{"id":856,"date":"2024-04-08T17:23:07","date_gmt":"2024-04-08T17:23:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=856"},"modified":"2024-08-04T10:43:56","modified_gmt":"2024-08-04T10:43:56","slug":"review-of-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/review-of-functions-fresh-take\/","title":{"raw":"Review of Functions: Fresh Take","rendered":"Review of Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Identify zeros of functions using equations, graphs, and tables<\/li>\r\n\t<li>Interpret graphs and tables to describe their behavior including properties of symmetry<\/li>\r\n\t<li>Make new functions from two or more given functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">A function is a special type of relation where each input is associated with exactly one output.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Functions are used to describe relationships between two sets.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The input of a function is called the independent variable, often denoted as [latex]x[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The output of a function is called the dependent variable, often denoted as [latex]y[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Function notation: [latex]y = f(x)[\/latex] is read as \"[latex]y[\/latex] equals [latex]f[\/latex] of [latex]x[\/latex].\"<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Functions can be represented using tables, graphs, or formulas (algebraic expressions).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Consider the function [latex]g(x) = x\u00b2 - 4x + 3[\/latex]<\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Evaluate [latex]g(3)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Create a small table of values for [latex]x = 0, 1, 2, 3, 4[\/latex]<\/li>\r\n<\/ol>\r\n<p><br \/>\r\n[reveal-answer q=\"826228\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"826228\"]<\/p>\r\n<ol>\r\n\t<li>\u00a0[latex]g(3) = 3\u00b2 - 4(3) + 3 = 9 - 12 + 3 = 0[\/latex]<\/li>\r\n\t<li>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>x<\/th>\r\n<th>g(x)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]0[\/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]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<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/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\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>The Domain and Range of a Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">The domain of a function is the complete set of possible input values ([latex]x[\/latex]-values).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The range of a function is the complete set of possible output values ([latex]y[\/latex]-values).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domain restrictions can occur due to mathematical limitations (e.g., division by zero, square roots of negative numbers).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Set notation and interval notation are used to describe domains and ranges.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The union symbol ([latex]\\cup[\/latex]) is used to describe domains and ranges with gaps or breaks.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Determine the domain and range of the function[latex] f(x) = |x + 1| - 3[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"143942\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"143942\"]<\/p>\r\n<ol>\r\n\t<li>Determine the domain:\r\n\r\n<p>The absolute value function is defined for all real numbers meaning there are no restrictions on the input. So the domain is:<\/p>\r\n<center>\u00a0[latex](-\\infty, \\infty)[\/latex] or [latex]{x | x \\text{ is any real number}}[\/latex]<\/center><\/li>\r\n\t<li>Determine the range:\r\n\r\n<p>The absolute value [latex]|x + 1|[\/latex] is always non-negative. The minimum value of [latex]|x + 1|[\/latex] is [latex]0[\/latex], which occurs when [latex]x = -1[\/latex].<\/p>\r\n<p>Subtracting [latex]3[\/latex] shifts the entire function down by [latex]3[\/latex] units. The minimum value of [latex]f(x)[\/latex] is therefore[latex] -3[\/latex].<\/p>\r\n\r\nThere is no upper limit to the function's value. Making the range:<br \/>\r\n<center>[latex][-3, \u221e)[\/latex] or [latex]{y | y \u2265 -3}[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Intercepts of a Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts are points where a function's graph crosses the [latex]x[\/latex]-axis ([latex]f(x) = 0[\/latex]).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts are also called zeros or roots of the function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The [latex]y[\/latex]-intercept is the point where a function's graph crosses the [latex]y[\/latex]-axis ([latex]x = 0[\/latex]).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">A function can have multiple [latex]x[\/latex]-intercepts but at most one y-intercept.