{"id":124,"date":"2023-06-21T13:22:35","date_gmt":"2023-06-21T13:22:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-and-solve-functions\/"},"modified":"2023-08-21T21:41:29","modified_gmt":"2023-08-21T21:41:29","slug":"evaluate-and-solve-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-and-solve-functions\/","title":{"raw":"\u25aa   Evaluating and Solving Functions","rendered":"\u25aa   Evaluating and Solving Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate and solve functions in algebraic form.<\/li>\r\n \t<li>Evaluate functions given tabular or graphical data.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\\left(x\\right)=5 - 3{x}^{2}[\/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.\r\n<div class=\"textbox\">\r\n<h3><strong>How To: EVALUATE A FUNCTION Given ITS FORMula.<\/strong><\/h3>\r\n<ol>\r\n \t<li>Replace the input variable in the formula with the value provided.<\/li>\r\n \t<li>Calculate the result.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Functions<\/h3>\r\nGiven the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], evaluate [latex]h\\left(4\\right)[\/latex].\r\n\r\n[reveal-answer q=\"768180\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"768180\"]\r\n<p style=\"text-align: left;\">To evaluate [latex]h\\left(4\\right)[\/latex], we substitute the value 4 for the input variable [latex]p[\/latex] in the given function.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}h\\left(p\\right)&amp;={p}^{2}+2p \\\\ h\\left(4\\right)&amp;={\\left(4\\right)}^{2}+2\\left(4\\right) \\\\ &amp;=16+8 \\\\ &amp;=24 \\end{align}[\/latex]<\/p>\r\nTherefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the video below for more examples of evaluating a function for specific values of the input.\r\n\r\nhttps:\/\/youtu.be\/Ehkzu5Uv7O0\r\n<div class=\"textbox examples\">\r\n<h3>evaluating functions<\/h3>\r\nWhen evaluating functions, it's handy to wrap the input variable in parentheses before making the substitution.\r\n\r\nEx. Given [latex]f(x)=x^2 - 8[\/latex], find [latex]f(-3)[\/latex]\r\n\r\n[latex]\\begin{align}f(x)&amp;=(x)^2 - 8 \\\\ &amp;= (-3)^2 - 8 \\\\ &amp;= 9 - 8 \\\\ &amp;= 1\\end{align}[\/latex]\r\n\r\nThe value of the function\u00a0[latex]f(x)=x^2 - 8[\/latex], at the input [latex]x=-3[\/latex], is [latex]1[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Functions at Specific Values<\/h3>\r\nFor the function, [latex]f\\left(x\\right)={x}^{2}+3x - 4[\/latex], evaluate each of the following.\r\n<ol>\r\n \t<li>[latex]f\\left(2\\right)[\/latex]<\/li>\r\n \t<li>[latex]f(a)[\/latex]<\/li>\r\n \t<li>[latex]f(a+h)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"645951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"645951\"]\r\n\r\nReplace the [latex]x[\/latex]\u00a0in the function with each specified value.\r\n<ol>\r\n \t<li>Because the input value is a number, 2, we can use algebra to simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&amp;={2}^{2}+3\\left(2\\right)-4 \\\\ &amp;=4+6 - 4 \\\\ &amp;=6\\hfill \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>In this case, the input value is a letter so we cannot simplify the answer any further.\r\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div><\/li>\r\n \t<li>With an input value of [latex]a+h[\/latex], we must use the distributive property.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}f\\left(a+h\\right)&amp;={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4 \\\\[2mm] &amp;={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\r\n<div style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[\/latex]<\/div>\r\nand we know that\r\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\r\nNow we combine the results and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}&amp;=\\dfrac{\\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\\right)-\\left({a}^{2}+3a - 4\\right)}{h} \\\\[2mm] &amp;=\\dfrac{2ah+{h}^{2}+3h}{h}\\\\[2mm] &amp;=\\frac{h\\left(2a+h+3\\right)}{h}&amp;&amp;\\text{Factor out }h. \\\\[2mm] &amp;=2a+h+3&amp;&amp;\\text{Simplify}.\\end{align}[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"300\"]1647[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], evaluate [latex]g\\left(5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"273881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"273881\"]\r\n\r\n[latex]g\\left(5\\right)=\\sqrt{5 - 4}=1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"130\"]97486[\/ohm_question]\r\n\r\n<\/div>\r\nIn addition to <strong>evaluating functions<\/strong> for a particular input, we can also <strong>solve functions<\/strong> for the input that creates a particular output.