{"id":705,"date":"2013-05-28T14:41:05","date_gmt":"2013-05-28T14:41:05","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=705"},"modified":"2026-04-23T11:59:23","modified_gmt":"2026-04-23T16:59:23","slug":"dependent-and-independent-variables-and-inverting-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/dependent-and-independent-variables-and-inverting-functions\/","title":{"rendered":"Dependent and Independent Variables with Inverting Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_pIBKFEJrNc8\" 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_pIBKFEJrNc8\" data-source-videoID=\"pIBKFEJrNc8\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Dependent and Independent Variables with Inverting Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Dependent and Independent Variables with Inverting Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_pIBKFEJrNc8:hover {cursor:pointer;} img#videoThumbnailImage_pIBKFEJrNc8 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1120-dependent-and-independent-variables-with-inverting-functions-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_pIBKFEJrNc8\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_pIBKFEJrNc8\" 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\",\"Dependent and Independent Variables with Inverting Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_pIBKFEJrNc8\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_pIBKFEJrNc8\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_pIBKFEJrNc8\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction 0N3_Function() {\n  var x = document.getElementById(\"0N3\");\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=\"0N3_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=\"0N3\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Dependent_and_Independent_Variables_with_Inverting_Function_Practice_Questions\" class=\"smooth-scroll\">Dependent and Independent Variables with Inverting 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=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hey, guys! Today, we\u2019re going to take a look at independent and dependent variables and then dive into the process of inverting functions.<\/p>\n<p>Let\u2019s step back and do a quick review of functions before we begin. Remember that a function relates inputs (\\(x\\)) and outputs (\\(y\\)). The output value will always depend on what you input. For a relationship to be a function, each input must have exactly one output.<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 40%\">\n<thead>\n<tr>\n<th>\\(x\\) (input)<\/th>\n<th>\\(y\\) (output)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nWe refer to these input and output values as either independent or dependent variables. We can plug in any value for \\(x\\), so it is our independent variable. The \\(y\\)-variable, however, will always depend on what was plugged in for \\(x\\). So, we refer to the \\(y\\)-variable as the dependent variable.<\/p>\n<p>Let\u2019s look at the function \\(f(x) = 2x + 4\\) as an example. This equation is written in what\u2019s called functional notation. \\(f(x)\\) essentially represents \\(y\\), just as it would in the linear equation \\(y = 2x + 4\\). \\(f(x)\\) is simply stating that the \\(y\\)-value is \u201ca function of \\(x\\),\u201d meaning that the \\(y\\)-value will depend on what the \\(x\\)-value is. <\/p>\n<p>This particular function could reflect a scenario where \\(x\\) represents how many plants you sell at a farmers market, and \\(f(x)\\), or \\(y\\), is how much money you will make. If you started with $4, and you sell each plant for $2, the money you earn (\\(y\\)) will depend on how many plants you sell (\\(x\\)). So in this scenario, the number of plants you sell is the independent variable, and the amount of money you make is the dependent variable. This makes sense because the amount of money you make should depend on how many plants you sell.<\/p>\n<p>We can use this function to solve for \\(y\\) in terms of \\(x\\). This means that we can choose any value for \\(x\\) and use it to find the value of \\(y\\).<\/p>\n<p>For example, let\u2019s say we started with the $4, and then we sell 26 plants at the farmers market. This means that \\(x\\) is equal to 26. Let\u2019s plug 26 in for \\(x\\), and then solve for \\(y\\) in terms of \\(x\\) to find out how much money we will make. <\/p>\n<div class=\"examplesentence\">\\(f(x) = 2x + 4\\)<br \/>\n\\(f(26) = 2(26) + 4\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, with a bit of algebra, we can simplify the right side of the equation.