{"id":29339,"date":"2023-05-12T18:11:25","date_gmt":"2023-05-12T18:11:25","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=29339"},"modified":"2023-11-15T16:40:18","modified_gmt":"2023-11-15T16:40:18","slug":"inverse-function-definition-types-examples-facts-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function","title":{"rendered":"Inverse Function &#8211; Definition, Types, Examples, Facts, FAQs"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-7a62c071-8afa-47c4-82c5-498f8296c7b7\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function#0-what-is-an-inverse-function>What Is an Inverse Function?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function#3-f-1-meaning-and-mapping-diagram>$f^{-1}$ Meaning and Mapping Diagram<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function#10-how-to-find-inverse-functions-using-algebra>How to Find Inverse Functions Using Algebra<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function#13-solved-examples-on-inverse-function>Solved Examples On Inverse Function<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function#14-practice-problems-on-inverse-function>Practice Problems On Inverse Function<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function#15-frequently-asked-questions-on-inverse-function>Frequently Asked Questions On Inverse Function<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-is-an-inverse-function\">What Is an Inverse Function?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>The inverse function <\/strong>$f^{-1}$<strong> undoes the action performed by the function f.&nbsp; We read <\/strong>$f^{-1}$<strong> as \u201cf inverse.\u201d<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">If $f^{-1}$ is an inverse of the function f, then f is an inverse function of $f^{-1}$. Thus, we can say that&nbsp; f and $f^{-1}$ reverse each other.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"313\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/relation-between-function-and-its-inverse.png\" alt=\"Relation between function and its inverse\" class=\"wp-image-35812\" title=\"Relation between function and its inverse\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/relation-between-function-and-its-inverse.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/relation-between-function-and-its-inverse-300x151.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">A function from set A to the set B, represented by $f:A \\rightarrow B$ is a relation from the set A (a set of inputs) to the set B (a set of possible outputs) such that every element in A is related to exactly one element from the set B.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Domain: <\/strong>The set of inputs\/ the set of values for which the function is defined\/ the set of values that we can plug into the function<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Range:<\/strong> The set of outputs\/ the set of resulting values&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The symbol for inverse function is $f^{-}1\\; (f$ with the exponent of $\\;\ufe63\\;1)$.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"1-inverse-function-definition-in-math\">Inverse Function Definition in Math<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">A function $g = f^{-1}$ is said to be an inverse function of a function $y = f(x)$ if whenever $f(x) = y$, we have $g(y) = f^{-1}(y) = x$.&nbsp; If f and g are inverse functions, then we have $f(x) = y$ if and only if $g(y) = x$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-inverse-function-formula\">Inverse Function Formula<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">If f maps x to $f(x)$, then $f^{-1}$ maps f(x) back to x. Thus, if you apply a function and then its inverse, you get the original value. Same applies if we reverse the order of the two functions. Thus, the inverse function formula can be given as<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f^{-1}(f(x)) = x$<strong>&nbsp;  <\/strong>\u2026for all x in the domain of f<\/p>\n\n\n\n<p class=\"eplus-wrapper\">and<\/p>\n\n\n\n<p class=\"eplus-wrapper\">&nbsp;$f(f^{-1}(y))=y$ \u2026for all y in the domain of $f^{-1}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>If domain <\/strong>$=$<strong> range, we can say that<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f^{-1}(f(x)) = f(f^{-1}(x)) = x$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-f-1-meaning-and-mapping-diagram\">$f^{-1}$ Meaning and Mapping Diagram<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Consider a mapping diagram of a function f and its inverse $f^{-1}$ .&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"570\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/mapping-diagram-of-inverse-function.png\" alt=\"Mapping diagram of inverse function\" class=\"wp-image-35813\" title=\"Mapping diagram of inverse function\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/mapping-diagram-of-inverse-function.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/mapping-diagram-of-inverse-function-300x276.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">What do you notice?<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The domain of f is the range of $f^{-1}$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The domain of $f^{-1}$ is the range of f.<\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>Action of the function&nbsp;&nbsp;f<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Action of the inverse function&nbsp;<\/strong>$f^{-1}$<strong>&nbsp;<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">f takes 1 to a.<\/td><td class=\"has-text-align-center\" data-align=\"center\">$f^{-1}$ takes a to 1.<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">f takes 2 to b.<\/td><td class=\"has-text-align-center\" data-align=\"center\">$f^{-1}$ takes b to 2.<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">f takes 3 to c.<\/td><td class=\"has-text-align-center\" data-align=\"center\">$f^{-1}$&nbsp; takes c to 3.<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">f takes 3 to d.<\/td><td class=\"has-text-align-center\" data-align=\"center\">$f^{-1}$&nbsp; takes d to 4.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-how-to-find-the-inverse-of-a-function\">How to Find the Inverse of a Function<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Let\u2019s understand the steps to find the inverse of a function with an example.