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving Functions<\/h3>\r\nGiven the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], solve for [latex]h\\left(p\\right)=3[\/latex].\r\n\r\n[reveal-answer q=\"119909\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"119909\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;h\\left(p\\right)=3\\\\ &amp;{p}^{2}+2p=3 &amp;&amp;\\text{Substitute the original function }h\\left(p\\right)={p}^{2}+2p. \\\\ &amp;{p}^{2}+2p - 3=0 &amp;&amp;\\text{Subtract 3 from each side}. \\\\ &amp;\\left(p+3\\text{)(}p - 1\\right)=0 &amp;&amp;\\text{Factor}. \\end{align}[\/latex]<\/p>\r\nIf [latex]\\left(p+3\\right)\\left(p - 1\\right)=0[\/latex], either [latex]\\left(p+3\\right)=0[\/latex] or [latex]\\left(p - 1\\right)=0[\/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[\/latex] in each case.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;p+3=0, &amp;&amp;p=-3 \\\\ &amp;p - 1=0, &amp;&amp;p=1\\hfill \\end{align}[\/latex]<\/p>\r\nThis gives us two solutions. The output [latex]h\\left(p\\right)=3[\/latex] when the input is either [latex]p=1[\/latex] or [latex]p=-3[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190959\/CNX_Precalc_Figure_01_01_0062.jpg\" alt=\"Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).\" width=\"487\" height=\"459\" \/>\r\n\r\nWe can also verify by graphing as in Figure 5. The graph verifies that [latex]h\\left(1\\right)=h\\left(-3\\right)=3[\/latex] and [latex]h\\left(4\\right)=24[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3><strong>How To: Solve a Function.<\/strong><\/h3>\r\n<ol>\r\n \t<li>Replace the output in the formula with the value provided.<\/li>\r\n \t<li>Solve for the input variable that makes the statement true.<\/li>\r\n<\/ol>\r\n<\/div>\r\nThe next video shows another example of how to\u00a0solve a function.\r\n\r\nhttps:\/\/youtu.be\/GLOmTED1UwA\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], solve [latex]g\\left(m\\right)=2[\/latex].\r\n\r\n[reveal-answer q=\"480629\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"480629\"]\r\n\r\n[latex]m=8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"300\"]15766[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Evaluating Functions Expressed in Formulas<\/h2>\r\nSome functions are defined by mathematical rules or procedures expressed in <strong>equation<\/strong> form. If it is possible to express the function output with a <strong>formula<\/strong> involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[\/latex] expresses a functional relationship between [latex]n[\/latex]\u00a0and [latex]p[\/latex]. We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>functions, Equations, and formulas<\/h3>\r\nWe've seen that an equation such as [latex]ax+by=c[\/latex] can be written in a different form by solving the equation for one of the variables. If we solve this linear equation for\u00a0<em>y\u00a0<\/em>it can be written in the slope-intercept form of a line, [latex]y = mx+b[\/latex].\r\n\r\nCertain formulas can be written in function form by solving for one of the variables. For instance, can you see how to solve the formula for a rectangle having a perimeter of 21 feet, [latex]21 = 2l + 2w[\/latex], for length?\r\n\r\n[reveal-answer q=\"773176\"]more[\/reveal-answer]\r\n[hidden-answer a=\"773176\"]\r\n\r\n[latex]\\begin{align} 21 &amp;= 2l + 2w \\\\ 21 - 2w &amp;= 2l \\\\ \\dfrac{21-2w}{2} &amp;= l \\end{align}[\/latex]\r\n\r\nWe can now declare a function, [latex]l = f(x)[\/latex] that returns an output length for a rectangle having a perimeter of 21 feet based on different width inputs.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function in equation form, write its algebraic formula.<\/h3>\r\n<ol>\r\n \t<li>Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves <em>only<\/em> the input variable.<\/li>\r\n \t<li>Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding an Equation of a Function<\/h3>\r\nExpress the relationship [latex]2n+6p=12[\/latex] as a function [latex]p=f\\left(n\\right)[\/latex], if possible.\r\n\r\n[reveal-answer q=\"938453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"938453\"]\r\n\r\nTo express the relationship in this form, we need to be able to write the relationship where [latex]p[\/latex] is a function of [latex]n[\/latex], which means writing it as [latex]p=[\/latex] expression involving [latex]n[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;2n+6p=12\\\\[1mm] &amp;6p=12 - 2n &amp;&amp;\\text{Subtract }2n\\text{ from both sides}. \\\\[1mm] &amp;p=\\frac{12 - 2n}{6} &amp;&amp;\\text{Divide both sides by 6 and simplify}. \\\\[1mm] &amp;p=\\frac{12}{6}-\\frac{2n}{6} \\\\[1mm] &amp;p=2-\\frac{1}{3}n \\end{align}[\/latex]<\/p>\r\nTherefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as\r\n<p style=\"text-align: center;\">[latex]p=f\\left(n\\right)=2-\\frac{1}{3}n[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nIt is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch this video to see another example of how to express an equation as a function.