{"id":36546,"date":"2017-12-27T19:25:33","date_gmt":"2017-12-27T19:25:33","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=36546"},"modified":"2026-03-26T10:03:49","modified_gmt":"2026-03-26T15:03:49","slug":"definition-of-a-function","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/definition-of-a-function\/","title":{"rendered":"What is a Function?"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_W4U6rjhmOyQ\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_W4U6rjhmOyQ\" data-source-videoID=\"W4U6rjhmOyQ\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"What is a Function? Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"What is a Function?\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_W4U6rjhmOyQ:hover {cursor:pointer;} img#videoThumbnailImage_W4U6rjhmOyQ {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1611-definition-of-a-function-1-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_W4U6rjhmOyQ\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_W4U6rjhmOyQ\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"What is a Function?\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_W4U6rjhmOyQ\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_W4U6rjhmOyQ\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_W4U6rjhmOyQ\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction qF2_Function() {\n  var x = document.getElementById(\"qF2\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"qF2_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"qF2\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#The_Basics_of_Relations\" class=\"smooth-scroll\">The Basics of Relations<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#What_is_a_Function\" class=\"smooth-scroll\">What is a Function?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Frequently_Asked_Questions\" class=\"smooth-scroll\">Frequently Asked Questions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Function_Practice_Questions\" class=\"smooth-scroll\">Function Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"FAQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"FAQs\">FAQs<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this overview of functions! In this video, we will be examining the nature of mathematical relationships between variables. As we work through some examples, our goal is to determine whether or not the criteria are met to define a special relationship known as a \u201cfunction.&#8221;<\/p>\n<h2><span id=\"The_Basics_of_Relations\" class=\"m-toc-anchor\"><\/span>The Basics of Relations<\/h2>\n<p>\nLet\u2019s take some time to review the basics of math rules, or relations.<\/p>\n<p>When we are working with an equation, the value of the variable \\(x\\) helps determine the value of the variable \\(y\\). <\/p>\n<div class=\"examplesentence\">\\(y=2x+3\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe \\(x\\)-value can be thought of as the <strong>input<\/strong> into the math rule that is determined by the equation, and the \\(y\\)-values that result are the <strong>output<\/strong>. The technical terms for all \\(x\\)-values and \\(y\\)-values are <strong>domain<\/strong> and <strong>range<\/strong>, respectively.<\/p>\n<p>So in this equation, the range, \\(y\\), is going to be the same as \\(2x+3\\).<\/p>\n<p>We can use a table to lay out some of the different outcomes of the equation. <\/p>\n<table class=\"ATable\" style=\"margin: auto;\">\n<thead>\n<tr style=\"height: 45px;\">\n<th style=\"vertical-align: middle;\"><strong>Domain<\/strong><\/th>\n<th style=\"vertical-align: middle;\"><strong>Range \\(\\hspace{20px}y=2x+3\\)<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height:50px\">\n<td>\\(-2\\)<\/td>\n<td>\\(2(-2)+3=\u00a0 -1\\)<\/td>\n<\/tr>\n<tr style=\"height:50px\">\n<td>\\(-1\\)<\/td>\n<td>\\(2(-1)+3=1\\)<\/td>\n<\/tr>\n<tr style=\"height:50px\">\n<td>\\(0\\)<\/td>\n<td>\\(2(0)+3=3\\)<\/td>\n<\/tr>\n<tr style=\"height:50px\">\n<td>\\(1\\)<\/td>\n<td>\\(2(1)+3=5\\)<\/td>\n<\/tr>\n<tr style=\"height:50px\">\n<td>\\(2\\)<\/td>\n<td>\\(2(2)+3=7\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nSo we have our equation here, and the different values that \\(x\\) could be listed here. For instance, if we find that \\(x\\) is -2, our problem would look like \\(2(-2)\\) (which is -4), plus 3, which equals -1.<\/p>\n<h3><span id=\"Ordered_Pairs\" class=\"m-toc-anchor\"><\/span>Ordered Pairs<\/h3>\n<p>\nA set of ordered pairs can also be used to show a mathematical relationship. The five elements of the domain and range from the table are shown here:<\/p>\n<div class=\"examplesentence\">\\((-2,-1), (-1,1), (0,3),\\)\\( (1,5), ( 2,7)\\)<\/div>\n<p>\n&nbsp;<br \/>\nIn words, each \\(x\\)-value is shown as an ordered pair with its corresponding \\(y\\)-value. These ordered pairs can be graphed on a coordinate plane to show the relationship visually.