<\/p>\n<div class=\"examplesentence\">\\(2 \\times 26 = 52\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen we add 4. This leaves us with \\(f(26) = 56\\). We have solved for \\(y\\) in terms of \\(x\\), or solved for \\(y\\) by plugging in a value for \\(x\\). We can see that when \\(x\\) is equal to 26, \\(y\\) will be 56. So if we sell 26 plants at the farmers market at the end of the day, we will have $56.<\/p>\n<p>With this understanding of independent and dependent variables, we\u2019re prepared to dive into the process of inverting functions.<\/p>\n<p>Up to this point we have used examples of functions where we solve for \\(y\\) in terms of \\(x\\). <\/p>\n<p>Now let\u2019s move on to the process of solving the inverse of a function. Finding the inverse of a function is essentially switching the input and the output. This means we are no longer using \\(x\\) to find \\(y\\). We will now be using \\(y\\) to find \\(x\\). <\/p>\n<p>Let\u2019s take it step by step to find the inverse of the function \\(f(x) = 3x &#8211; 4\\). <\/p>\n<p>Since we\u2019re looking to find the inverse, we\u2019ll need to write it like this:<\/p>\n<div class=\"examplesentence\">\\(f^{-1} (x)=\\) ?<\/div>\n<p>\n&nbsp;<br \/>\nFinding the inverse of a function will be a four-step process.<\/p>\n<h3><span id=\"Step_1\" class=\"m-toc-anchor\"><\/span>Step 1<\/h3>\n<p>\nStep 1 is to write the function as a linear equation. So in this case we will write:<\/p>\n<div class=\"examplesentence\">\\(f(x) = 3x &#8211; 4\\)<br \/>\n\\(y = 3x &#8211; 4\\)<\/div>\n<p>\n&nbsp;<br \/>\nBecause we are switching the input and the output, we want to be working with \\(x\\) and \\(y\\) instead of \\(x\\) and \\(f(x)\\). This will make the switch easier to see visually. <\/p>\n<h3><span id=\"Step_2\" class=\"m-toc-anchor\"><\/span>Step 2<\/h3>\n<p>\nStep 2 is to simply swap \\(x\\) and \\(y\\). This inverts the scenario from \\(x\\) being the input to now \\(y\\) being the input.<\/p>\n<p>Now the function should look like this:<\/p>\n<div class=\"examplesentence\">\\(y=3x-4\\)  <br \/>\n\\(x=3y-4\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Step_3\" class=\"m-toc-anchor\"><\/span>Step 3<\/h3>\n<p>\nStep 3 is simply to solve for \\(y\\). To do that, let\u2019s use inverse operations to isolate the variable. We\u2019ll add 4 to both sides, and then divide by 3.<\/p>\n<div class=\"examplesentence\">\\(x = 3y &#8211; 4\\:\\) now becomes \\(\\:y = x + 43\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Step_4\" class=\"m-toc-anchor\"><\/span>Step 4<\/h3>\n<p>\nOur last step is to write the equation in inverse notation:<\/p>\n<div class=\"examplesentence\">\\(f^{-1} (x)=\\frac{x+4}{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd there you have it! The inverse of the function \\(f(x) = 3x &#8211; 4\\) is \\(f -1(x)= x + 43\\).<\/p>\n<p>It is important to note that when you graph a function and its inverse on the same coordinate grid, an important relationship should be visible. You should notice that the two lines are reflected over the line \\(y=x\\). So essentially, when you\u2019re finding the inverse of a function, you are simply reflecting that function over the line \\(y=x\\).<\/p>\n<p>Let\u2019s graph the function and its inverse from the previous example so we can see this relationship. <\/p>\n<div class=\"examplesentence\">\n\\(f(x) = 3x &#8211; 4\\)\n<p style=\"margin-top: -1em; margin-bottom: 1em; font-size: 90%; font-style: italic\">(the original function)<\/p>\n\\(f -1(x) = x + 43\\)\n<p style=\"margin-top: -1em; margin-bottom: 0em; font-size: 90%; font-style: italic\">(inverse of the function)<\/p>\n<\/div>\n<p>\n&nbsp;<br \/>\nIn our graph we can see the red line representing the line \\(y=x\\). The blue line represents our original function, and the green line is the inverse of that function. When we graph the function and its inverse we are able to see visually that the lines are reflected over the red line \\(y=x\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/02\/graphing-a-function-and-the-inverse.png\" alt=\"\" width=\"338\" height=\"364\"\/><\/p>\n<p>That\u2019s all there is to it! I hope this review was helpful. Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Dependent_and_Independent_Variables_with_Inverting_Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Dependent and Independent Variables with Inverting 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 values represents one set of output for the function \\(f(x)=-3x+2\\)? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\({-1, -2, -5, -8}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\({-1, 0, 1, 2}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\({1, -2, -5, -8}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-4\">\\({5, 2, -1, -4}\\)<\/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>The output is the result of substituting an input for \\(x\\) and evaluating. The resulting evaluation is the output of the function.