r\n<th>[latex]3[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g(x)[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]-2[\/latex]<\/th>\r\n<th>[latex]-1[\/latex]<\/th>\r\n<th>[latex]0[\/latex]<\/th>\r\n<th>[latex]1[\/latex]<\/th>\r\n<th>[latex]2[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]h(x)[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li><img class=\"alignnone size-medium wp-image-6119\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193222\/Algebraic_Operations_7_33-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" \/><\/li>\r\n \t<li><img class=\"alignnone size-medium wp-image-6120\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193246\/Algebraic_Operations_8_35-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" \/><\/li>\r\n \t<li><img class=\"alignnone size-medium wp-image-6121\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193247\/Algebraic_Operations_9_37-165x300.jpeg\" alt=\"\" width=\"165\" height=\"300\" \/><\/li>\r\n \t<li><img class=\"alignnone size-medium wp-image-6122\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193248\/Algebraic_Operations_10_39-297x300.jpeg\" alt=\"\" width=\"297\" height=\"300\" \/><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"18\">\r\n \t<li><img class=\"alignnone size-medium wp-image-6123\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193249\/Algebraic_Operations_11_41.jpeg\" alt=\"\" width=\"225\" height=\"225\" \/><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"19\">\r\n \t<li><img class=\"alignnone size-medium wp-image-6124\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193250\/Algebraic_Operations_12_43-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" \/><\/li>\r\n \t<li><img class=\"alignnone size-medium wp-image-6125\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193251\/Algebraic_Operations_13_45.jpeg\" alt=\"\" width=\"184\" height=\"191\" \/><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the function is odd, even, or neither.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"21\">\r\n \t<li>[latex]f(x) = 3x^4[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\dfrac{1}{x} + 3x[\/latex]<\/li>\r\n \t<li>[latex]g(x) = 2x^4[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex]f[\/latex].\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"24\">\r\n \t<li>[latex]g(x) = -f(x)[\/latex]<\/li>\r\n \t<li>[latex]g(x) = 4f(x)[\/latex]<\/li>\r\n \t<li>[latex]g(x) = f(5x)[\/latex]<\/li>\r\n \t<li>[latex]g(x) = f\\left(\\dfrac{1}{3}x\\right)[\/latex]<\/li>\r\n \t<li>[latex]g(x) = 3f(-x)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, write a formula for the function [latex]g[\/latex] that results when the graph of a given toolkit function is transformed as described.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"29\">\r\n \t<li>The graph of [latex]f(x) = |x|[\/latex] is reflected over the [latex]y[\/latex]-axis and horizontally compressed by a factor of [latex]\\dfrac{1}{4}[\/latex].<\/li>\r\n \t<li>The graph of [latex]f(x) = \\dfrac{1}{x^2}[\/latex] is vertically compressed by a factor of [latex]\\dfrac{1}{3}[\/latex], then shifted to the left [latex]2[\/latex] units and down [latex]3[\/latex] units.<\/li>\r\n \t<li>The graph of [latex]f(x) = x^2[\/latex] is vertically compressed by a factor of [latex]\\dfrac{1}{2}[\/latex], then shifted to the right [latex]5[\/latex] units and up [latex]1[\/latex] unit.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"32\">\r\n \t<li>[latex]g(x) = 4(x + 1)^2 - 5[\/latex]<\/li>\r\n \t<li>[latex]h(x) = -2 |x - 4| + 3[\/latex]<\/li>\r\n \t<li>[latex]m(x) = \\dfrac{1}{2} x^3[\/latex]<\/li>\r\n \t<li>[latex]p(x) = \\left(\\dfrac{1}{3} x\\right)^3 - 3[\/latex]<\/li>\r\n \t<li>[latex]a(x) = \\sqrt{-x + 4}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graph below to sketch the given transformations.\r\n<img class=\"alignnone size-medium wp-image-6126\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"37\">\r\n \t<li>[latex]g(x) = -f(x)[\/latex]<\/li>\r\n \t<li>[latex]g(x) = f(x - 2)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<h2><span data-sheets-root=\"1\">Inverse Functions<\/span><\/h2>\r\nFor the following exercises, find [latex]f^{-1}(x)[\/latex] for each function.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol>\r\n \t<li>[latex]f(x) = x + 3[\/latex]<\/li>\r\n \t<li>[latex]f(x) = 2 - x[\/latex]<\/li>\r\n \t<li>[latex]f(x) = \\dfrac{x}{x + 2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, find a domain on which each function [latex]f[\/latex] is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of [latex]f[\/latex] restricted to that domain.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"4\">\r\n \t<li>[latex]f(x) = (x + 7)^2[\/latex]<\/li>\r\n \t<li>[latex]f(x) = x^2 - 5[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use function composition to verify that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are inverse functions.