<\/p>\n<p>The relation, \\(y=2x+3\\), is graphed here:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65161 size-medium\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-1-300x298.png\" alt=\"y = 2x + 3 as a graph\" width=\"300\" height=\"298\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-1-300x298.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-1-150x150.png 150w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-1-768x762.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-1.png 850w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The domain values are found on the \\(x\\)-axis, and the graph takes shape according to the range values on the \\(y\\)-axis.<\/p>\n<h3><span id=\"Mapping\" class=\"m-toc-anchor\"><\/span>Mapping<\/h3>\n<p>\nAnother convenient visual representation of a relationship is called a <strong>mapping<\/strong>, which shows a direct correspondence between the domain and range elements. Here are two examples:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65171 size-large\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/map-A-and-Map-B-1024x409.png\" alt=\"mapping, input and output\" width=\"1024\" height=\"409\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/map-A-and-Map-B-1024x409.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/map-A-and-Map-B-300x120.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/map-A-and-Map-B-768x307.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/map-A-and-Map-B-1536x614.png 1536w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/map-A-and-Map-B.png 1558w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>In Map A, each input value maps to one output value, but in Map B, the domain value \u201c2\u201d maps to two output values, 20 and 40. This simple image plainly shows the criteria to determine whether the relation given is a function.<\/p>\n<p>By definition, a <strong>relation<\/strong> is defined as a function if each element of the domain maps to one, and only one, element of the range. Not all relations are functions, but functions are a subset of relations. We have covered several representations of relations in this video. The following image shows four representations that show a relation that does not qualify as a function:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65168 size-large\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-1024x374.png\" alt=\"Ordered pairs, table, graph, mapping diagram\" width=\"1024\" height=\"374\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-1024x374.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-300x110.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-768x281.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-1536x561.png 1536w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs.png 1798w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h3><span id=\"Vertical_Line_Test\" class=\"m-toc-anchor\"><\/span>Vertical Line Test<\/h3>\n<p>\nAt this point, we will focus on the graph of this relation to introduce the vertical line test, which allows you to quickly determine the function status of any graph that you are given.<\/p>\n<p>In short, if you draw a vertical line through a graph and hit only ONE point, then that graph represents a function. In the example we just showed, if you draw a vertical line through the domain value of -2, the line will pass through range values of 2 and -2.  <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65164\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-2-253x300.png\" alt=\"vertical line test\" width=\"253\" height=\"300\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-2-253x300.png 253w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-2-768x912.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graph-2.png 816w\" sizes=\"auto, (max-width: 253px) 100vw, 253px\" \/><\/p>\n<p>Therefore, the relation fails the vertical line test. Here are more examples of this convenient tool to determine a function from a graph.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65165 size-large\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-2-1024x397.png\" alt=\"function graphs\" width=\"1024\" height=\"397\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-2-1024x397.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-2-300x116.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-2-768x298.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-2-1536x595.png 1536w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-2.png 1801w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>Think you\u2019ve got the hang of it? <\/p>\n<h2><span id=\"What_is_a_Function\" class=\"m-toc-anchor\"><\/span>What is a Function?<\/h2>\n<p>\nLet\u2019s go through a few problems together and see.