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts provide information about the shape of the function's graph.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The [latex]y[\/latex]-intercept represents the function's output when the input is zero.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572228574\">Consider the function [latex]f(x)=\\sqrt{x+3}+1[\/latex].<\/p>\r\n<ol id=\"fs-id1170571053593\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Find all zeros of [latex]f[\/latex].<\/li>\r\n\t<li>Find the [latex]y[\/latex]-intercept (if any).<\/li>\r\n\t<li>Sketch a graph of [latex]f[\/latex].<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170572168266\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572168266\"]<\/p>\r\n<ol id=\"fs-id1170572168266\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>To find the zeros, solve [latex]\\sqrt{x+3}+1=0[\/latex]. This equation implies [latex]\\sqrt{x+3}=-1[\/latex]. Since [latex]\\sqrt{x+3}\\ge 0[\/latex] for all [latex]x[\/latex], this equation has no solutions, and therefore [latex]f[\/latex] has no zeros.<\/li>\r\n\t<li>The [latex]y[\/latex]-intercept is given by [latex](0,f(0))=(0,\\sqrt{3}+1)[\/latex].<\/li>\r\n\t<li>To graph this function, we make a table of values. Since we need [latex]x+3\\ge 0[\/latex], we need to choose values of [latex]x\\ge -3[\/latex]. We choose values that make the square-root function easy to evaluate.<br \/>\r\n<table id=\"fs-id1170572247842\" class=\"column-header\" summary=\"A table with 2 rows and 3 columns. The first row is labeled \u201cx\u201d and has the values \u201c-3; -2; 1\u201d. The second row is labeled \u201cf(x)\u201d and has the values \u201c1; 2; 3\u201d.\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]f(x)[\/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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572205983\">Making use of the table and knowing that, since the function is a square root, the graph of [latex]f[\/latex] should be similar to the graph of [latex]y=\\sqrt{x}[\/latex], we sketch the graph (Figure 9).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"315\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202126\/CNX_Calc_Figure_01_01_008.jpg\" alt=\"An image of a graph. The y axis runs from -2 to 4 and the x axis runs from -3 to 2. The graph is of the function \u201cf(x) = (square root of x + 3) + 1\u201d, which is an increasing curved function that starts at the point (-3, 1). There are 3 points plotted on the function at (-3, 1), (-2, 2), and (1, 3). The function has a y intercept at (0, 1 + square root of 3).\" width=\"315\" height=\"272\" \/> Figure 9. The graph of [latex]f(x)=\\sqrt{x+3}+1[\/latex] has a [latex]y[\/latex]-intercept but no [latex]x[\/latex]-intercepts.[\/caption]\r\n\r\n<div class=\"wp-caption-text\">\u00a0<\/div>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572549227\">Find the zeros of [latex]f(x)=x^3-5x^2+6x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"098321\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"098321\"]<\/p>\r\n<p id=\"fs-id1165042323386\">Factor the polynomial.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572216813\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572216813\"]<\/p>\r\n<p id=\"fs-id1170572216813\">[latex]x=0,2,3[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2 class=\"entry-title\">Symmetry of Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Function Symmetry:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Symmetry about the [latex]y[\/latex]-axis: [latex]f(x) = f(-x)[\/latex] (even functions)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Symmetry about the origin: [latex]f(-x) = -f(x)[\/latex] (odd functions)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Some functions are neither even nor odd<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Absolute Value Function:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Defined as [latex]f(x) = |x| = x [\/latex]if [latex]x \u2265 0[\/latex], and [latex]-x[\/latex] if [latex]x &lt; 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Always outputs non-negative values<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Symmetric about the [latex]y[\/latex]-axis (even function)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202138\/CNX_Calc_Figure_01_01_012.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201c(a), symmetry about the y-axis\u201d and is of the curved function \u201cf(x) = (x to the 4th) - 2(x squared) - 3\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. This function decreases until it hits the point (-1, -4), which is minimum of the function. Then the graph increases to the point (0,3), which is a local maximum. Then the the graph decreases until it hits the point (1, -4), before it increases again. The second graph is labeled \u201c(b), symmetry about the origin\u201d and is of the curved function \u201cf(x) = x cubed - 4x\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. The graph of the function starts at the x intercept at (-2, 0) and increases until the approximate point of (-1.2, 3.1). The function then decreases, passing through the origin, until it hits the approximate point of (1.2, -3.1). The function then begins to increase again and has another x intercept at (2, 0).