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"6\">\r\n \t<li>[latex]f(x) = \\sqrt[3]{x - 1}[\/latex] and [latex]g(x) = x^3 + 1[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use a graphing utility to determine whether each function is one-to-one.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"7\">\r\n \t<li>[latex]f(x) = \\sqrt{x}[\/latex]<\/li>\r\n \t<li>[latex]f(x) = -5x + 1[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the graph represents a one-to-one function.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"9\">\r\n \t<li><img class=\"alignnone size-full wp-image-6128\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200322\/inverse_functions_1_23.jpeg\" alt=\"\" width=\"252\" height=\"253\" \/><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graph of f shown in Figure 11.\r\n<img class=\"alignnone size-medium wp-image-6129\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200326\/inverse_functions_2_25-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"10\">\r\n \t<li>Find [latex]f(0)[\/latex].<\/li>\r\n \t<li>Find [latex]f^{-1}(0)[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graph of the one-to-one function shown in Figure 12.\r\n<img class=\"alignnone size-full wp-image-6130\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200452\/inverse_functions_3_29.jpeg\" alt=\"\" width=\"255\" height=\"254\" \/>\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"12\">\r\n \t<li>Sketch the graph of [latex]f^{-1}[\/latex].<\/li>\r\n \t<li>If the complete graph of [latex]f[\/latex] is shown, find the domain of [latex]f[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, evaluate or solve, assuming that the function [latex]f[\/latex] is one-to-one.\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li>If [latex]f(6) = 7[\/latex], find [latex]f^{-1}(7)[\/latex].<\/li>\r\n \t<li>If [latex]f^{-1}(-4) = -8[\/latex], find [latex]f(-8)[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the values listed in Table 6 to evaluate or solve.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"16\">\r\n \t<li>Find [latex]f(1)[\/latex].<\/li>\r\n \t<li>Find [latex]f^{-1}(0)[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nUse the tabular representation of [latex]f[\/latex] in Table 7 to create a table for [latex]f^{-1}(x)[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]13[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"14\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol start=\"18\">\r\n \t<li>To convert from [latex]x[\/latex] degrees Celsius to [latex]y[\/latex] degrees Fahrenheit, we use the formula [latex]f(x) = \\dfrac{9}{5} x + 32[\/latex]. Find the inverse function, if it exists, and explain its meaning.<\/li>\r\n \t<li>A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, [latex]t[\/latex], in hours given by [latex]d(t) = 50t[\/latex]. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function [latex]t(d)[\/latex]. Find [latex]t(180)[\/latex] and interpret its meaning.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>","rendered":"<h2><span data-sheets-root=\"1\">Combinations and Compositions of Functions<\/span><\/h2>\n<p>For the following exercises, determine the domain for each function in interval notation.<\/p>\n<ol>\n<li>Given [latex]f(x) = x^2 + 2x[\/latex] and [latex]g(x) = 6 - x^2[\/latex], find [latex]f + g[\/latex], [latex]f - g[\/latex], [latex]fg[\/latex], and [latex]\\dfrac{f}{g}[\/latex].<\/li>\n<li>Given [latex]f(x) = 2x^2 + 4x[\/latex] and [latex]g(x) = \\dfrac{1}{2x}[\/latex], find [latex]f + g[\/latex], [latex]f - g[\/latex], [latex]fg[\/latex], and [latex]\\dfrac{f}{g}[\/latex].<\/li>\n<li>Given [latex]f(x) = 3x^2[\/latex] and [latex]g(x) = \\sqrt{x - 5}[\/latex], find [latex]f + g[\/latex], [latex]f - g[\/latex], [latex]fg[\/latex], and [latex]\\dfrac{f}{g}[\/latex].<\/li>\n<\/ol>\n<p>For the following exercise, find the indicated function given [latex]f(x) = 2x^2 + 1[\/latex] and [latex]g(x) = 3x - 5[\/latex].<\/p>\n<ol start=\"4\">\n<li>\n<ol type=\"a\">\n<li>[latex]f(g(2))[\/latex]<\/li>\n<li>[latex]f(g(x))[\/latex]<\/li>\n<li>[latex]g(f(x))[\/latex]<\/li>\n<li>[latex](g \\circ g)(x)[\/latex]<\/li>\n<li>[latex](f \\circ f)(-2)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use each pair of functions to find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex]. Simplify your answers.<\/p>\n<ol start=\"10\">\n<li>Given [latex]f(x) = \\sqrt{x} + 2[\/latex] and [latex]g(x) = x^2 + 3[\/latex], find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex].<\/li>\n<li>Given [latex]f(x) = \\sqrt[3]{x}[\/latex] and [latex]g(x) = \\dfrac{x + 1}{x^3}[\/latex], find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex].