{"id":203,"date":"2024-10-18T21:13:04","date_gmt":"2024-10-18T21:13:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/function-basics-fresh-take\/"},"modified":"2024-10-21T13:50:59","modified_gmt":"2024-10-21T13:50:59","slug":"function-basics-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/function-basics-fresh-take\/","title":{"raw":"Function Basics: Fresh Take","rendered":"Function Basics: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Define what a function is and apply the vertical line test to identify functions<\/li>\n\t<li>Use function notation to represent and evaluate functions<\/li>\n\t<li>Recognize the graphs of fundamental toolkit functions<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Determining Whether a Relation Represents a Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>A <strong>function<\/strong> is like a magical box that takes something in and spits something else out. In math terms, it's a special relationship where each input (or \"independent variable\") is connected to exactly one output (or \"dependent variable\"). So, if you feed the function a number, it will give you back another number, but it won't ever get confused and give you two or more numbers for the same input.<\/p>\n<ul>\n\t<li><strong>Input<\/strong>: Also known as the independent variable, often labeled [latex]x[\/latex].<\/li>\n\t<li><strong>Output<\/strong>: Also known as the dependent variable, often labeled [latex]y[\/latex].<\/li>\n\t<li><strong>Domain<\/strong>: The set of all possible inputs.<\/li>\n\t<li><strong>Range<\/strong>: The set of all possible outputs.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox questionHelp\">\n<p id=\"fs-id1165137635406\"><strong>How to: Given a relationship between two quantities, determine whether the relationship is a function.<\/strong><\/p>\n<ol id=\"fs-id1165134065124\" type=\"1\">\n\t<li>Identify the input values.<\/li>\n\t<li>Identify the output values.<\/li>\n\t<li>If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">The coffee shop menu, shown below, consists of items and their prices.<center><img class=\"aligncenter wp-image-9161\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211302\/fb8854bea4c2687291cdc81c609909bada5f3c07.png\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"500\" height=\"241\"><\/center>\n<p>&nbsp;<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n\t<li>Is price a function of the item?<\/li>\n\t<li>Is the item a function of the price?<\/li>\n<\/ol>\n\n\n[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]\n\n\n<ol style=\"list-style-type: lower-alpha;\">\n\t<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices.\n\n\n<p>&nbsp;<\/p>\n<center><img class=\"aligncenter wp-image-9162\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211303\/fb8854bea4c2687291cdc81c609909bada5f3c07-1.png\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"500\" height=\"241\"><\/center>\n<p>&nbsp;<\/p>\n\n\nEach item on the menu has only one price, so the price is a function of the item.<\/li>\n\t<li>Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See the image below.&nbsp;\n\n\n<p>&nbsp;<\/p>\n<center><img class=\"aligncenter wp-image-9163\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211304\/4ca8a33070cb29d5a0d84af71c52fecedce72117.png\" alt=\"Association of the prices to the donuts.\" width=\"500\" height=\"241\"><\/center>\n<p>&nbsp;<\/p>\n\n\n&nbsp;Therefore, the item is a not a function of price.<\/li>\n<\/ol>\n\n\n[\/hidden-answer]<\/section>\n<section class=\"textbox example\">\n<p>Find the domain and range of the relation and determine whether it is&nbsp;a function.<\/p>\n<p style=\"text-align: center;\">[latex]\\{(\u22123, 4),(\u22122, 4),( \u22121, 4),(2, 4),(3, 4)\\}[\/latex]<\/p>\n\n\n[reveal-answer q=\"587362\"]Show Solution[\/reveal-answer] [hidden-answer a=\"587362\"] \n<p>Domain: {[latex]-3, -2, -1, 2, 3[\/latex]}<\/p>\n<p>Range: {[latex]4[\/latex]}<\/p>\n<p>To help you determine whether this is a function, you could reorganize the information by creating a table.<\/p>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<th style=\"text-align: center;\" scope=\"row\"><i>x<\/i><\/th>\n<th style=\"text-align: center;\"><i>y<\/i><\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]\u22123[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Each input has only one output, and the fact that it is the same output [latex](4)[\/latex] does not matter. This relation is a function.<\/p>\n\n\n[\/hidden-answer]<\/section>\n<h2>Verifying a Function Using the Vertical Line Test<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The <strong>vertical line test<\/strong> is a quick way to determine if a graph represents a function. If any vertical line intersects the graph more than once, it's not a function.<\/p>\n<p>When you're looking at a graph and wondering if it represents a function, try imagining or drawing vertical lines across the graph. If any of these lines touch the graph at more than one point, then what you're looking at isn't a function.