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=lHTLjfPpFyQ\r\n\r\nSometimes a relationship between variables cannot be expressed as a function. See the example below for more information.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Expressing the Equation of a Circle as a Function<\/h3>\r\nDoes the equation [latex]{x}^{2}+{y}^{2}=1[\/latex] represent a function with [latex]x[\/latex] as input and [latex]y[\/latex] as output? If so, express the relationship as a function [latex]y=f\\left(x\\right)[\/latex].\r\n\r\n[reveal-answer q=\"557070\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"557070\"]\r\n\r\nFirst we subtract [latex]{x}^{2}[\/latex] from both sides.\r\n<p style=\"text-align: center;\">[latex]{y}^{2}=1-{x}^{2}[\/latex]<\/p>\r\nWe now try to solve for [latex]y[\/latex] in this equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\pm \\sqrt{1-{x}^{2}} \\\\[1mm] &amp;=\\sqrt{1-{x}^{2}}\\hspace{3mm}\\text{and}\\hspace{3mm}-\\sqrt{1-{x}^{2}} \\end{align}[\/latex]<\/p>\r\nWe get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\\left(x\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIf [latex]x - 8{y}^{3}=0[\/latex], express [latex]y[\/latex] as a function of [latex]x[\/latex].\r\n\r\n[reveal-answer q=\"933974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"933974\"][latex]y=f\\left(x\\right)=\\cfrac{\\sqrt[3]{x}}{2}[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"320\"]111699[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?<\/strong>\r\n\r\n<em>Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y}[\/latex], if we want to express [latex]y[\/latex] as a function of [latex]x[\/latex], there is no simple algebraic formula involving only [latex]x[\/latex] that equals [latex]y[\/latex]. However, each [latex]x[\/latex] does determine a unique value for [latex]y[\/latex], and there are mathematical procedures by which [latex]y[\/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[\/latex] as a function of [latex]x[\/latex], even though the formula cannot be written explicitly.<\/em>\r\n\r\n<\/div>\r\n<h2>Evaluating a Function Given in Tabular Form<\/h2>\r\nAs we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.\r\n\r\nThe function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.\r\n<table summary=\"Six rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th>Pet<\/th>\r\n<th>Memory span in hours<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Puppy<\/td>\r\n<td>0.008<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Adult dog<\/td>\r\n<td>0.083<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cat<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Goldfish<\/td>\r\n<td>2160<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Beta fish<\/td>\r\n<td>3600<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAt times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[\/latex].\r\n\r\nThe <strong>domain<\/strong> of the function is the type of pet and the range is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of \"goldfish.\" We would write [latex]P\\left(\\text{goldfish}\\right)=2160[\/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[\/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function represented by a table, identify specific output and input values.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Find the given input in the row (or column) of input values.<\/li>\r\n \t<li>Identify the corresponding output value paired with that input value.<\/li>\r\n \t<li>Find the given output values in the row (or column) of output values, noting every time that output value appears.<\/li>\r\n \t<li>Identify the input value(s) corresponding to the given output value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating and Solving a Tabular Function<\/h3>\r\nUsing the table below,\r\n<ol>\r\n \t<li>Evaluate [latex]g\\left(3\\right)[\/latex].<\/li>\r\n \t<li>Solve [latex]g\\left(n\\right)=6[\/latex].<\/li>\r\n<\/ol>\r\n<table summary=\"Two rows and six columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"15206\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15206\"]\r\n<ul>\r\n \t<li>Evaluating [latex]g\\left(3\\right)[\/latex] means determining the output value of the function [latex]g[\/latex] for the input value of [latex]n=3[\/latex]. The table output value corresponding to [latex]n=3[\/latex] is 7, so [latex]g\\left(3\\right)=7[\/latex].