<\/li>\n<li>Given [latex]f(x) = \\dfrac{1}{x - 4}[\/latex] and [latex]g(x) = \\dfrac{2}{x} + 4[\/latex], find [latex]f(g(x))[\/latex] and [latex]g(f(x))[\/latex].<\/li>\n<\/ol>\n<p>For the following exercises, use each set of functions to find [latex]f(g(h(x)))[\/latex]. Simplify your answers.<\/p>\n<ol start=\"13\">\n<li>Given [latex]f(x) = x^2 + 1[\/latex], [latex]g(x) = \\dfrac{1}{x}[\/latex], and [latex]h(x) = x + 3[\/latex], find [latex]f(g(h(x)))[\/latex].<\/li>\n<li>Given [latex]f(x) = \\sqrt{2 - 4x}[\/latex] and [latex]g(x) = -\\dfrac{3}{x}[\/latex], find the following:\n<ol type=\"a\">\n<li>[latex](g \\circ f)(x)[\/latex]<\/li>\n<li>The domain of [latex](g \\circ f)(x)[\/latex] in interval notation<\/li>\n<\/ol>\n<\/li>\n<li>Given the functions [latex]p(x) = \\dfrac{1}{\\sqrt{x}}[\/latex] and [latex]m(x) = x^2 - 4[\/latex], state the domain of each of the following functions using interval notation:\n<ol type=\"a\">\n<li>[latex]\\dfrac{p(x)}{m(x)}[\/latex]<\/li>\n<li>[latex]p(m(x))[\/latex]<\/li>\n<li>[latex]m(p(x))[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>For [latex]f(x) = \\dfrac{1}{x}[\/latex] and [latex]g(x) = \\sqrt{x - 1}[\/latex], write the domain of [latex](f \\circ g)(x)[\/latex] in interval notation.<\/li>\n<\/ol>\n<p>For the following exercises, find functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] so the given function can be expressed as [latex]h(x) = f(g(x))[\/latex].<\/p>\n<ol start=\"22\">\n<li>[latex]h(x) = (x - 5)^3[\/latex]<\/li>\n<li>[latex]h(x) = \\dfrac{4}{(x+2)^2}[\/latex]<\/li>\n<li>[latex]h(x) = \\sqrt[3]{\\dfrac{1}{2x - 3}}[\/latex]<\/li>\n<li>[latex]h(x) = \\sqrt[4]{\\dfrac{3x - 2}{x + 5}}[\/latex]<\/li>\n<li>[latex]h(x) = \\sqrt{2x + 6}[\/latex]<\/li>\n<li>[latex]h(x) = \\sqrt[3]{x - 1}[\/latex]<\/li>\n<li>[latex]h(x) = \\dfrac{1}{(x - 2)^3}[\/latex]<\/li>\n<li>[latex]h(x) = \\sqrt{\\dfrac{2x-1}{3x+4}}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the graphs of [latex]f[\/latex], shown in the first graph, and [latex]g[\/latex], shown in the second graph, to evaluate the expressions.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6111\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190149\/Algebraic_Operations_1-300x173.jpeg\" alt=\"\" width=\"300\" height=\"173\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190149\/Algebraic_Operations_1-300x173.jpeg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190149\/Algebraic_Operations_1-65x38.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190149\/Algebraic_Operations_1-225x130.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190149\/Algebraic_Operations_1-350x202.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190149\/Algebraic_Operations_1.jpeg 488w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6113\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190201\/Algebraic_Operations_2-300x173.jpeg\" alt=\"\" width=\"300\" height=\"173\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190201\/Algebraic_Operations_2-300x173.jpeg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190201\/Algebraic_Operations_2-65x38.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190201\/Algebraic_Operations_2-225x130.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190201\/Algebraic_Operations_2-350x202.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190201\/Algebraic_Operations_2.jpeg 488w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol start=\"30\">\n<li>[latex]f(g(1))[\/latex]<\/li>\n<li>[latex]g(f(0))[\/latex]<\/li>\n<li>[latex]f(f(4))[\/latex]<\/li>\n<li>[latex]g(g(0))[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use graphs of [latex]f(x)[\/latex], [latex]g(x)[\/latex], and [latex]h(x)[\/latex], to evaluate the expressions.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6114\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190559\/Algebraic_Operations_3-294x300.jpeg\" alt=\"\" width=\"294\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190559\/Algebraic_Operations_3-294x300.jpeg 294w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190559\/Algebraic_Operations_3-65x66.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190559\/Algebraic_Operations_3-225x229.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190559\/Algebraic_Operations_3.jpeg 309w\" sizes=\"(max-width: 294px) 100vw, 294px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6115\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190600\/Algebraic_Operations_4.jpeg\" alt=\"\" width=\"244\" height=\"250\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190600\/Algebraic_Operations_4.jpeg 244w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190600\/Algebraic_Operations_4-65x67.