\" width=\"731\" height=\"426\" \/><\/p>\r\n<p>Figure 13. (a) A graph that is symmetric about the [latex]y[\/latex]-axis. (b) A graph that is symmetric about the origin.<\/p>\r\n<p id=\"fs-id1170572173116\">If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function [latex]f[\/latex] has symmetry? Looking at Figure 13(a) again, we see that since [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis, if the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,y)[\/latex] is on the graph. In other words, [latex]f(\u2212x)=f(x)[\/latex]. If a function [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>even function<\/strong>, which has symmetry about the [latex]y[\/latex]-axis. For example, [latex]f(x)=x^2[\/latex] is even because<\/p>\r\n<div id=\"fs-id1170572552270\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(\u2212x)=(\u2212x)^2=x^2=f(x)[\/latex].<\/div>\r\n<p id=\"fs-id1170572552328\">In contrast, looking at Figure 13(b) again, if a function [latex]f[\/latex] is symmetric about the origin, then whenever the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,\u2212y)[\/latex] is also on the graph. In other words, [latex]f(\u2212x)=\u2212f(x)[\/latex]. If [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>odd function<\/strong>, which has symmetry about the origin. For example, [latex]f(x)=x^3[\/latex] is odd because<\/p>\r\n<center>[latex]f(\u2212x)=(\u2212x)^3=\u2212x^3=\u2212f(x)[\/latex].<\/center>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572478824\">Determine whether [latex]f(x)=4x^3-5x[\/latex] is even, odd, or neither.<\/p>\r\n<p>[reveal-answer q=\"935578\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"935578\"]<\/p>\r\n<p>Compare [latex]f(\u2212x)[\/latex] with [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572547379\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572547379\"]<\/p>\r\n<p id=\"fs-id1170572547379\">[latex]f(x)[\/latex] is odd.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>Absolute Value Functions<\/h3>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572548622\">For the function [latex]f(x)=|x+2|-4[\/latex], find the domain and range.<\/p>\r\n<p>[reveal-answer q=\"882365\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"882365\"]<\/p>\r\n<p id=\"fs-id1165043422402\">[latex]|x+2|\\ge 0[\/latex] for all real numbers [latex]x[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572176864\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572176864\"]<\/p>\r\n<p id=\"fs-id1170572176864\">Domain = [latex](\u2212\\infty ,\\infty )[\/latex], range = [latex]\\{y|y\\ge -4\\}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Composing Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">Combining Functions with Mathematical Operators:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Sum: [latex](f + g)(x) = f(x) + g(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Difference :[latex] (f - g)(x) = f(x) - g(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Product: [latex] (f \u00b7 g)(x) = f(x) \u00b7 g(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Quotient: [latex](\\frac{f}{g})(x) = \\frac{f(x)}{g(x)}\\text{, where }g(x) \u2260 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Composite Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex](f \u2218 g)(x) = f(g(x))[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The output of g becomes the input of f<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Order matters: [latex](f \u2218 g)(x) \u2260 (g \u2218 f)(x)[\/latex] in general<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p>In general [latex]f\\circ g[\/latex] and [latex]g\\circ f[\/latex] are different functions. In other words in many cases [latex]f\\left(g\\left(x\\right)\\right)\\ne g\\left(f\\left(x\\right)\\right)[\/latex] for all [latex]x[\/latex].<\/p>\r\n<p>For example if [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]g\\left(x\\right)=x+2[\/latex], then<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(g\\left(x\\right)\\right)&amp;=f\\left(x+2\\right) \\\\[2mm] &amp;={\\left(x+2\\right)}^{2} \\\\[2mm] &amp;={x}^{2}+4x+4\\hfill \\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">but<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}g\\left(f\\left(x\\right)\\right)&amp;=g\\left({x}^{2}\\right) \\\\[2mm] \\text{ }&amp;={x}^{2}+2\\hfill \\end{align}[\/latex]<\/p>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572481359\">Consider the functions [latex]f(x)=x^2+1[\/latex] and [latex]g(x)=\\frac{1}{x}[\/latex].