<\/p>\n<p>Is this a function?<\/p>\n<div class=\"examplesentence\">33\\((\u22122 , 2), (\u22123 , 3), (\u22124 , 4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nIn this set of ordered pairs, the domain values are \\({-2, -3, -4}\\) and the range values are \\({2, 3, 4}\\).  Because each domain value is paired with only one range value, this relation is a function. What about this set?<\/p>\n<div class=\"examplesentence\">\\((3, 2), (3, 3), (2, 4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nIn this set, the domain values are \\({3, 2}\\), and the range values are \\({2, 3, 4}\\). Because the domain value of 3 is paired with 2 and 3 in the range, this relation is NOT a function. <\/p>\n<p>Here\u2019s one more set to figure out: <\/p>\n<div class=\"examplesentence\">\\((1, 2), (2, 2), (3, 2),\\)\\( (4, 2), (5, 2)\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>What do you think? Is this a function or just a relation?<\/p>\n<p>The domain of this relation is \\({1, 2, 3, 4, 5}\\) and the range is \\({2}\\). Because each element of the domain is paired with \u201c2\u201d, it satisfies the definition of a function. It is allowable for each element of the domain to pair with the SAME element of the range.<\/p>\n<p>Remember the vertical line test? Which one of these graphs passes that test if we were to draw a vertical line through it?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65167 size-large\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-3-1024x508.png\" alt=\"vertical line test two graphs\" width=\"1024\" height=\"508\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-3-1024x508.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-3-300x149.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-3-768x381.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/graphs-3.png 1283w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>The first graph here passes the test, because the vertical line only passes through one point. In the second graph, the vertical line passes through two points, so it is not a function.<\/p>\n<p>Now that we\u2019ve gone through several different representations of functions, you should be well on your way to knocking out more complex equations down the road.<\/p>\n<p>Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"FAQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Frequently_Asked_Questions\" class=\"m-toc-anchor\"><\/span>Frequently Asked Questions<\/h2>\n<div class=\"faq-list\">\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is a function in math?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>A function is a relation where one quantity depends on the other.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you find the domain of a function?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Find the domain of a function by considering what numbers could be inputs for the function.<\/p>\n<div class=\"lightbulb-example-2\"><span class=\"lightbulb-icon\">\ud83d\udca1<\/span><span class=\"faq-example-question\">Example: What is the domain of \\(f(x)=\\dfrac{x+7}{x-3}\\)?<\/span><\/p>\n<hr style=\"padding: 0; margin-top: -0.2em; margin-bottom: 1.2em\">A denominator can never be 0, so \\(x\\) can be any number except 3: <\/p>\n<p style=\"text-align: center; margin-bottom: 0em\">\\(x=\\dfrac{R}{3}\\)<\/p>\n<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you find the inverse of a function?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>To find the inverse of a function, change \\(f(x)\\) or \\(y\\) to \\(f^{-1} (x)\\) and swap the positions of that and \\(x\\). Then, solve for \\(f^{-1} (x)\\).<\/p>\n<div class=\"lightbulb-example-2\"><span class=\"lightbulb-icon\">\ud83d\udca1<\/span><span class=\"faq-example-question\">Example: What is the inverse of \\(f(x)=4x+1\\)?<\/span><\/p>\n<hr style=\"padding: 0; margin-top: -0.2em; margin-bottom: 1.2em\">First, change \\(f(x)\\) to \\(f^{-1} (x)\\) and swap positions of that and \\(x\\).<\/p>\n<p style=\"text-align:center;\">\\(x=4f^{-1} (x)+1\\)<\/p>\n<p> Then, solve for \\(f^{-1} (x)\\).<\/p>\n<p style=\"text-align:center; margin-bottom: 0em; line-height: 50px\">\\(x-1=f^{-1} (x)\\)<br \/> \\(\\dfrac{x-1}{4}=f^{-1} (x)\\)<\/p>\n<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you find the range of a function?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Find the range of a function by determining what the outputs of a function are.<\/p>\n<div class=\"lightbulb-example-2\"><span class=\"lightbulb-icon\">\ud83d\udca1<\/span><span class=\"faq-example-question\">Example: What is the range of the function \\(f(x)=x^2+x+4\\)?<\/span><\/p>\n<hr style=\"padding: 0; margin-top: -0.2em; margin-bottom: 1.2em\">The range is \\([4,\\infty )\\) because the lowest value of \\(f(x)\\) for any given \\(x\\)-value is 4, but \\(f(x)\\) can be any value greater than 4.