<\/p>\n<\/div>\n<section class=\"textbox example\">Does the graph below represent a function? <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\"> [reveal-answer q=\"783855\"]Show Solution[\/reveal-answer] [hidden-answer a=\"783855\"] Yes. [\/hidden-answer]<\/section>\n<p>For more examples of the vertical line test, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5Z8DaZPJLKY?si=_RiPwpHxZnk02Ena\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Use+the+Vertical+Line+Test+to+Determine+if+a+Graph+Represents+a+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Determining Whether a Function is One-to-One<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>A function is <strong>one-to-one<\/strong> if each output corresponds to exactly one input. This ensures that the function has a unique inverse.<\/p>\n<p>Think of a one-to-one function like a VIP list at a party. Each name on the list gets you one, and only one, ticket in. No name repeats, and no ticket can be used twice.<\/p>\n<\/div>\n<section class=\"textbox example\">Is the area of a circle a function of its radius? If yes, is the function one-to-one? [reveal-answer q=\"380432\"]Show Solution[\/reveal-answer] [hidden-answer a=\"380432\"] A circle of radius [latex]r[\/latex] has a unique area measure given by [latex]A=\\pi {r}^{2}[\/latex], so for any input, [latex]r[\/latex], there is only one output, [latex]A[\/latex]. The area is a function of radius [latex]r[\/latex]. If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[\/latex] is given by the formula [latex]A=\\pi {r}^{2}[\/latex]. Because areas and radii are positive numbers, there is exactly one solution: [latex]r=\\sqrt{\\frac{A}{\\pi }}[\/latex]. So the area of a circle is a one-to-one function of the circle\u2019s radius. [\/hidden-answer]<\/section>\n<p>For an overview of one-to-one functions, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328524&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Muf7hNZN9xw&amp;video_target=tpm-plugin-t7wubt0r-Muf7hNZN9xw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Overview+of+one+to+one+functions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cOverview of one to one functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>The Horizontal Line Test<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>To check if a function is one-to-one, use the <strong>horizontal line test<\/strong>. If any horizontal line intersects the graph more than once, the function is not one-to-one.<\/p>\n<p>Similar to the vertical line test, but this time you're drawing horizontal lines. If any line touches the graph more than once, the function isn't one-to-one, meaning it doesn't have a unique inverse.<\/p>\n<\/div>\n<p>For more examples of the horizontal line test, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tbSGdcSN8RE?si=e3uL9p49M1BgIiIx\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Determine+if+the+Graph+of+a+Relation+is+a+One-to-One+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Determine if the Graph of a Relation is a One-to-One Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Representing Functions Using Tables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<strong>How to: Given a table of input and output values, determine whether the table represents a function.<\/strong>\n<ol id=\"fs-id1165137629040\" type=\"1\">\n\t<li>Identify the input and output values.<\/li>\n\t<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\n<\/ol>\n<\/div>\n<h2>Function Notation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Function Notation:<\/strong> The function is often denoted as [latex]f(x)[\/latex], where [latex]x[\/latex] is the input and [latex]f(x)[\/latex] is the output. This notation is a shorthand way of saying \"the function [latex]f[\/latex] acting on [latex]x[\/latex] gives [latex]f(x)[\/latex].\"<\/p>\n<p><strong>Independent and Dependent Variables:<\/strong> In function notation, [latex]x[\/latex] is the independent variable (the input), and [latex]f(x)[\/latex] is the dependent variable (the output). The output depends on what you put in as the input.<\/p>\n<p><strong>Flexibility in Naming:<\/strong> Function notation allows you to use different names like [latex]f(x)[\/latex], [latex]g(x)[\/latex], or [latex]c(x)[\/latex] to distinguish between multiple functions. Functions can have inputs that are not just numbers. For example, the input could be the name of a month, and the output could be the number of days in that month.<\/p>\n<\/div>\n<section class=\"textbox example\">A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent? [reveal-answer q=\"226737\"]Show Solution[\/reveal-answer] [hidden-answer a=\"226737\"] When we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers, [latex]N[\/latex], is [latex]300[\/latex]. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were [latex]300[\/latex] police officers in the town. [\/hidden-answer]<\/section>\n<p>In the following videos we show two more examples of how to express a relationship using function notation.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lF0fzdaxU_8?si=vHolO-huQgF2Ihgl\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Function+Notation+Application+Problem.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Function Notation Application Problem\u201d here (opens in new window).<\/a><\/p>\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/nAF_GZFwU1g?si=8Fk_XuRL99vEiQnN\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Function+Notation+Application.