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190600\/Algebraic_Operations_4-225x231.jpeg 225w\" sizes=\"(max-width: 244px) 100vw, 244px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6116\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190601\/Algebraic_Operations_5-300x176.jpeg\" alt=\"\" width=\"300\" height=\"176\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190601\/Algebraic_Operations_5-300x176.jpeg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190601\/Algebraic_Operations_5-65x38.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190601\/Algebraic_Operations_5-225x132.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190601\/Algebraic_Operations_5-350x206.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31190601\/Algebraic_Operations_5.jpeg 432w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol start=\"34\">\n<li>[latex]g(f(2))[\/latex]<\/li>\n<li>[latex]f(g(1))[\/latex]<\/li>\n<li>[latex]h(f(2))[\/latex]<\/li>\n<li>[latex]f(g(f(-2)))[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the function values for [latex]f[\/latex] and [latex]g[\/latex] shown in the table below to evaluate each expression.<\/p>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)[\/latex]<\/th>\n<th>[latex]g(x)[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"38\">\n<li>[latex]f(g(5))[\/latex]<\/li>\n<li>[latex]g(f(3))[\/latex]<\/li>\n<li>[latex]f(f(1))[\/latex]<\/li>\n<li>[latex]g(g(6))[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the function values for [latex]f[\/latex] and [latex]g[\/latex] shown in the table below to evaluate the expressions.<\/p>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)[\/latex]<\/th>\n<th>[latex]g(x)[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]11[\/latex]<\/td>\n<td>[latex]-8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"42\">\n<li>[latex](f \\circ g)(1)[\/latex]<\/li>\n<li>[latex](g \\circ f)(3)[\/latex]<\/li>\n<li>[latex](f \\circ f)(3)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use each pair of functions to find [latex]f(g(0))[\/latex] and [latex]g(f(0))[\/latex].<\/p>\n<ol start=\"45\">\n<li>[latex]f(x) = 5x + 7[\/latex], [latex]g(x) = 4 - 2x^2[\/latex]<\/li>\n<li>[latex]f(x) = \\dfrac{1}{x+2}[\/latex], [latex]g(x) = 4x + 3[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the functions [latex]f(x) = 2x^2 + 1[\/latex] and [latex]g(x) = 3x + 5[\/latex] to evaluate or find the composite function as indicated.<\/p>\n<ol start=\"47\">\n<li>[latex]f(g(x))[\/latex]<\/li>\n<li>[latex](g \\circ g)(x)[\/latex]<\/li>\n<\/ol>\n<p>Real-World Applications.<\/p>\n<ol start=\"49\">\n<li>The function [latex]A(d)[\/latex] gives the pain level on a scale of [latex]0[\/latex] to [latex]10[\/latex] 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(t)[\/latex]. Which of the following would you do in order to determine when the patient will be at a pain level of [latex]4[\/latex]?\n<ol type=\"a\">\n<li>Evaluate [latex]A(m(4))[\/latex].<\/li>\n<li>Evaluate [latex]m(A(4))[\/latex].<\/li>\n<li>Solve [latex]A(m(t)) = 4[\/latex].<\/li>\n<li>Solve [latex]m(A(d)) = 4[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>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(t) = 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].<\/li>\n<li>Use the function you found in the previous exercise to find the total area burned after [latex]5[\/latex] minutes.<\/li>\n<li>The number of bacteria in a refrigerated food product is given by [latex]N(T) = 23T^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(t) = 5t + 1.5[\/latex], where [latex]t[\/latex] is the time in hours.\n\n\n<ol type=\"a\">\n<li>Find the composite function [latex]N(T(t))[\/latex].<\/li>\n<li>Find the time (round to two decimal places) when the bacteria count reaches [latex]6752[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h2><span data-sheets-root=\"1\">Transformations of Functions<\/span><\/h2>\n<p>For the following exercises, write a formula for the function obtained when the graph is shifted as described.<\/p>\n<ol>\n<li>[latex]f(x) = |x|[\/latex] is shifted down 3 units and to the right 1 unit.<\/li>\n<li>[latex]f(x) = \\dfrac{1}{x^2}[\/latex] is shifted up 2 units and to the left 4 units.<\/li>\n<\/ol>\n<p>For the following exercises, describe how the graph of the function is a transformation of the graph of the original function [latex]f[\/latex].<\/p>\n<ol start=\"3\">\n<li>[latex]y = f(x + 43)[\/latex]<\/li>\n<li>[latex]y = f(x - 4)[\/latex]<\/li>\n<li>[latex]y = f(x) + 8[\/latex]<\/li>\n<li>[latex]y = f(x) - 7[\/latex]<\/li>\n<li>[latex]y = f(x + 4) - 1[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, determine the interval(s) on which the function is increasing and decreasing.