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let us consider a function $f(x) = ax + b$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 1: For the given function, replace <\/strong>$f(x)$<strong> by y. In other words, substitute <\/strong>$f(x) = y$<strong>.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Put $f(x) = y$ in $f(x) = ax + b$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The result is $y = ax + b$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 2:&nbsp; Replace <\/strong><strong>x<\/strong><strong> with <\/strong><strong>y<\/strong><strong>. Replace <\/strong><strong>y<\/strong><strong> with <\/strong><strong>x<\/strong><strong>.&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">&nbsp;In the function $y = ax + b$, we interchange x and y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The result is $x = ay + b$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 3: Solve the expression to write it in terms of <\/strong>y<strong>.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x = ay + b \\Rightarrow y = $\\frac{x \\;-\\; b}{a}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The result is $y = \\frac{x \\;-\\; b}{a}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 4: Replace <\/strong>$y = f^{-1}(x)$<strong>.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The result is $f^{-1}(x) = \\frac{x \\;-\\; ba}$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, $f ^{-1}(x)= \\frac{x \\;-\\; b}{a} is an inverse function of $y = f(x) = ax + b$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-inverses-of-some-common-functions\">Inverses of Some Common Functions<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Some common inverse functions are given below:<\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>Function<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Inverse Function<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Conditions&nbsp;<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">$ax + b$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$x \\;-\\; ba$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$a \\neq 0$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\frac{1}{x}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\frac{1}{y}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">x and y not equal to 0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$x^2$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\sqrt{y}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$x$ and $y \u2265 0$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$e^x$<\/td><td class=\"has-text-align-center\" data-align=\"center\">ln(y)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$y &gt; 0$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">sin(x)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$sin^{-1}(y)$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\frac{-\\pi}{2}$ to $\\frac{-\\pi}{2}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">cos(x)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$cos^{-1}(y)$<\/td><td class=\"has-text-align-center\" data-align=\"center\">0 to<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">tan(x)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$tan^{-1}(y)$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\frac{\\pi-}{2}$ to $\\frac{-\\pi}{2}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-does-every-function-have-an-inverse\">Does Every Function Have an Inverse?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Not every function has an inverse. A function has an inverse if and only if it is one-to-one (or bijective). A function f has an inverse function if and only if, for every element y in its range, there is only one value of x in its domain for which $f(x) = y$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">It means that for each output, there is precisely only one input. The function does not take the same value twice. In other words, a function f is one-to-one if, for every a and b in its domain, $f(a) = f(b)$ implies $a = b$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-inverse-function-graph\">Inverse Function Graph<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>The graphs of f and <\/strong>$f\\;-\\;1$<strong>are symmetric over the line <\/strong>$x = y$<strong>. <\/strong>The functions f and $f\\;-\\;1$ are mirror images of each other on a graph since the roles of these two variables are reversed. Thus, we can identify whether two functions are inverses of each other simply by checking if they are symmetric over $x = y$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-what-are-the-types-of-inverse-functions\">What Are the Types of Inverse Functions?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The different types of inverse functions include inverse trigonometric functions, inverse of rational functions, inverse hyperbolic functions, and inverse of logarithmic functions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-inverse-of-rational-functions\">Inverse of Rational Functions<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">A rational function is an algebraic function such that both numerators and denominators are polynomials. It is a function of the form $f(x) = \\frac{P(x)}{Q(x)}$ where $Q(x) \\neq 0$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The steps to find the inverse of a rational function are:<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 1:<\/strong> Substitute $f(x) = y$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 2:<\/strong> Interchange x and y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 3:<\/strong> Solve for y and express it in terms of x.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 4:<\/strong> Replace y with $f^{-1}(x)$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The final result we get is the inverse function.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Example: Find the inverse of <\/strong>$f(x) = \\frac{x + 1}{5 \\;-\\; 2x}. (x \\neq \\frac{5}{2}$ and $x \\neq \\frac{-1}{2})$<strong>.