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFunction Notation Application\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Finding Input and Output Values of a Function<\/h2>\n<h3>Evaluation of Functions in Algebraic Forms<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Evaluating Functions:<\/strong> To evaluate a function, substitute the given input into the function's formula and perform the calculations. For example, if [latex]f(x)=x^2\u22128[\/latex], and you want to find [latex]f(\u22123)[\/latex], you'd calculate [latex](\u22123)^2\u22128=1[\/latex].<\/p>\n<p><strong>Solving for Input:<\/strong> If you know the output and want to find the input, set the output equal to the function's formula and solve for the input. For instance, if [latex]h(p)=p^2+2p[\/latex] and [latex]h(p)=3[\/latex], you'd solve the equation [latex]p^2+2p=3[\/latex] to find the value of [latex]p[\/latex].<\/p>\n<\/div>\n<section class=\"textbox example\">Given the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], evaluate [latex]h\\left(4\\right)[\/latex]. [reveal-answer q=\"768180\"]Show Solution[\/reveal-answer] [hidden-answer a=\"768180\"]\n\n\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)&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>\n\n\nTherefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[\/latex]. [\/hidden-answer]<\/section>\n<section class=\"textbox example\">Given the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], evaluate [latex]g\\left(5\\right)[\/latex]. [reveal-answer q=\"273881\"]Show Solution[\/reveal-answer] [hidden-answer a=\"273881\"] [latex]g\\left(5\\right)=\\sqrt{5 - 4}=1[\/latex] [\/hidden-answer]<\/section>\n<p>Watch the video below for more examples of evaluating a function for specific values of the input.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Ehkzu5Uv7O0?si=NLcF2ZsKaIVgPEvj\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Evaluating+Functions+Using+Function+Notation+(L9.3).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluating Functions Using Function Notation (L9.3)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The next video shows another example of how to solve a function.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/GLOmTED1UwA?si=kbCsHbm8keEDPCW5\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Find+Function+Inputs+for+a+Given+Quadratic+Function+Output.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find Function Inputs for a Given Quadratic Function Output\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Evaluating a Function Given in Tabular Form<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Functions can be represented in tables, which can be particularly useful when equations are not the best fit.<\/p>\n<p>The domain is the type of input, and the range is the output, often a real number.<\/p>\n<p>To find the output for a given input, look for the input in the table and identify the corresponding output.<\/p>\n<p>To find the input for a given output, scan the output column and note all instances of that output, then find the corresponding input(s).<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Using the table below, evaluate [latex]g(1)[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>[latex]n[\/latex]<\/strong><\/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<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>[latex]g(n)[\/latex]<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]7[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n[reveal-answer q=\"15206\"]Show Solution[\/reveal-answer] [hidden-answer a=\"15206\"] [latex]g(1) = 8[\/latex] [\/hidden-answer]<\/section>\n<section>Watch the following video for more on evaluating a function given a table.\n\n\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/meqZdQkoNOQ?si=H1Ozm0Pw9_9lFfQv\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Evaluate+a+Function+and+Solve+for+a+Function+Value+Given+a+Table.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Evaluate a Function and Solve for a Function Value Given a Table\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Finding Function Values from a Graph<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Graphs can also be used to evaluate functions.<\/p>\n<p>The graph represents a set of ordered pairs [latex](x,y)[\/latex] or [latex](x,f(x))[\/latex], where [latex]x[\/latex] is the input and [latex]f(x)[\/latex] is the output.<\/p>\n<\/div>\n<section class=\"textbox example\">Using the graph, solve [latex]f\\left(x\\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\"> [reveal-answer q=\"529772\"]Show Solution[\/reveal-answer] [hidden-answer a=\"529772\"] [latex]x=0[\/latex] or [latex]x=2[\/latex] [\/hidden-answer]<\/section>\n\n\nWatch the following video for more on finding function values given a graph.\n\n\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/HRCD79-uXKo?si=6qHwr1y4Vfcc4aQw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Determine+a+Function+Value+From+a+Graph.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Determine a Function Value From a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Identifying Basic Toolkit Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>In this section, we dive into the world of \"Toolkit Functions.\" Think of these as the ABCs of functions\u2014your go-to set of basic functions that you'll encounter throughout your studies. These toolkit functions come in various shapes and forms, from simple constant functions to more complex cube root functions. Knowing these functions by heart, including their graphs and equations, will give you a solid foundation for understanding more complex functions later on.