<\/p>\r\n<ol id=\"fs-id1170572482111\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Find [latex](g\\circ f)(x)[\/latex] and state its domain and range.<\/li>\r\n\t<li>Evaluate [latex](g\\circ f)(4)[\/latex] and [latex](g\\circ f)\\left(-\\frac{1}{2}\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170572222946\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572222946\"]<\/p>\r\n<ol id=\"fs-id1170572222946\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>We can find the formula for [latex](g\\circ f)(x)[\/latex] in two different ways. We could write<br \/>\r\n<div id=\"fs-id1170571109518\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex](g\\circ f)(x)=g(f(x))=g(x^2+1)=\\dfrac{1}{x^2+1}[\/latex].<\/div>\r\n<p>Alternatively, we could write<\/p>\r\n<div id=\"fs-id1170573521539\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex](g\\circ f)(x)=g(f(x))=\\dfrac{1}{f(x)}=\\dfrac{1}{x^2+1}[\/latex].<\/div>\r\n<p>Since [latex]x^2+1\\ne 0[\/latex] for all real numbers [latex]x[\/latex], the domain of [latex](g\\circ f)(x)[\/latex] is the set of all real numbers.<\/p>\r\n<p>Since [latex]0&lt;\\frac{1}{(x^2+1)}\\le 1[\/latex], the range is, at most, the interval [latex](0,1][\/latex]. To show that the range is this entire interval, we let [latex]y=\\frac{1}{(x^2+1)}[\/latex] and solve this equation for [latex]x[\/latex] to show that for all [latex]y[\/latex] in the interval [latex](0,1][\/latex], there exists a real number [latex]x[\/latex] such that [latex]y=\\frac{1}{(x^2+1)}[\/latex].<\/p>\r\n<p>Solving this equation for [latex]x[\/latex], we see that [latex]x^2+1=\\frac{1}{y}[\/latex], which implies that<\/p>\r\n<div id=\"fs-id1170573368782\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]x=\u00b1\\sqrt{\\frac{1}{y}-1}[\/latex].<\/div>\r\n<p>If [latex]y[\/latex] is in the interval [latex](0,1][\/latex], the expression under the radical is nonnegative, and therefore there exists a real number [latex]x[\/latex] such that [latex]\\frac{1}{(x^2+1)}=y[\/latex]. We conclude that the range of [latex]g\\circ f[\/latex] is the interval [latex](0,1][\/latex].<\/p>\r\n<\/li>\r\n\t<li>[latex](g\\circ f)(4)=g(f(4))=g(4^2+1)=g(17)=\\frac{1}{17}[\/latex]<br \/>\r\n[latex](g\\circ f)(-\\frac{1}{2})=g(f(-\\frac{1}{2}))=g((-\\frac{1}{2})^2+1)=g(\\frac{5}{4})=\\frac{4}{5}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572483665\">Let [latex]f(x)=2-5x[\/latex]<\/p>\r\n<p>Let [latex]g(x)=\\sqrt{x}[\/latex]<\/p>\r\n<p>Find [latex](f\\circ g)(x)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572481428\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572481428\"]<\/p>\r\n<p id=\"fs-id1170572481428\">[latex](f\\circ g)(x)=2-5\\sqrt{x}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572173799\">Consider the functions [latex]f[\/latex] and [latex]g[\/latex] described below.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]x[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]f(x)[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]x[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]-4[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]g(x)[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol id=\"fs-id1170572242030\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Evaluate [latex](f\\circ f)(3)[\/latex] and [latex](f\\circ f)(1)[\/latex].<\/li>\r\n\t<li>State the domain and range of [latex](f\\circ f)(x)[\/latex].<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170572552174\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572552174\"]<\/p>\r\n<ol id=\"fs-id1170572552174\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>[latex](f\\circ f)(3)=f(f(3))=f(-2)=4[\/latex]<br \/>\r\n[latex](f\\circ f)(1)=f(f(1))=f(-2)=4[\/latex]<\/li>\r\n\t<li>The domain of [latex]f\\circ f[\/latex] is the set [latex]\\{-3,-2,-1,0,1,2,3,4\\}[\/latex]. Since the range of [latex]f[\/latex] is the set [latex]\\{-2,0,2,4\\}[\/latex], the range of [latex]f\\circ f[\/latex] is the set [latex]\\{0,4\\}[\/latex].<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572480218\">If items are on sale for [latex]10\\%[\/latex] off their original price, and a customer has a coupon for an additional [latex]30\\%[\/latex] off, what will be the final price for an item that is originally [latex]x[\/latex] dollars, after applying the coupon to the sale price?<\/p>\r\n<p>[reveal-answer q=\"324566\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"324566\"]<\/p>\r\n<p>The sale price of an item with an original price of [latex]x[\/latex] dollars is [latex]f(x)=0.90x[\/latex]. The coupon price for an item that is [latex]y[\/latex] dollars is [latex]g(y)=0.70y[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572213178\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572213178\"]<\/p>\r\n<p id=\"fs-id1170572213178\">[latex](g\\circ f)(x)=0.63x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify zeros of functions using equations, graphs, and tables<\/li>\n<li>Interpret graphs and tables to describe their behavior including properties of symmetry<\/li>\n<li>Make new functions from two or more given functions<\/li>\n<\/ul>\n<\/section>\n<h2>Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">A function is a special type of relation where each input is associated with exactly one output.