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, $y = \\frac{x + 1}{5 \\;-\\; 2x}$ \u2026 $x\\neq \\frac{5}{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Interchange x and y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x = \\frac{y + 1}{5 \\;-\\; 2y}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$5x\\;-\\;2xy = y + 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$5x\\;-\\;1 = 2xy + y$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$5x\\;-\\;1 = y(2x + 1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$y = \\frac{5x \\;-\\; 1}{2x + 1}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f^{-1}(x) = \\frac{5x \\;-\\; 1}{2x + 1}$  \u2026 $x \\neq \\frac{-1}{2}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"10-how-to-find-inverse-functions-using-algebra\">How to Find Inverse Functions Using Algebra<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">We can replace f(x) with y and solve the <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/number-sense\/expression\">algebraic expression<\/a> in terms of x.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Consider an example.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Function: $f(x) = 5x \\;-\\; 9$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Replace the f(x) with y: $y = 5x \\;-\\; 9$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Add 9 to both sides: $y + 9 = 5x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Divide both sides of the equation by 5: $\\frac{y + 9}{5} = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Exchange the sides: $x = \\frac{y + 9}{5}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Solve the expression replacing x with $f^{-1}(y) = \\frac{y + 9}{5}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"11-facts-about-inverse-function\">Facts about Inverse Function<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The inverse of f(x) is $f^{-1}(y)$.<br>Inverse function $f^{-1}(x)$ is not the same as the reciprocal $\\frac{1}{f(x)}$.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>Horizontal line test:<\/strong> If a horizontal line intersects the graph of a function at more than two points, the function does not have an inverse. If the horizontal line intersects the graph only at a single point, the graph is one-to-one.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Inverse Logarithmic Functions and Inverse Exponential Function: The natural log functions are inverse of the exponential functions.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">If f and g are inverse functions of each other, then $f(g(x)) = g(f(x) = x$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Sometimes we have to restrict the domain of a function so that the inverse function can be defined. For example, the function $f(x) =  x^2\\;-\\;5$ does not have an inverse if the domain is the set of real numbers. However, if the domain is restricted to $x \\leq \\;-\\;5$, the function becomes one-to-one and has an inverse.<br><\/li>\n\n\n\n<li class=\"eplus-wrapper\">The natural logarithmic function is the inverse of the exponential function. The inverse of the exponential function $y = ax$ is $x = a^y$. Note that the logarithmic function $y = log_a x$ is equivalent to the exponential equation $x = a^y$.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"12-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In this article, we learnt about Inverse functions, their graphs, and steps for finding inverse functions. Let\u2019s solve a few solved examples and practice problems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"13-solved-examples-on-inverse-function\">Solved Examples On Inverse Function<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>1. What is the inverse of the function <\/strong>$f(x) = x + 1$<strong>?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Given function: <\/strong>$f(x) = x + 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Replace f(x) by y.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$y = x + 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Interchange x and y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>&nbsp;<\/strong>$x = y + 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Solve for y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$y = x\\;-\\;1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Replace<strong> <\/strong>y by $f^{-1}(x)$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f^{-1}(x) = x\\;-\\;1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>2. Find the inverse of <\/strong>$f(x) = x$<strong>?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Given function: $f(x) = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Replace f(x) by y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$y = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Interchange x and y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x = y$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Solve for y.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$y = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Replace<strong> <\/strong>y by $f^{-1}(x)$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f^{-1}(x) = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The inverse of an identity function is the identity function itself.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>3. What is the inverse of a function <\/strong>$g(x) = 5(x + 3)$<strong>?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$g(x) = 5(x + 3)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$y = 5(x + 3)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x = 5(y + 3)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x = 5y + 15$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$y = \\frac{x \\;-\\; 15}{5}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$g\\;-\\;1(x) = \\frac{x \\;-\\; 15}{5}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>4. If <\/strong>$g(x) = 8\\;-\\;\\frac{x}{3}$<strong>, and&nbsp; <\/strong>$f(x) = 24\\;-\\;3x$<strong> , show that f and g are inverses.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>i) <\/strong>$g(x) = 8\\;-\\;\\frac{x}{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f(g(x)) = f(8\\;-\\;\\frac{x}{3})$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f(g(x)) = 24\\;-\\;3(8\\;-\\;\\frac{x}{3})$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f(g(x))= 24\\;-\\;24 + x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f(g(x)) = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>ii) <\/strong>$f(x) = 24\\;-\\;3x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$g(f(x)) = g(24\\;-\\;3x)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$g(f(x)) = 8\\;-\\;\\frac{(24\\;-\\;3x)}{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$g(f(x)) = 8\\;-\\;8 + x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$g(f(x)) = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">From (i) and (ii), we have $f(g(x)) = x = g(f(x))$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, f and g are inverses.