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"3\">Toolkit Functions<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Name<\/th>\n<th style=\"text-align: center;\">Function<\/th>\n<th style=\"text-align: center;\">Graph<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Constant<\/td>\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\n<td><span id=\"fs-id1165137643159\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005019\/CNX_Precalc_Figure_01_01_018n.jpg\" alt=\"Graph of a constant function.\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Identity\/Linear<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<td><span id=\"fs-id1165137811013\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_019n.jpg\" alt=\"Graph of a straight line.\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Absolute value<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td><span id=\"fs-id1165135195221\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_020n.jpg\" alt=\"Graph of absolute function.\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Quadratic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105554.png\"><img class=\"alignnone size-full wp-image-16077\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211257\/Screenshot-2023-09-27-105554.png\" alt=\"Graph of a parabola.\" width=\"567\" height=\"348\"><\/a><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Cubic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td><span id=\"fs-id1165137722123\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Reciprocal<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td><span id=\"fs-id1165134544980\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005021\/CNX_Precalc_Figure_01_01_023n.jpg\" alt=\"Graph of f(x)=1\/x.\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Reciprocal squared<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105645.png\"><img class=\"alignnone size-full wp-image-16079\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211258\/Screenshot-2023-09-27-105645.png\" alt=\"Graph of f(x)=1\/x^2.\" width=\"562\" height=\"348\"><\/a><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Square root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105142.png\"><img class=\"alignnone wp-image-16071 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211259\/Screenshot-2023-09-27-105142.png\" alt=\"Graph of f(x)=sqrt(x).\" width=\"565\" height=\"349\"><\/a><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Cube root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td><span id=\"fs-id1165137838612\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005021\/CNX_Precalc_Figure_01_01_026n.jpg\" alt=\"Graph of f(x)=x^(1\/3).\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Define what a function is and apply the vertical line test to identify functions<\/li>\n<li>Use function notation to represent and evaluate functions<\/li>\n<li>Recognize the graphs of fundamental toolkit functions<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Determining Whether a Relation Represents a Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>A <strong>function<\/strong> is like a magical box that takes something in and spits something else out. In math terms, it&#8217;s a special relationship where each input (or &#8220;independent variable&#8221;) is connected to exactly one output (or &#8220;dependent variable&#8221;). So, if you feed the function a number, it will give you back another number, but it won&#8217;t ever get confused and give you two or more numbers for the same input.<\/p>\n<ul>\n<li><strong>Input<\/strong>: Also known as the independent variable, often labeled [latex]x[\/latex].<\/li>\n<li><strong>Output<\/strong>: Also known as the dependent variable, often labeled [latex]y[\/latex].<\/li>\n<li><strong>Domain<\/strong>: The set of all possible inputs.<\/li>\n<li><strong>Range<\/strong>: The set of all possible outputs.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox questionHelp\">\n<p id=\"fs-id1165137635406\"><strong>How to: Given a relationship between two quantities, determine whether the relationship is a function.<\/strong><\/p>\n<ol id=\"fs-id1165134065124\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Identify the output values.<\/li>\n<li>If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">The coffee shop menu, shown below, consists of items and their prices.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9161\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211302\/fb8854bea4c2687291cdc81c609909bada5f3c07.png\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"500\" height=\"241\" \/><\/div>\n<p>&nbsp;<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Is price a function of the item?<\/li>\n<li>Is the item a function of the price?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214538\">Show Solution<\/button> <\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices.\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9162\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211303\/fb8854bea4c2687291cdc81c609909bada5f3c07-1.png\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"500\" height=\"241\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Each item on the menu has only one price, so the price is a function of the item.