<\/li>\r\n \t<li>Solving [latex]g\\left(n\\right)=6[\/latex] means identifying the input values, [latex]n[\/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[\/latex] and [latex]n=4[\/latex].<\/li>\r\n<\/ul>\r\n<table summary=\"Two rows and six columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen we input 2 into the function [latex]g[\/latex], our output is 6. When we input 4 into the function [latex]g[\/latex], our output is also 6.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the table from the previous example, evaluate [latex]g\\left(1\\right)[\/latex] .\r\n\r\n[reveal-answer q=\"724802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"724802\"][latex]g\\left(1\\right)=8[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"400\"]3751[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Finding Function Values from a Graph<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>ordered pairs of inputs and outputs<\/h3>\r\nWe can view a function as a set of inputs and their corresponding outputs. That is, we can see a function as a set of ordered pairs, [latex]\\left(x, y \\right).[\/latex]\r\n\r\nRemember that, in function notation, [latex]y = f(x)[\/latex], so the ordered pairs containing inputs and outputs can be written in the form of (<em>input<\/em>,\u00a0<em>output<\/em>) or [latex]\\left(x, f(x)\\right)[\/latex].\r\n\r\n<\/div>\r\nEvaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Reading Function Values from a Graph<\/h3>\r\nGiven the graph below,\r\n<ol>\r\n \t<li>Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\r\n \t<li>Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191001\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/>\r\n[reveal-answer q=\"915833\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915833\"]\r\n<ol>\r\n \t<li>To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the [latex]y[\/latex]-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex].<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/li>\r\n \t<li>To solve [latex]f\\left(x\\right)=4[\/latex], we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4[\/latex], we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex]. These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]x=-1[\/latex] or [latex]x=3[\/latex]. This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4[\/latex], or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex] See the graph below.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191006\/CNX_Precalc_Figure_01_01_0092.jpg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the graph, solve [latex]f\\left(x\\right)=1[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/>\r\n[reveal-answer q=\"529772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"529772\"]\r\n\r\n[latex]x=0[\/latex] or [latex]x=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"480\"]2471[\/ohm_question]\r\n\r\n[ohm_question height=\"470\"]2886[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nYou can use an online graphing calculator to graph functions, find function values, and evaluate functions. Watch this short tutorial to learn how to within Desmos. Other online graphing tools will be slightly different.\r\nhttps:\/\/youtu.be\/jACDzJ-rmsM\r\nNow try the following with an online graphing calculator:\r\n<ol>\r\n \t<li>Graph the function [latex]f(x) = -\\frac{1}{2}x^2+x+4[\/latex] using function notation.<\/li>\r\n \t<li>Evaluate the function at [latex]x=1[\/latex]<\/li>\r\n \t<li>Make a table of values that references the function. Include at least the interval [latex][-5,5][\/latex] for [latex]x[\/latex]-values.<\/li>\r\n \t<li>Solve the function for [latex]f(0)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate and solve functions in algebraic form.<\/li>\n<li>Evaluate functions given tabular or graphical data.<\/li>\n<\/ul>\n<\/div>\n<p>When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\\left(x\\right)=5 - 3{x}^{2}[\/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.<\/p>\n<div class=\"textbox\">\n<h3><strong>How To: EVALUATE A FUNCTION Given ITS FORMula.<\/strong><\/h3>\n<ol>\n<li>Replace the input variable in the formula with the value provided.<\/li>\n<li>Calculate the result.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Functions<\/h3>\n<p>Given the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], evaluate [latex]h\\left(4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q768180\">Show Solution<\/span><\/p>\n<div id=\"q768180\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">To evaluate [latex]h\\left(4\\right)[\/latex], we substitute the value 4 for the input variable [latex]p[\/latex] in the given function.