<\/li>\n<li class=\"whitespace-normal break-words\">Functions are used to describe relationships between two sets.<\/li>\n<li class=\"whitespace-normal break-words\">The input of a function is called the independent variable, often denoted as [latex]x[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">The output of a function is called the dependent variable, often denoted as [latex]y[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Function notation: [latex]y = f(x)[\/latex] is read as &#8220;[latex]y[\/latex] equals [latex]f[\/latex] of [latex]x[\/latex].&#8221;<\/li>\n<li class=\"whitespace-normal break-words\">Functions can be represented using tables, graphs, or formulas (algebraic expressions).<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Consider the function [latex]g(x) = x\u00b2 - 4x + 3[\/latex]<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Evaluate [latex]g(3)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Create a small table of values for [latex]x = 0, 1, 2, 3, 4[\/latex]<\/li>\n<\/ol>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q826228\">Show Answer<\/button><\/p>\n<div id=\"q826228\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a0[latex]g(3) = 3\u00b2 - 4(3) + 3 = 9 - 12 + 3 = 0[\/latex]<\/li>\n<li>\n<table>\n<thead>\n<tr>\n<th>x<\/th>\n<th>g(x)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/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<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>The Domain and Range of a Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">The domain of a function is the complete set of possible input values ([latex]x[\/latex]-values).<\/li>\n<li class=\"whitespace-normal break-words\">The range of a function is the complete set of possible output values ([latex]y[\/latex]-values).<\/li>\n<li class=\"whitespace-normal break-words\">Domain restrictions can occur due to mathematical limitations (e.g., division by zero, square roots of negative numbers).<\/li>\n<li class=\"whitespace-normal break-words\">Set notation and interval notation are used to describe domains and ranges.<\/li>\n<li class=\"whitespace-normal break-words\">The union symbol ([latex]\\cup[\/latex]) is used to describe domains and ranges with gaps or breaks.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Determine the domain and range of the function[latex]f(x) = |x + 1| - 3[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q143942\">Show Answer<\/button><\/p>\n<div id=\"q143942\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Determine the domain:\n<p>The absolute value function is defined for all real numbers meaning there are no restrictions on the input. So the domain is:<\/p>\n<div style=\"text-align: center;\">\u00a0[latex](-\\infty, \\infty)[\/latex] or [latex]{x | x \\text{ is any real number}}[\/latex]<\/div>\n<\/li>\n<li>Determine the range:\n<p>The absolute value [latex]|x + 1|[\/latex] is always non-negative. The minimum value of [latex]|x + 1|[\/latex] is [latex]0[\/latex], which occurs when [latex]x = -1[\/latex].<\/p>\n<p>Subtracting [latex]3[\/latex] shifts the entire function down by [latex]3[\/latex] units. The minimum value of [latex]f(x)[\/latex] is therefore[latex]-3[\/latex].<\/p>\n<p>There is no upper limit to the function&#8217;s value. Making the range:<\/p>\n<div style=\"text-align: center;\">[latex][-3, \u221e)[\/latex] or [latex]{y | y \u2265 -3}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Intercepts of a Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts are points where a function&#8217;s graph crosses the [latex]x[\/latex]-axis ([latex]f(x) = 0[\/latex]).<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts are also called zeros or roots of the function.<\/li>\n<li class=\"whitespace-normal break-words\">The [latex]y[\/latex]-intercept is the point where a function&#8217;s graph crosses the [latex]y[\/latex]-axis ([latex]x = 0[\/latex]).<\/li>\n<li class=\"whitespace-normal break-words\">A function can have multiple [latex]x[\/latex]-intercepts but at most one y-intercept.<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts provide information about the shape of the function&#8217;s graph.<\/li>\n<li class=\"whitespace-normal break-words\">The [latex]y[\/latex]-intercept represents the function&#8217;s output when the input is zero.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572228574\">Consider the function [latex]f(x)=\\sqrt{x+3}+1[\/latex].<\/p>\n<ol id=\"fs-id1170571053593\" style=\"list-style-type: lower-alpha;\">\n<li>Find all zeros of [latex]f[\/latex].<\/li>\n<li>Find the [latex]y[\/latex]-intercept (if any).