<\/p>\n<p>The best way to see the output is by graphing the function, which will show that 5, 2, \u22121, and \u22124 are the outputs for the inputs \u22121, 0, 1, and 2, respectively. In other words, if you plug in \u22121, 0, 1, and 2 into the function for \\(x\\), you will get 5, 2, \u22121, and \u22124.<\/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 \/>\nThe graph of function \\(g(x)\\) and its inverse, \\(g^{-1}(x)\\), are reflected over which line?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(x\\)-axis<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\(y\\)-axis<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">\\(y=x\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(y=-x\\)<\/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>When a function and its inverse are graphed, it is possible to see that the graph of the inverse is reflected over the line \\(y=x\\). <\/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 \/>\nThe function \\(t(x)=5x\u201315\\), where \\(x\\) is the number of crates of eggs, represents Kian\u2019s total daily income from selling crates of eggs at the farmer\u2019s market. What is Kian\u2019s income if he sells 28 crates of eggs in one day?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">$65<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">$125<\/div><div class=\"PQ\"  id=\"PQ-3-3\">$140<\/div><div class=\"PQ\"  id=\"PQ-3-4\">$155<\/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>Since the function represents Kian\u2019s income and the input, \\(x\\), represents the number or crates of eggs, we can substitute 28 for \\(x\\) and evaluate to find Kian\u2019s daily income at the farmer\u2019s market. <\/p>\n<p style=\"text-align:center; line-height: 35px\">\n\\(t(x)=5x\u201315\\)<br \/>\n\\(t(x)=5(28)\u201315\\)<br \/>\n\\(t(x)=140\u201315\\)<br \/>\n\\(t(x)=125\\)<\/p>\n<p>Therefore, Kian\u2019s daily income for selling 28 crates of eggs is $125.<\/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 shows the inverse of the function \\(f(x)=\\frac{1}{2}x-3\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">\\(f^{-1}(x)=2x-3\\)<\/div><div class=\"PQ\"  id=\"PQ-4-2\">\\(f^{-1}(x)=2x+3\\)<\/div><div class=\"PQ\"  id=\"PQ-4-3\">\\(f^{-1}(x)=2x-6\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">\\(f^{-1}(x)=2x+6\\)<\/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 find the inverse of a function, we use the following steps. First, we replace the function notation with \\(y\\). Next, we switch the \\(x\\) and \\(y\\) and solve for \\(y\\).<\/p>\n<p style=\"text-align:center; line-height: 40px\">\n\\(f(x)=\\frac{1}{2}x-3\\)<br \/>\n\\(y=\\frac{1}{2}x-3\\)<br \/>\n\\(x=\\frac{1}{2}y-3\\)<br \/>\n\\(x+3=\\frac{1}{2}y\\)<br \/>\n\\(2(x+3=\\frac{1}{2}y)\\)<br \/>\n\\(2x+6=y\\)<br \/>\n\\(f^{-1}(x)=2x+6\\)<\/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 \/>\nNadya uses the function of \\(I(x)=0.05x+20\\) to calculate her monthly car insurance, where \\(x\\) represents the number of miles, she drives each month. How much is her July bill if she drives 45 miles?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">$22.25<\/div><div class=\"PQ\"  id=\"PQ-5-2\">$42.50<\/div><div class=\"PQ\"  id=\"PQ-5-3\">$65.05<\/div><div class=\"PQ\"  id=\"PQ-5-4\">$245.00<\/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>We can find Nadya\u2019s July car insurance bill by substituting 45 for \\(x\\) and evaluating.<\/p>\n<p style=\"text-align:center; line-height: 35px\">\n\\(I(x)=0.05x+20\\)<br \/>\n\\(I(x)=0.05(45)+20\\)<br \/>\n\\(I(x)=2.25+20\\)<br \/>\n\\(I(x)=22.25\\)\n<\/p>\n<p>Therefore, the total car insurance bill for driving 45 miles is $22.25.<\/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-i\/\">Return to Algebra I Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":100231,"parent":0,"menu_order":44,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-705","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":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/705","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=705"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/705\/revisions"}],"predecessor-version":[{"id":292157,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/705\/revisions\/292157"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100231"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=705"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}