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}h\\left(p\\right)&={p}^{2}+2p \\\\ h\\left(4\\right)&={\\left(4\\right)}^{2}+2\\left(4\\right) \\\\ &=16+8 \\\\ &=24 \\end{align}[\/latex]<\/p>\n<p>Therefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the video below for more examples of evaluating a function for specific values of the input.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Evaluating Functions Using Function Notation (L9.3)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/Ehkzu5Uv7O0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox examples\">\n<h3>evaluating functions<\/h3>\n<p>When evaluating functions, it&#8217;s handy to wrap the input variable in parentheses before making the substitution.<\/p>\n<p>Ex. Given [latex]f(x)=x^2 - 8[\/latex], find [latex]f(-3)[\/latex]<\/p>\n<p>[latex]\\begin{align}f(x)&=(x)^2 - 8 \\\\ &= (-3)^2 - 8 \\\\ &= 9 - 8 \\\\ &= 1\\end{align}[\/latex]<\/p>\n<p>The value of the function\u00a0[latex]f(x)=x^2 - 8[\/latex], at the input [latex]x=-3[\/latex], is [latex]1[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Functions at Specific Values<\/h3>\n<p>For the function, [latex]f\\left(x\\right)={x}^{2}+3x - 4[\/latex], evaluate each of the following.<\/p>\n<ol>\n<li>[latex]f\\left(2\\right)[\/latex]<\/li>\n<li>[latex]f(a)[\/latex]<\/li>\n<li>[latex]f(a+h)[\/latex]<\/li>\n<li>[latex]\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q645951\">Show Solution<\/span><\/p>\n<div id=\"q645951\" class=\"hidden-answer\" style=\"display: none\">\n<p>Replace the [latex]x[\/latex]\u00a0in the function with each specified value.<\/p>\n<ol>\n<li>Because the input value is a number, 2, we can use algebra to simplify.\n<div style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&={2}^{2}+3\\left(2\\right)-4 \\\\ &=4+6 - 4 \\\\ &=6\\hfill \\end{align}[\/latex]<\/div>\n<\/li>\n<li>In this case, the input value is a letter so we cannot simplify the answer any further.\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\n<\/li>\n<li>With an input value of [latex]a+h[\/latex], we must use the distributive property.\n<div style=\"text-align: center;\">[latex]\\begin{align}f\\left(a+h\\right)&={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4 \\\\[2mm] &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\n<div style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[\/latex]<\/div>\n<p>and we know that<\/p>\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\n<p>Now we combine the results and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}&=\\dfrac{\\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\\right)-\\left({a}^{2}+3a - 4\\right)}{h} \\\\[2mm] &=\\dfrac{2ah+{h}^{2}+3h}{h}\\\\[2mm] &=\\frac{h\\left(2a+h+3\\right)}{h}&&\\text{Factor out }h. \\\\[2mm] &=2a+h+3&&\\text{Simplify}.\\end{align}[\/latex]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm1647\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1647&theme=oea&iframe_resize_id=ohm1647&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], evaluate [latex]g\\left(5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q273881\">Show Solution<\/span><\/p>\n<div id=\"q273881\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g\\left(5\\right)=\\sqrt{5 - 4}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm97486\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=97486&theme=oea&iframe_resize_id=ohm97486&show_question_numbers\" width=\"100%\" height=\"130\"><\/iframe><\/p>\n<\/div>\n<p>In addition to <strong>evaluating functions<\/strong> for a particular input, we can also <strong>solve functions<\/strong> for the input that creates a particular output.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Functions<\/h3>\n<p>Given the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], solve for [latex]h\\left(p\\right)=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q119909\">Show Solution<\/span><\/p>\n<div id=\"q119909\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}&h\\left(p\\right)=3\\\\ &{p}^{2}+2p=3 &&\\text{Substitute the original function }h\\left(p\\right)={p}^{2}+2p. \\\\ &{p}^{2}+2p - 3=0 &&\\text{Subtract 3 from each side}. \\\\ &\\left(p+3\\text{)(}p - 1\\right)=0 &&\\text{Factor}. \\end{align}[\/latex]<\/p>\n<p>If [latex]\\left(p+3\\right)\\left(p - 1\\right)=0[\/latex], either [latex]\\left(p+3\\right)=0[\/latex] or [latex]\\left(p - 1\\right)=0[\/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[\/latex] in each case.