<\/li>\n<li>Sketch a graph of [latex]f[\/latex].<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572168266\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572168266\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572168266\" style=\"list-style-type: lower-alpha;\">\n<li>To find the zeros, solve [latex]\\sqrt{x+3}+1=0[\/latex]. This equation implies [latex]\\sqrt{x+3}=-1[\/latex]. Since [latex]\\sqrt{x+3}\\ge 0[\/latex] for all [latex]x[\/latex], this equation has no solutions, and therefore [latex]f[\/latex] has no zeros.<\/li>\n<li>The [latex]y[\/latex]-intercept is given by [latex](0,f(0))=(0,\\sqrt{3}+1)[\/latex].<\/li>\n<li>To graph this function, we make a table of values. Since we need [latex]x+3\\ge 0[\/latex], we need to choose values of [latex]x\\ge -3[\/latex]. We choose values that make the square-root function easy to evaluate.<br \/>\n<table id=\"fs-id1170572247842\" class=\"column-header\" summary=\"A table with 2 rows and 3 columns. The first row is labeled \u201cx\u201d and has the values \u201c-3; -2; 1\u201d. The second row is labeled \u201cf(x)\u201d and has the values \u201c1; 2; 3\u201d.\">\n<tbody>\n<tr valign=\"top\">\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]f(x)[\/latex]<\/td>\n<td>[latex]1[\/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 id=\"fs-id1170572205983\">Making use of the table and knowing that, since the function is a square root, the graph of [latex]f[\/latex] should be similar to the graph of [latex]y=\\sqrt{x}[\/latex], we sketch the graph (Figure 9).<\/p>\n<figure style=\"width: 315px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202126\/CNX_Calc_Figure_01_01_008.jpg\" alt=\"An image of a graph. The y axis runs from -2 to 4 and the x axis runs from -3 to 2. The graph is of the function \u201cf(x) = (square root of x + 3) + 1\u201d, which is an increasing curved function that starts at the point (-3, 1). There are 3 points plotted on the function at (-3, 1), (-2, 2), and (1, 3). The function has a y intercept at (0, 1 + square root of 3).\" width=\"315\" height=\"272\" \/><figcaption class=\"wp-caption-text\">Figure 9. The graph of [latex]f(x)=\\sqrt{x+3}+1[\/latex] has a [latex]y[\/latex]-intercept but no [latex]x[\/latex]-intercepts.<\/figcaption><\/figure>\n<div class=\"wp-caption-text\">\u00a0<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572549227\">Find the zeros of [latex]f(x)=x^3-5x^2+6x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q098321\">Hint<\/button><\/p>\n<div id=\"q098321\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042323386\">Factor the polynomial.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572216813\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572216813\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572216813\">[latex]x=0,2,3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2 class=\"entry-title\">Symmetry of Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Function Symmetry:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Symmetry about the [latex]y[\/latex]-axis: [latex]f(x) = f(-x)[\/latex] (even functions)<\/li>\n<li class=\"whitespace-normal break-words\">Symmetry about the origin: [latex]f(-x) = -f(x)[\/latex] (odd functions)<\/li>\n<li class=\"whitespace-normal break-words\">Some functions are neither even nor odd<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Value Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Defined as [latex]f(x) = |x| = x[\/latex]if [latex]x \u2265 0[\/latex], and [latex]-x[\/latex] if [latex]x < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Always outputs non-negative values<\/li>\n<li class=\"whitespace-normal break-words\">Symmetric about the [latex]y[\/latex]-axis (even function)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202138\/CNX_Calc_Figure_01_01_012.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201c(a), symmetry about the y-axis\u201d and is of the curved function \u201cf(x) = (x to the 4th) - 2(x squared) - 3\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. This function decreases until it hits the point (-1, -4), which is minimum of the function. Then the graph increases to the point (0,3), which is a local maximum. Then the the graph decreases until it hits the point (1, -4), before it increases again. The second graph is labeled \u201c(b), symmetry about the origin\u201d and is of the curved function \u201cf(x) = x cubed - 4x\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. The graph of the function starts at the x intercept at (-2, 0) and increases until the approximate point of (-1.2, 3.1). The function then decreases, passing through the origin, until it hits the approximate point of (1.2, -3.1). The function then begins to increase again and has another x intercept at (2, 0).\" width=\"731\" height=\"426\" \/><\/p>\n<p>Figure 13. (a) A graph that is symmetric about the [latex]y[\/latex]-axis. (b) A graph that is symmetric about the origin.