<\/p>\n<ol start=\"8\">\n<li>[latex]g(x) = 5(x + 3)^2 - 2[\/latex]<\/li>\n<li>[latex]k(x) = -3\\sqrt{x - 1}[\/latex]<\/li>\n<\/ol>\n<ol start=\"10\">\n<li>Use the graph of [latex]f(x) = 2^x[\/latex] to sketch a graph of the transformation of [latex]f(x)[\/latex], [latex]h(x) = 2^x - 3[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6118\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31192452\/Algebraic_Operations_6-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31192452\/Algebraic_Operations_6-288x300.jpeg 288w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31192452\/Algebraic_Operations_6-65x68.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31192452\/Algebraic_Operations_6-225x234.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31192452\/Algebraic_Operations_6-350x365.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31192452\/Algebraic_Operations_6.jpeg 357w\" sizes=\"(max-width: 288px) 100vw, 288px\" \/><\/li>\n<\/ol>\n<p>For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.<\/p>\n<ol start=\"11\">\n<li>[latex]f(t) = (t + 1)^2 - 3[\/latex]<\/li>\n<li>[latex]k(x) = (x - 2)^3 - 1[\/latex]<\/li>\n<\/ol>\n<ol start=\"13\">\n<li>Tabular representations for the functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are given below. Write [latex]g(x)[\/latex] and [latex]h(x)[\/latex] as transformations of [latex]f(x)[\/latex].<br \/>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]-2[\/latex]<\/th>\n<th>[latex]-1[\/latex]<\/th>\n<th>[latex]0[\/latex]<\/th>\n<th>[latex]1[\/latex]<\/th>\n<th>[latex]2[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>[latex]f(x)[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]-1[\/latex]<\/th>\n<th>[latex]0[\/latex]<\/th>\n<th>[latex]1[\/latex]<\/th>\n<th>[latex]2[\/latex]<\/th>\n<th>[latex]3[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>[latex]g(x)[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]-2[\/latex]<\/th>\n<th>[latex]-1[\/latex]<\/th>\n<th>[latex]0[\/latex]<\/th>\n<th>[latex]1[\/latex]<\/th>\n<th>[latex]2[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>[latex]h(x)[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<p>For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6119\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193222\/Algebraic_Operations_7_33-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193222\/Algebraic_Operations_7_33-295x300.jpeg 295w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193222\/Algebraic_Operations_7_33-65x66.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193222\/Algebraic_Operations_7_33-225x229.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193222\/Algebraic_Operations_7_33.jpeg 312w\" sizes=\"(max-width: 295px) 100vw, 295px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6120\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193246\/Algebraic_Operations_8_35-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193246\/Algebraic_Operations_8_35-295x300.jpeg 295w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193246\/Algebraic_Operations_8_35-65x66.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193246\/Algebraic_Operations_8_35-225x229.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193246\/Algebraic_Operations_8_35.jpeg 312w\" sizes=\"(max-width: 295px) 100vw, 295px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6121\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193247\/Algebraic_Operations_9_37-165x300.jpeg\" alt=\"\" width=\"165\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193247\/Algebraic_Operations_9_37-165x300.jpeg 165w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193247\/Algebraic_Operations_9_37-65x118.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193247\/Algebraic_Operations_9_37.jpeg 225w\" sizes=\"(max-width: 165px) 100vw, 165px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6122\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193248\/Algebraic_Operations_10_39-297x300.jpeg\" alt=\"\" width=\"297\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193248\/Algebraic_Operations_10_39-297x300.jpeg 297w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193248\/Algebraic_Operations_10_39-65x66.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193248\/Algebraic_Operations_10_39-225x227.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193248\/Algebraic_Operations_10_39-350x354.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193248\/Algebraic_Operations_10_39.jpeg 375w\" sizes=\"(max-width: 297px) 100vw, 297px\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"18\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6123\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193249\/Algebraic_Operations_11_41.jpeg\" alt=\"\" width=\"225\" height=\"225\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193249\/Algebraic_Operations_11_41.