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&p+3=0, &&p=-3 \\\\ &p - 1=0, &&p=1\\hfill \\end{align}[\/latex]<\/p>\n<p>This gives us two solutions. The output [latex]h\\left(p\\right)=3[\/latex] when the input is either [latex]p=1[\/latex] or [latex]p=-3[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190959\/CNX_Precalc_Figure_01_01_0062.jpg\" alt=\"Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).\" width=\"487\" height=\"459\" \/><\/p>\n<p>We can also verify by graphing as in Figure 5. The graph verifies that [latex]h\\left(1\\right)=h\\left(-3\\right)=3[\/latex] and [latex]h\\left(4\\right)=24[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3><strong>How To: Solve a Function.<\/strong><\/h3>\n<ol>\n<li>Replace the output in the formula with the value provided.<\/li>\n<li>Solve for the input variable that makes the statement true.<\/li>\n<\/ol>\n<\/div>\n<p>The next video shows another example of how to\u00a0solve a function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Find Function Inputs for a Given Quadratic Function Output\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GLOmTED1UwA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], solve [latex]g\\left(m\\right)=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q480629\">Show Solution<\/span><\/p>\n<div id=\"q480629\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]m=8[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm15766\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15766&theme=oea&iframe_resize_id=ohm15766&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluating Functions Expressed in Formulas<\/h2>\n<p>Some functions are defined by mathematical rules or procedures expressed in <strong>equation<\/strong> form. If it is possible to express the function output with a <strong>formula<\/strong> involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[\/latex] expresses a functional relationship between [latex]n[\/latex]\u00a0and [latex]p[\/latex]. We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>functions, Equations, and formulas<\/h3>\n<p>We&#8217;ve seen that an equation such as [latex]ax+by=c[\/latex] can be written in a different form by solving the equation for one of the variables. If we solve this linear equation for\u00a0<em>y\u00a0<\/em>it can be written in the slope-intercept form of a line, [latex]y = mx+b[\/latex].<\/p>\n<p>Certain formulas can be written in function form by solving for one of the variables. For instance, can you see how to solve the formula for a rectangle having a perimeter of 21 feet, [latex]21 = 2l + 2w[\/latex], for length?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q773176\">more<\/span><\/p>\n<div id=\"q773176\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align} 21 &= 2l + 2w \\\\ 21 - 2w &= 2l \\\\ \\dfrac{21-2w}{2} &= l \\end{align}[\/latex]<\/p>\n<p>We can now declare a function, [latex]l = f(x)[\/latex] that returns an output length for a rectangle having a perimeter of 21 feet based on different width inputs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function in equation form, write its algebraic formula.<\/h3>\n<ol>\n<li>Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves <em>only<\/em> the input variable.<\/li>\n<li>Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding an Equation of a Function<\/h3>\n<p>Express the relationship [latex]2n+6p=12[\/latex] as a function [latex]p=f\\left(n\\right)[\/latex], if possible.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q938453\">Show Solution<\/span><\/p>\n<div id=\"q938453\" class=\"hidden-answer\" style=\"display: none\">\n<p>To express the relationship in this form, we need to be able to write the relationship where [latex]p[\/latex] is a function of [latex]n[\/latex], which means writing it as [latex]p=[\/latex] expression involving [latex]n[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&2n+6p=12\\\\[1mm] &6p=12 - 2n &&\\text{Subtract }2n\\text{ from both sides}. \\\\[1mm] &p=\\frac{12 - 2n}{6} &&\\text{Divide both sides by 6 and simplify}. \\\\[1mm] &p=\\frac{12}{6}-\\frac{2n}{6} \\\\[1mm] &p=2-\\frac{1}{3}n \\end{align}[\/latex]<\/p>\n<p>Therefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as<\/p>\n<p style=\"text-align: center;\">[latex]p=f\\left(n\\right)=2-\\frac{1}{3}n[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch this video to see another example of how to express an equation as a function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Write a Linear Relation as a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lHTLjfPpFyQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Sometimes a relationship between variables cannot be expressed as a function. See the example below for more information.