<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is a linear function?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>A linear function is any function that graphs as a line.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Function Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich set of ordered pairs shows a relation that is a function?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">\\((1,-3)\\), \\((2,-4)\\), \\((0,2)\\), \\((-1,5)\\), \\((-2,8)\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\((1,1)\\), \\((1,2)\\), \\((1,3)\\), \\((1,4)\\), \\((1,5)\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\((0,-3)\\), \\((1,4)\\), \\((2,-5)\\), \\((3,-6)\\), \\((0,3)\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\((-5,-5)\\), \\((-4,-4)\\), \\((-5,-3)\\), \\((-4,-2)\\), \\((-5,-1)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>By definition, a relation is a function if each element of the domain maps to only one element in the range. In other words, a relation is a function if each \\(x\\)-value has only one \\(y\\)-value.<\/p>\n<p>Looking at the relations in each choice, the only relation where the domain value (\\(x\\)-value) does not repeat is \\((1,-3)\\), \\((2,-4)\\), \\((0,2)\\), \\((-1,5)\\), \\((-2,8)\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich equation shows a relation that is NOT a function? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(y=x\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\(y=x^2\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">\\(x=y^2\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(x^3=y\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To see if the equations are a function, we can graph them and see if they pass the vertical line test. It is clear upon further examination that the only graph that does not pass the vertical line test is \\(x=y^2\\).<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-111036 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-1-1.png\" alt=\"several different colored lines graphed\" width=\"706\" height=\"468\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-1-1.png 1715w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-1-1-300x199.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-1-1-1024x679.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-1-1-768x510.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-1-1-1536x1019.png 1536w\" sizes=\"auto, (max-width: 706px) 100vw, 706px\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich relation is a function? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\(x=-3y^2+5\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\((-3{,}2),(-2{,}1),(-1{,}0),\\)\\((0{,}0),(-1{,}1)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-3\"><table class=\"ATable\" style=\"width: 20%; margin-bottom: 0em\">\r\n<thead>\r\n<tr>\r\n<th>\\(x\\)<\/th>\r\n<th>\\(y\\)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"background-color: white\">\\(-2\\)<\/td>\r\n<td style=\"background-color: white\">\\(-6\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(-1\\)<\/td>\r\n<td style=\"background-color: white\">\\(-3\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(0\\)<\/td>\r\n<td style=\"background-color: white\">\\(0\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(1\\)<\/td>\r\n<td style=\"background-color: white\">\\(3\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div><div class=\"PQ\"  id=\"PQ-3-4\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-2-e1770149273670.png\" alt=\"graph, red sideways parabola, vertex at (0, 0), opens to the right\" width=\"302.1\" height=\"305.4\" class=\"alignnone size-full wp-image-111033\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-2-e1770149273670.png 1007w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-2-e1770149273670-297x300.png 297w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-2-e1770149273670-768x776.png 768w\" sizes=\"(max-width: 1007px) 100vw, 1007px\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The graph of the equation in Choice A and the graph in Choice D are both sideways parabolas that fail the vertical line test. In Choice B, \u20131 maps to both 0 and 1, which means it is not a function. Choice C is the only relation where each of the values of the domain maps to one unique value in the range.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich table of values shows a relation that is a function?