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193249\/Algebraic_Operations_11_41-150x150.jpeg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193249\/Algebraic_Operations_11_41-65x65.jpeg 65w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"19\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6124\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193250\/Algebraic_Operations_12_43-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193250\/Algebraic_Operations_12_43-300x272.jpeg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193250\/Algebraic_Operations_12_43-65x59.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193250\/Algebraic_Operations_12_43-225x204.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193250\/Algebraic_Operations_12_43.jpeg 312w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6125\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193251\/Algebraic_Operations_13_45.jpeg\" alt=\"\" width=\"184\" height=\"191\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193251\/Algebraic_Operations_13_45.jpeg 184w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31193251\/Algebraic_Operations_13_45-65x67.jpeg 65w\" sizes=\"(max-width: 184px) 100vw, 184px\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, determine whether the function is odd, even, or neither.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"21\">\n<li>[latex]f(x) = 3x^4[\/latex]<\/li>\n<li>[latex]h(x) = \\dfrac{1}{x} + 3x[\/latex]<\/li>\n<li>[latex]g(x) = 2x^4[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex]f[\/latex].<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"24\">\n<li>[latex]g(x) = -f(x)[\/latex]<\/li>\n<li>[latex]g(x) = 4f(x)[\/latex]<\/li>\n<li>[latex]g(x) = f(5x)[\/latex]<\/li>\n<li>[latex]g(x) = f\\left(\\dfrac{1}{3}x\\right)[\/latex]<\/li>\n<li>[latex]g(x) = 3f(-x)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, write a formula for the function [latex]g[\/latex] that results when the graph of a given toolkit function is transformed as described.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"29\">\n<li>The graph of [latex]f(x) = |x|[\/latex] is reflected over the [latex]y[\/latex]-axis and horizontally compressed by a factor of [latex]\\dfrac{1}{4}[\/latex].<\/li>\n<li>The graph of [latex]f(x) = \\dfrac{1}{x^2}[\/latex] is vertically compressed by a factor of [latex]\\dfrac{1}{3}[\/latex], then shifted to the left [latex]2[\/latex] units and down [latex]3[\/latex] units.<\/li>\n<li>The graph of [latex]f(x) = x^2[\/latex] is vertically compressed by a factor of [latex]\\dfrac{1}{2}[\/latex], then shifted to the right [latex]5[\/latex] units and up [latex]1[\/latex] unit.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"32\">\n<li>[latex]g(x) = 4(x + 1)^2 - 5[\/latex]<\/li>\n<li>[latex]h(x) = -2 |x - 4| + 3[\/latex]<\/li>\n<li>[latex]m(x) = \\dfrac{1}{2} x^3[\/latex]<\/li>\n<li>[latex]p(x) = \\left(\\dfrac{1}{3} x\\right)^3 - 3[\/latex]<\/li>\n<li>[latex]a(x) = \\sqrt{-x + 4}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use the graph below to sketch the given transformations.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6126\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79-298x300.jpeg 298w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79-150x150.jpeg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79-65x65.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79-225x226.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79-350x352.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31195051\/Algebraic_Operations_14_79.jpeg 459w\" sizes=\"(max-width: 298px) 100vw, 298px\" \/><\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"37\">\n<li>[latex]g(x) = -f(x)[\/latex]<\/li>\n<li>[latex]g(x) = f(x - 2)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h2><span data-sheets-root=\"1\">Inverse Functions<\/span><\/h2>\n<p>For the following exercises, find [latex]f^{-1}(x)[\/latex] for each function.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol>\n<li>[latex]f(x) = x + 3[\/latex]<\/li>\n<li>[latex]f(x) = 2 - x[\/latex]<\/li>\n<li>[latex]f(x) = \\dfrac{x}{x + 2}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, find a domain on which each function [latex]f[\/latex] is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of [latex]f[\/latex] restricted to that domain.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"4\">\n<li>[latex]f(x) = (x + 7)^2[\/latex]<\/li>\n<li>[latex]f(x) = x^2 - 5[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use function composition to verify that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are inverse functions.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"6\">\n<li>[latex]f(x) = \\sqrt[3]{x - 1}[\/latex] and [latex]g(x) = x^3 + 1[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"7\">\n<li>[latex]f(x) = \\sqrt{x}[\/latex]<\/li>\n<li>[latex]f(x) = -5x + 1[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, determine whether the graph represents a one-to-one function.