<\/li>\n<li>Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See the image below.&nbsp;\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9163\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211304\/4ca8a33070cb29d5a0d84af71c52fecedce72117.png\" alt=\"Association of the prices to the donuts.\" width=\"500\" height=\"241\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;Therefore, the item is a not a function of price.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the domain and range of the relation and determine whether it is&nbsp;a function.<\/p>\n<p style=\"text-align: center;\">[latex]\\{(\u22123, 4),(\u22122, 4),( \u22121, 4),(2, 4),(3, 4)\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q587362\">Show Solution<\/button> <\/p>\n<div id=\"q587362\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain: {[latex]-3, -2, -1, 2, 3[\/latex]}<\/p>\n<p>Range: {[latex]4[\/latex]}<\/p>\n<p>To help you determine whether this is a function, you could reorganize the information by creating a table.<\/p>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<th style=\"text-align: center;\" scope=\"row\"><i>x<\/i><\/th>\n<th style=\"text-align: center;\"><i>y<\/i><\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]\u22123[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" scope=\"row\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Each input has only one output, and the fact that it is the same output [latex](4)[\/latex] does not matter. This relation is a function.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Verifying a Function Using the Vertical Line Test<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>The <strong>vertical line test<\/strong> is a quick way to determine if a graph represents a function. If any vertical line intersects the graph more than once, it&#8217;s not a function.<\/p>\n<p>When you&#8217;re looking at a graph and wondering if it represents a function, try imagining or drawing vertical lines across the graph. If any of these lines touch the graph at more than one point, then what you&#8217;re looking at isn&#8217;t a function.<\/p>\n<\/div>\n<section class=\"textbox example\">Does the graph below represent a function? <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/> <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q783855\">Show Solution<\/button> <\/p>\n<div id=\"q783855\" class=\"hidden-answer\" style=\"display: none\"> Yes. <\/div>\n<\/div>\n<\/section>\n<p>For more examples of the vertical line test, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5Z8DaZPJLKY?si=_RiPwpHxZnk02Ena\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Use+the+Vertical+Line+Test+to+Determine+if+a+Graph+Represents+a+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Determining Whether a Function is One-to-One<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>A function is <strong>one-to-one<\/strong> if each output corresponds to exactly one input. This ensures that the function has a unique inverse.<\/p>\n<p>Think of a one-to-one function like a VIP list at a party. Each name on the list gets you one, and only one, ticket in. No name repeats, and no ticket can be used twice.<\/p>\n<\/div>\n<section class=\"textbox example\">Is the area of a circle a function of its radius? If yes, is the function one-to-one? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q380432\">Show Solution<\/button> <\/p>\n<div id=\"q380432\" class=\"hidden-answer\" style=\"display: none\"> A circle of radius [latex]r[\/latex] has a unique area measure given by [latex]A=\\pi {r}^{2}[\/latex], so for any input, [latex]r[\/latex], there is only one output, [latex]A[\/latex]. The area is a function of radius [latex]r[\/latex]. If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[\/latex] is given by the formula [latex]A=\\pi {r}^{2}[\/latex]. Because areas and radii are positive numbers, there is exactly one solution: [latex]r=\\sqrt{\\frac{A}{\\pi }}[\/latex]. So the area of a circle is a one-to-one function of the circle\u2019s radius. <\/div>\n<\/div>\n<\/section>\n<p>For an overview of one-to-one functions, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328524&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Muf7hNZN9xw&amp;video_target=tpm-plugin-t7wubt0r-Muf7hNZN9xw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Overview+of+one+to+one+functions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cOverview of one to one functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>The Horizontal Line Test<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>To check if a function is one-to-one, use the <strong>horizontal line test<\/strong>. If any horizontal line intersects the graph more than once, the function is not one-to-one.<\/p>\n<p>Similar to the vertical line test, but this time you&#8217;re drawing horizontal lines. If any line touches the graph more than once, the function isn&#8217;t one-to-one, meaning it doesn&#8217;t have a unique inverse.<\/p>\n<\/div>\n<p>For more examples of the horizontal line test, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tbSGdcSN8RE?si=e3uL9p49M1BgIiIx\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Determine+if+the+Graph+of+a+Relation+is+a+One-to-One+Function.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Determine if the Graph of a Relation is a One-to-One Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Representing Functions Using Tables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>How to: Given a table of input and output values, determine whether the table represents a function.