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Expressing the Equation of a Circle as a Function<\/h3>\n<p>Does the equation [latex]{x}^{2}+{y}^{2}=1[\/latex] represent a function with [latex]x[\/latex] as input and [latex]y[\/latex] as output? If so, express the relationship as a function [latex]y=f\\left(x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q557070\">Show Solution<\/span><\/p>\n<div id=\"q557070\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we subtract [latex]{x}^{2}[\/latex] from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]{y}^{2}=1-{x}^{2}[\/latex]<\/p>\n<p>We now try to solve for [latex]y[\/latex] in this equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\pm \\sqrt{1-{x}^{2}} \\\\[1mm] &=\\sqrt{1-{x}^{2}}\\hspace{3mm}\\text{and}\\hspace{3mm}-\\sqrt{1-{x}^{2}} \\end{align}[\/latex]<\/p>\n<p>We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>If [latex]x - 8{y}^{3}=0[\/latex], express [latex]y[\/latex] as a function of [latex]x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933974\">Show Solution<\/span><\/p>\n<div id=\"q933974\" class=\"hidden-answer\" style=\"display: none\">[latex]y=f\\left(x\\right)=\\cfrac{\\sqrt[3]{x}}{2}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm111699\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111699&theme=oea&iframe_resize_id=ohm111699&show_question_numbers\" width=\"100%\" height=\"320\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?<\/strong><\/p>\n<p><em>Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y}[\/latex], if we want to express [latex]y[\/latex] as a function of [latex]x[\/latex], there is no simple algebraic formula involving only [latex]x[\/latex] that equals [latex]y[\/latex]. However, each [latex]x[\/latex] does determine a unique value for [latex]y[\/latex], and there are mathematical procedures by which [latex]y[\/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[\/latex] as a function of [latex]x[\/latex], even though the formula cannot be written explicitly.<\/em><\/p>\n<\/div>\n<h2>Evaluating a Function Given in Tabular Form<\/h2>\n<p>As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.<\/p>\n<p>The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.<\/p>\n<table summary=\"Six rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Pet<\/th>\n<th>Memory span in hours<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Puppy<\/td>\n<td>0.008<\/td>\n<\/tr>\n<tr>\n<td>Adult dog<\/td>\n<td>0.083<\/td>\n<\/tr>\n<tr>\n<td>Cat<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>Goldfish<\/td>\n<td>2160<\/td>\n<\/tr>\n<tr>\n<td>Beta fish<\/td>\n<td>3600<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[\/latex].<\/p>\n<p>The <strong>domain<\/strong> of the function is the type of pet and the range is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of &#8220;goldfish.&#8221; We would write [latex]P\\left(\\text{goldfish}\\right)=2160[\/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[\/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a function represented by a table, identify specific output and input values.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Find the given input in the row (or column) of input values.<\/li>\n<li>Identify the corresponding output value paired with that input value.<\/li>\n<li>Find the given output values in the row (or column) of output values, noting every time that output value appears.<\/li>\n<li>Identify the input value(s) corresponding to the given output value.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating and Solving a Tabular Function<\/h3>\n<p>Using the table below,<\/p>\n<ol>\n<li>Evaluate [latex]g\\left(3\\right)[\/latex].<\/li>\n<li>Solve [latex]g\\left(n\\right)=6[\/latex].<\/li>\n<\/ol>\n<table summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15206\">Show Solution<\/span><\/p>\n<div id=\"q15206\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Evaluating [latex]g\\left(3\\right)[\/latex] means determining the output value of the function [latex]g[\/latex] for the input value of [latex]n=3[\/latex]. The table output value corresponding to [latex]n=3[\/latex] is 7, so [latex]g\\left(3\\right)=7[\/latex].<\/li>\n<li>Solving [latex]g\\left(n\\right)=6[\/latex] means identifying the input values, [latex]n[\/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[\/latex] and [latex]n=4[\/latex].<\/li>\n<\/ul>\n<table summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When we input 2 into the function [latex]g[\/latex], our output is 6. When we input 4 into the function [latex]g[\/latex], our output is also 6.