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"9\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6128\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200322\/inverse_functions_1_23.jpeg\" alt=\"\" width=\"252\" height=\"253\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200322\/inverse_functions_1_23.jpeg 252w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200322\/inverse_functions_1_23-150x150.jpeg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200322\/inverse_functions_1_23-65x65.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200322\/inverse_functions_1_23-225x226.jpeg 225w\" sizes=\"(max-width: 252px) 100vw, 252px\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use the graph of f shown in Figure 11.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6129\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200326\/inverse_functions_2_25-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200326\/inverse_functions_2_25-288x300.jpeg 288w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200326\/inverse_functions_2_25-65x68.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200326\/inverse_functions_2_25-225x234.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200326\/inverse_functions_2_25-350x365.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200326\/inverse_functions_2_25.jpeg 357w\" sizes=\"(max-width: 288px) 100vw, 288px\" \/><\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"10\">\n<li>Find [latex]f(0)[\/latex].<\/li>\n<li>Find [latex]f^{-1}(0)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use the graph of the one-to-one function shown in Figure 12.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6130\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200452\/inverse_functions_3_29.jpeg\" alt=\"\" width=\"255\" height=\"254\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200452\/inverse_functions_3_29.jpeg 255w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200452\/inverse_functions_3_29-150x150.jpeg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200452\/inverse_functions_3_29-65x65.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/09\/31200452\/inverse_functions_3_29-225x224.jpeg 225w\" sizes=\"(max-width: 255px) 100vw, 255px\" \/><\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"12\">\n<li>Sketch the graph of [latex]f^{-1}[\/latex].<\/li>\n<li>If the complete graph of [latex]f[\/latex] is shown, find the domain of [latex]f[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, evaluate or solve, assuming that the function [latex]f[\/latex] is one-to-one.<\/p>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li>If [latex]f(6) = 7[\/latex], find [latex]f^{-1}(7)[\/latex].<\/li>\n<li>If [latex]f^{-1}(-4) = -8[\/latex], find [latex]f(-8)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>For the following exercises, use the values listed in Table 6 to evaluate or solve.<\/p>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<th>[latex]f(x)[\/latex]<\/th>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"16\">\n<li>Find [latex]f(1)[\/latex].<\/li>\n<li>Find [latex]f^{-1}(0)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>Use the tabular representation of [latex]f[\/latex] in Table 7 to create a table for [latex]f^{-1}(x)[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]13[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<\/tr>\n<tr>\n<th>[latex]f(x)[\/latex]<\/th>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"14\">\n<li style=\"list-style-type: none;\">\n<ol start=\"18\">\n<li>To convert from [latex]x[\/latex] degrees Celsius to [latex]y[\/latex] degrees Fahrenheit, we use the formula [latex]f(x) = \\dfrac{9}{5} x + 32[\/latex]. Find the inverse function, if it exists, and explain its meaning.<\/li>\n<li>A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, [latex]t[\/latex], in hours given by [latex]d(t) = 50t[\/latex]. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function [latex]t(d)[\/latex]. Find [latex]t(180)[\/latex] and interpret its meaning.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":142,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3903"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3903\/revisions"}],"predecessor-version":[{"id":6269,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3903\/revisions\/6269"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/142"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3903\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=3903"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=3903"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=3903"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=3903"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}