<\/strong><\/p>\n<ol id=\"fs-id1165137629040\" type=\"1\">\n<li>Identify the input and output values.<\/li>\n<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\n<\/ol>\n<\/div>\n<h2>Function Notation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Function Notation:<\/strong> The function is often denoted as [latex]f(x)[\/latex], where [latex]x[\/latex] is the input and [latex]f(x)[\/latex] is the output. This notation is a shorthand way of saying &#8220;the function [latex]f[\/latex] acting on [latex]x[\/latex] gives [latex]f(x)[\/latex].&#8221;<\/p>\n<p><strong>Independent and Dependent Variables:<\/strong> In function notation, [latex]x[\/latex] is the independent variable (the input), and [latex]f(x)[\/latex] is the dependent variable (the output). The output depends on what you put in as the input.<\/p>\n<p><strong>Flexibility in Naming:<\/strong> Function notation allows you to use different names like [latex]f(x)[\/latex], [latex]g(x)[\/latex], or [latex]c(x)[\/latex] to distinguish between multiple functions. Functions can have inputs that are not just numbers. For example, the input could be the name of a month, and the output could be the number of days in that month.<\/p>\n<\/div>\n<section class=\"textbox example\">A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q226737\">Show Solution<\/button> <\/p>\n<div id=\"q226737\" class=\"hidden-answer\" style=\"display: none\"> When we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers, [latex]N[\/latex], is [latex]300[\/latex]. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were [latex]300[\/latex] police officers in the town. <\/div>\n<\/div>\n<\/section>\n<p>In the following videos we show two more examples of how to express a relationship using function notation.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lF0fzdaxU_8?si=vHolO-huQgF2Ihgl\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Function+Notation+Application+Problem.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Function Notation Application Problem\u201d here (opens in new window).<\/a><\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/nAF_GZFwU1g?si=8Fk_XuRL99vEiQnN\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Function+Notation+Application.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFunction Notation Application\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Finding Input and Output Values of a Function<\/h2>\n<h3>Evaluation of Functions in Algebraic Forms<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Evaluating Functions:<\/strong> To evaluate a function, substitute the given input into the function&#8217;s formula and perform the calculations. For example, if [latex]f(x)=x^2\u22128[\/latex], and you want to find [latex]f(\u22123)[\/latex], you&#8217;d calculate [latex](\u22123)^2\u22128=1[\/latex].<\/p>\n<p><strong>Solving for Input:<\/strong> If you know the output and want to find the input, set the output equal to the function&#8217;s formula and solve for the input. For instance, if [latex]h(p)=p^2+2p[\/latex] and [latex]h(p)=3[\/latex], you&#8217;d solve the equation [latex]p^2+2p=3[\/latex] to find the value of [latex]p[\/latex].<\/p>\n<\/div>\n<section class=\"textbox example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q768180\">Show Solution<\/button> <\/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><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q273881\">Show Solution<\/button> <\/p>\n<div id=\"q273881\" class=\"hidden-answer\" style=\"display: none\"> [latex]g\\left(5\\right)=\\sqrt{5 - 4}=1[\/latex] <\/div>\n<\/div>\n<\/section>\n<p>Watch the video below for more examples of evaluating a function for specific values of the input.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Ehkzu5Uv7O0?si=NLcF2ZsKaIVgPEvj\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Evaluating+Functions+Using+Function+Notation+(L9.3).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluating Functions Using Function Notation (L9.3)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The next video shows another example of how to solve a function.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/GLOmTED1UwA?si=kbCsHbm8keEDPCW5\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Find+Function+Inputs+for+a+Given+Quadratic+Function+Output.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find Function Inputs for a Given Quadratic Function Output\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Evaluating a Function Given in Tabular Form<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Functions can be represented in tables, which can be particularly useful when equations are not the best fit.<\/p>\n<p>The domain is the type of input, and the range is the output, often a real number.<\/p>\n<p>To find the output for a given input, look for the input in the table and identify the corresponding output.<\/p>\n<p>To find the input for a given output, scan the output column and note all instances of that output, then find the corresponding input(s).