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the table from the previous example, evaluate [latex]g\\left(1\\right)[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q724802\">Show Solution<\/span><\/p>\n<div id=\"q724802\" class=\"hidden-answer\" style=\"display: none\">[latex]g\\left(1\\right)=8[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3751\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3751&theme=oea&iframe_resize_id=ohm3751&show_question_numbers\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<h2>Finding Function Values from a Graph<\/h2>\n<div class=\"textbox examples\">\n<h3>ordered pairs of inputs and outputs<\/h3>\n<p>We can view a function as a set of inputs and their corresponding outputs. That is, we can see a function as a set of ordered pairs, [latex]\\left(x, y \\right).[\/latex]<\/p>\n<p>Remember that, in function notation, [latex]y = f(x)[\/latex], so the ordered pairs containing inputs and outputs can be written in the form of (<em>input<\/em>,\u00a0<em>output<\/em>) or [latex]\\left(x, f(x)\\right)[\/latex].<\/p>\n<\/div>\n<p>Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Reading Function Values from a Graph<\/h3>\n<p>Given the graph below,<\/p>\n<ol>\n<li>Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\n<li>Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191001\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q915833\">Show Solution<\/span><\/p>\n<div id=\"q915833\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the [latex]y[\/latex]-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/li>\n<li>To solve [latex]f\\left(x\\right)=4[\/latex], we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4[\/latex], we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex]. These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]x=-1[\/latex] or [latex]x=3[\/latex]. This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4[\/latex], or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex] See the graph below.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191006\/CNX_Precalc_Figure_01_01_0092.jpg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the graph, solve [latex]f\\left(x\\right)=1[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q529772\">Show Solution<\/span><\/p>\n<div id=\"q529772\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=0[\/latex] or [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm2471\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2471&theme=oea&iframe_resize_id=ohm2471&show_question_numbers\" width=\"100%\" height=\"480\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2886\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2886&theme=oea&iframe_resize_id=ohm2886&show_question_numbers\" width=\"100%\" height=\"470\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>You can use an online graphing calculator to graph functions, find function values, and evaluate functions. Watch this short tutorial to learn how to within Desmos. Other online graphing tools will be slightly different.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-4\" title=\"Learn Desmos: Function Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jACDzJ-rmsM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nNow try the following with an online graphing calculator:<\/p>\n<ol>\n<li>Graph the function [latex]f(x) = -\\frac{1}{2}x^2+x+4[\/latex] using function notation.<\/li>\n<li>Evaluate the function at [latex]x=1[\/latex]<\/li>\n<li>Make a table of values that references the function. Include at least the interval [latex][-5,5][\/latex] for [latex]x[\/latex]-values.<\/li>\n<li>Solve the function for [latex]f(0)[\/latex]<\/li>\n<\/ol>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-124\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 111699. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 1647. <strong>Authored by<\/strong>: WebWork-Rochester, mb Lippman,David, mb Sousa,James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 97486. <strong>Authored by<\/strong>: Carmichael,Patrick. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15766, 2886. <strong>Authored by<\/strong>: Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 3751. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 2471. <strong>Authored by<\/strong>: Greg Langkamp. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Learn Desmos: Functions. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jACDzJ-rmsM\">https:\/\/youtu.be\/jACDzJ-rmsM<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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