<\/p>\n<p id=\"fs-id1170572173116\">If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function [latex]f[\/latex] has symmetry? Looking at Figure 13(a) again, we see that since [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis, if the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,y)[\/latex] is on the graph. In other words, [latex]f(\u2212x)=f(x)[\/latex]. If a function [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>even function<\/strong>, which has symmetry about the [latex]y[\/latex]-axis. For example, [latex]f(x)=x^2[\/latex] is even because<\/p>\n<div id=\"fs-id1170572552270\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(\u2212x)=(\u2212x)^2=x^2=f(x)[\/latex].<\/div>\n<p id=\"fs-id1170572552328\">In contrast, looking at Figure 13(b) again, if a function [latex]f[\/latex] is symmetric about the origin, then whenever the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,\u2212y)[\/latex] is also on the graph. In other words, [latex]f(\u2212x)=\u2212f(x)[\/latex]. If [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>odd function<\/strong>, which has symmetry about the origin. For example, [latex]f(x)=x^3[\/latex] is odd because<\/p>\n<div style=\"text-align: center;\">[latex]f(\u2212x)=(\u2212x)^3=\u2212x^3=\u2212f(x)[\/latex].<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572478824\">Determine whether [latex]f(x)=4x^3-5x[\/latex] is even, odd, or neither.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q935578\">Hint<\/button><\/p>\n<div id=\"q935578\" class=\"hidden-answer\" style=\"display: none\">\n<p>Compare [latex]f(\u2212x)[\/latex] with [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572547379\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572547379\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572547379\">[latex]f(x)[\/latex] is odd.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3>Absolute Value Functions<\/h3>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572548622\">For the function [latex]f(x)=|x+2|-4[\/latex], find the domain and range.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q882365\">Hint<\/button><\/p>\n<div id=\"q882365\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043422402\">[latex]|x+2|\\ge 0[\/latex] for all real numbers [latex]x[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572176864\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572176864\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572176864\">Domain = [latex](\u2212\\infty ,\\infty )[\/latex], range = [latex]\\{y|y\\ge -4\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Composing Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ol>\n<li class=\"whitespace-normal break-words\">Combining Functions with Mathematical Operators:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sum: [latex](f + g)(x) = f(x) + g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Difference :[latex](f - g)(x) = f(x) - g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Product: [latex](f \u00b7 g)(x) = f(x) \u00b7 g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quotient: [latex](\\frac{f}{g})(x) = \\frac{f(x)}{g(x)}\\text{, where }g(x) \u2260 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Composite Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](f \u2218 g)(x) = f(g(x))[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The output of g becomes the input of f<\/li>\n<li class=\"whitespace-normal break-words\">Order matters: [latex](f \u2218 g)(x) \u2260 (g \u2218 f)(x)[\/latex] in general<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>In general [latex]f\\circ g[\/latex] and [latex]g\\circ f[\/latex] are different functions. In other words in many cases [latex]f\\left(g\\left(x\\right)\\right)\\ne g\\left(f\\left(x\\right)\\right)[\/latex] for all [latex]x[\/latex].<\/p>\n<p>For example if [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]g\\left(x\\right)=x+2[\/latex], then<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(g\\left(x\\right)\\right)&=f\\left(x+2\\right) \\\\[2mm] &={\\left(x+2\\right)}^{2} \\\\[2mm] &={x}^{2}+4x+4\\hfill \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left;\">but<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}g\\left(f\\left(x\\right)\\right)&=g\\left({x}^{2}\\right) \\\\[2mm] \\text{ }&={x}^{2}+2\\hfill \\end{align}[\/latex]<\/p>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572481359\">Consider the functions [latex]f(x)=x^2+1[\/latex] and [latex]g(x)=\\frac{1}{x}[\/latex].<\/p>\n<ol id=\"fs-id1170572482111\" style=\"list-style-type: lower-alpha;\">\n<li>Find [latex](g\\circ f)(x)[\/latex] and state its domain and range.<\/li>\n<li>Evaluate [latex](g\\circ f)(4)[\/latex] and [latex](g\\circ f)\\left(-\\frac{1}{2}\\right)[\/latex].