<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>Using the table below, evaluate [latex]g(1)[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>[latex]n[\/latex]<\/strong><\/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<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>[latex]g(n)[\/latex]<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]7[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q15206\">Show Solution<\/button> <\/p>\n<div id=\"q15206\" class=\"hidden-answer\" style=\"display: none\"> [latex]g(1) = 8[\/latex] <\/div>\n<\/div>\n<\/section>\n<section>Watch the following video for more on evaluating a function given a table.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/meqZdQkoNOQ?si=H1Ozm0Pw9_9lFfQv\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Evaluate+a+Function+and+Solve+for+a+Function+Value+Given+a+Table.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Evaluate a Function and Solve for a Function Value Given a Table\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Finding Function Values from a Graph<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>Graphs can also be used to evaluate functions.<\/p>\n<p>The graph represents a set of ordered pairs [latex](x,y)[\/latex] or [latex](x,f(x))[\/latex], where [latex]x[\/latex] is the input and [latex]f(x)[\/latex] is the output.<\/p>\n<\/div>\n<section class=\"textbox example\">Using the graph, solve [latex]f\\left(x\\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\" \/> <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q529772\">Show Solution<\/button> <\/p>\n<div id=\"q529772\" class=\"hidden-answer\" style=\"display: none\"> [latex]x=0[\/latex] or [latex]x=2[\/latex] <\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more on finding function values given a graph.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/HRCD79-uXKo?si=6qHwr1y4Vfcc4aQw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Determine+a+Function+Value+From+a+Graph.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Determine a Function Value From a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Identifying Basic Toolkit Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>In this section, we dive into the world of &#8220;Toolkit Functions.&#8221; Think of these as the ABCs of functions\u2014your go-to set of basic functions that you&#8217;ll encounter throughout your studies. These toolkit functions come in various shapes and forms, from simple constant functions to more complex cube root functions. Knowing these functions by heart, including their graphs and equations, will give you a solid foundation for understanding more complex functions later on.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"3\">Toolkit Functions<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Name<\/th>\n<th style=\"text-align: center;\">Function<\/th>\n<th style=\"text-align: center;\">Graph<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Constant<\/td>\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\n<td><span id=\"fs-id1165137643159\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005019\/CNX_Precalc_Figure_01_01_018n.jpg\" alt=\"Graph of a constant function.\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Identity\/Linear<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<td><span id=\"fs-id1165137811013\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_019n.jpg\" alt=\"Graph of a straight line.\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Absolute value<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td><span id=\"fs-id1165135195221\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_020n.jpg\" alt=\"Graph of absolute function.\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Quadratic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105554.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-16077\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211257\/Screenshot-2023-09-27-105554.png\" alt=\"Graph of a parabola.\" width=\"567\" height=\"348\" \/><\/a><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Cubic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td><span id=\"fs-id1165137722123\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Reciprocal<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td><span id=\"fs-id1165134544980\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005021\/CNX_Precalc_Figure_01_01_023n.jpg\" alt=\"Graph of f(x)=1\/x.\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Reciprocal squared<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105645.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-16079\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211258\/Screenshot-2023-09-27-105645.png\" alt=\"Graph of f(x)=1\/x^2.\" width=\"562\" height=\"348\" \/><\/a><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Square root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105142.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16071 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/59\/2024\/10\/18211259\/Screenshot-2023-09-27-105142.png\" alt=\"Graph of f(x)=sqrt(x).\" width=\"565\" height=\"349\" \/><\/a><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Cube root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td><span id=\"fs-id1165137838612\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005021\/CNX_Precalc_Figure_01_01_026n.jpg\" alt=\"Graph of f(x)=x^(1\/3).\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Algebra and Trigonometry 2e\",\"author\":\"Jay 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