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\"><table class=\"ATable\" style=\"width: 20%; margin-bottom: 0em\">\r\n<thead>\r\n<tr>\r\n<th>\\(x\\)<\/th>\r\n<th>\\(y\\)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"background-color: white\">\\(0\\)<\/td>\r\n<td style=\"background-color: white\">\\(2\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(1\\)<\/td>\r\n<td style=\"background-color: white\">\\(3\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(2\\)<\/td>\r\n<td style=\"background-color: white\">\\(4\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(2\\)<\/td>\r\n<td style=\"background-color: white\">\\(5\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\"><table class=\"ATable\" style=\"width: 20%; margin-bottom: 0em\">\r\n<thead>\r\n<tr>\r\n<th>\\(x\\)<\/th>\r\n<th>\\(y\\)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"background-color: white\">\\(2\\)<\/td>\r\n<td style=\"background-color: white\">\\(2\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(3\\)<\/td>\r\n<td style=\"background-color: white\">\\(3\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(4\\)<\/td>\r\n<td style=\"background-color: white\">\\(4\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(5\\)<\/td>\r\n<td style=\"background-color: white\">\\(5\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div><div class=\"PQ\"  id=\"PQ-4-3\"><table class=\"ATable\" style=\"width: 20%; margin-bottom: 0em\">\r\n<thead>\r\n<tr>\r\n<th>\\(x\\)<\/th>\r\n<th>\\(y\\)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"background-color: white\">\\(-1\\)<\/td>\r\n<td style=\"background-color: white\">\\(4\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(-2\\)<\/td>\r\n<td style=\"background-color: white\">\\(6\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(-2\\)<\/td>\r\n<td style=\"background-color: white\">\\(-6\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(-1\\)<\/td>\r\n<td style=\"background-color: white\">\\(-4\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div><div class=\"PQ\"  id=\"PQ-4-4\"><table class=\"ATable\" style=\"width: 20%; margin-bottom: 0em\">\r\n<thead>\r\n<tr>\r\n<th>\\(x\\)<\/th>\r\n<th>\\(y\\)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"background-color: white\">\\(7\\)<\/td>\r\n<td style=\"background-color: white\">\\(3\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(5\\)<\/td>\r\n<td style=\"background-color: white\">\\(1\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(3\\)<\/td>\r\n<td style=\"background-color: white\">\\(-1\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"background-color: white\">\\(7\\)<\/td>\r\n<td style=\"background-color: white\">\\(-3\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To determine if the relation presented in the tables is a function, we look at the domain values (\\(x\\)-values) to see if they map to one unique range value. The only relation that meets this requirement is Choice B, where each value of the domain is mapped to a unique range value.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich shows the graph of a function?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-3.png\" alt=\"graph, purple squiggle, passes through the points (negative 1, negative 1.5), (0, 0), and (1, 1.5)\" width=\"354\" height=\"355\" \/><\/div><div class=\"PQ\"  id=\"PQ-5-2\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-4.png\" alt=\"graph, sideways parabola, points at (negative 3, 0), (0, 1.75), and (0, negative 1.75)\" width=\"354\" height=\"355\" \/><\/div><div class=\"PQ\"  id=\"PQ-5-3\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-5.png\" alt=\"graph, sideways absolute value function, points at (negative 3, 0), (0, 3), and (0, negative 3)\" width=\"354\" height=\"355\" \/><\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\"> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Definition-of-a-Function-PQ-6.png\" alt=\"coordinate grid, green parabola graphed, vertex at (0, -3), x-intercepts at x equals negative 1.75 and x equals 1.75\" width=\"354\" height=\"355\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Since we have already been given the graphs of the relations, we can apply the vertical line test to determine which relation is a function. The only graph that passes the vertical line test is Choice D. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-ii\/\">Return to Algebra II Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra II Videos<\/p>\n","protected":false},"author":1,"featured_media":100606,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-36546","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-functions-videos","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/36546","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=36546"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/36546\/revisions"}],"predecessor-version":[{"id":280133,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/36546\/revisions\/280133"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100606"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=36546"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}