<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572222946\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572222946\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572222946\" style=\"list-style-type: lower-alpha;\">\n<li>We can find the formula for [latex](g\\circ f)(x)[\/latex] in two different ways. We could write\n<div id=\"fs-id1170571109518\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex](g\\circ f)(x)=g(f(x))=g(x^2+1)=\\dfrac{1}{x^2+1}[\/latex].<\/div>\n<p>Alternatively, we could write<\/p>\n<div id=\"fs-id1170573521539\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex](g\\circ f)(x)=g(f(x))=\\dfrac{1}{f(x)}=\\dfrac{1}{x^2+1}[\/latex].<\/div>\n<p>Since [latex]x^2+1\\ne 0[\/latex] for all real numbers [latex]x[\/latex], the domain of [latex](g\\circ f)(x)[\/latex] is the set of all real numbers.<\/p>\n<p>Since [latex]0<\\frac{1}{(x^2+1)}\\le 1[\/latex], the range is, at most, the interval [latex](0,1][\/latex]. To show that the range is this entire interval, we let [latex]y=\\frac{1}{(x^2+1)}[\/latex] and solve this equation for [latex]x[\/latex] to show that for all [latex]y[\/latex] in the interval [latex](0,1][\/latex], there exists a real number [latex]x[\/latex] such that [latex]y=\\frac{1}{(x^2+1)}[\/latex].<\/p>\n<p>Solving this equation for [latex]x[\/latex], we see that [latex]x^2+1=\\frac{1}{y}[\/latex], which implies that<\/p>\n<div id=\"fs-id1170573368782\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]x=\u00b1\\sqrt{\\frac{1}{y}-1}[\/latex].<\/div>\n<p>If [latex]y[\/latex] is in the interval [latex](0,1][\/latex], the expression under the radical is nonnegative, and therefore there exists a real number [latex]x[\/latex] such that [latex]\\frac{1}{(x^2+1)}=y[\/latex]. We conclude that the range of [latex]g\\circ f[\/latex] is the interval [latex](0,1][\/latex].<\/p>\n<\/li>\n<li>[latex](g\\circ f)(4)=g(f(4))=g(4^2+1)=g(17)=\\frac{1}{17}[\/latex]<br \/>\n[latex](g\\circ f)(-\\frac{1}{2})=g(f(-\\frac{1}{2}))=g((-\\frac{1}{2})^2+1)=g(\\frac{5}{4})=\\frac{4}{5}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572483665\">Let [latex]f(x)=2-5x[\/latex]<\/p>\n<p>Let [latex]g(x)=\\sqrt{x}[\/latex]<\/p>\n<p>Find [latex](f\\circ g)(x)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572481428\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572481428\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572481428\">[latex](f\\circ g)(x)=2-5\\sqrt{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572173799\">Consider the functions [latex]f[\/latex] and [latex]g[\/latex] described below.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-3[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]g(x)[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1170572242030\" style=\"list-style-type: lower-alpha;\">\n<li>Evaluate [latex](f\\circ f)(3)[\/latex] and [latex](f\\circ f)(1)[\/latex].<\/li>\n<li>State the domain and range of [latex](f\\circ f)(x)[\/latex].<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572552174\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572552174\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572552174\" style=\"list-style-type: lower-alpha;\">\n<li>[latex](f\\circ f)(3)=f(f(3))=f(-2)=4[\/latex]<br \/>\n[latex](f\\circ f)(1)=f(f(1))=f(-2)=4[\/latex]<\/li>\n<li>The domain of [latex]f\\circ f[\/latex] is the set [latex]\\{-3,-2,-1,0,1,2,3,4\\}[\/latex]. Since the range of [latex]f[\/latex] is the set [latex]\\{-2,0,2,4\\}[\/latex], the range of [latex]f\\circ f[\/latex] is the set [latex]\\{0,4\\}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572480218\">If items are on sale for [latex]10\\%[\/latex] off their original price, and a customer has a coupon for an additional [latex]30\\%[\/latex] off, what will be the final price for an item that is originally [latex]x[\/latex] dollars, after applying the coupon to the sale price?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q324566\">Hint<\/button><\/p>\n<div id=\"q324566\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sale price of an item with an original price of [latex]x[\/latex] dollars is [latex]f(x)=0.90x[\/latex]. The coupon price for an item that is [latex]y[\/latex] dollars is [latex]g(y)=0.70y[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572213178\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572213178\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572213178\">[latex](g\\circ f)(x)=0.63x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":143,"module-header":"fresh_take","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/856"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/856\/revisions"}],"predecessor-version":[{"id":3623,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/856\/revisions\/3623"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/856\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=856"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=856"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=856"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=856"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}