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"14-practice-problems-on-inverse-function\">Practice Problems On Inverse Function<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Inverse Function - Definition, Types, Examples, Facts, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">$f^{-1}(f(x))=$<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">y<\/div><div class=\"spq_answer_block\" data-value=\"1\">f(x)<\/div><div class=\"spq_answer_block\" data-value=\"2\">x<\/div><div class=\"spq_answer_block\" data-value=\"3\">f(y)<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: x<br\/>$f^{-1}(f(x)) = x$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">A function has an inverse if and only if it is<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">bijective<\/div><div class=\"spq_answer_block\" data-value=\"1\">one-one<\/div><div class=\"spq_answer_block\" data-value=\"2\">onto<\/div><div class=\"spq_answer_block\" data-value=\"3\">many to one<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: bijective<br\/>A function has an inverse if and only if it is bijective (one-to-one).<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">If f and g are inverse functions, then we have $f(x) = y$ if and only if<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$f(y) = x$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$y = x$ <\/div><div class=\"spq_answer_block\" data-value=\"2\">$g(y) = x$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$f(x) = g(y)$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $g(y) = x$<br\/>If f and g are inverse functions, then we have $f(x) = y$ if and only if $g(y) = x$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">The graphs of a function and its inverse are symmetric over<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">the x-axis<\/div><div class=\"spq_answer_block\" data-value=\"1\">the y-axis<\/div><div class=\"spq_answer_block\" data-value=\"2\">the line $x = y$<\/div><div class=\"spq_answer_block\" data-value=\"3\">the line $x = 1$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: the line $x = y$<br\/>The graph of a function and its inverse are symmetric over the line $x = y$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">What is the inverse of $f(x) = x + 5$?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$f^{-1}(x) = x\\;-\\;5$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$f^{-1}(x) = x + 5$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$f\\;-\\;1(x) = 5x$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$f^{-1}(x) = \\frac{x}{5}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $f^{-1}(x) = x\\;-\\;5$<br\/>$f(x) = x + 5$<br>\r\n$y = x + 5$<br>\r\n$x = y + 5$<br>\r\n$y = x \\;-\\; 5$<br>\r\nThus, $f^{-1}(x) = x\\;-\\;5$<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" 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\"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$f(y) = x$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If f and g are inverse functions, then we have $$f(x) = y$$ if and only if $$g(y) = x$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$y = x$$ \",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If f and g are inverse functions, then we have $$f(x) = y$$ if and only if $$g(y) = x$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$f(x) = g(y)$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If f and g are inverse functions, then we have $$f(x) = y$$ if and only if $$g(y) = x$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$g(y) = x$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"If f and g are inverse functions, then we have $$f(x) = y$$ if and only if $$g(y) = x$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"If f and g are inverse functions, then we have $$f(x) = y$$ if and only if $$g(y) = x$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The graphs of a function and its inverse are symmetric over\",\n                    \"text\": \"The graphs of a function and its inverse are symmetric over\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The graph of a function and its inverse are symmetric over the line $$x = y$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"the x-axis\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The graph of a function and its inverse are symmetric over the line $$x = y$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"the y-axis\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The graph of a function and its inverse are symmetric over the line $$x = y$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"the line $$x = 1$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The graph of a function and its inverse are symmetric over the line $$x = y$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"the line $$x = y$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The graph of a function and its inverse are symmetric over the line $$x = y$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The graph of a function and its inverse are symmetric over the line $$x = y$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the inverse of $$f(x) = x + 5$$?\",\n                    \"text\": \"What is the inverse of $$f(x) = x + 5$$?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$f(x) = x + 5$$<br>\r\n$$y = x + 5$$<br>\r\n$$x = y + 5$$<br>\r\n$$y = x \\\\;-\\\\; 5$$<br>\r\nThus, $$f^{-1}(x) = x\\\\;-\\\\;5$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$f^{-1}(x) = x + 5$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$f(x) = x + 5$$<br>\r\n$$y = x + 5$$<br>\r\n$$x = y + 5$$<br>\r\n$$y = x \\\\;-\\\\; 5$$<br>\r\nThus, $$f^{-1}(x) = x\\\\;-\\\\;5$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$f\\\\;-\\\\;1(x) = 5x$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$f(x) = x + 5$$<br>\r\n$$y = x + 5$$<br>\r\n$$x = y + 5$$<br>\r\n$$y = x \\\\;-\\\\; 5$$<br>\r\nThus, $$f^{-1}(x) = x\\\\;-\\\\;5$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$f^{-1}(x) = \\\\frac{x}{5}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$f(x) = x + 5$$<br>\r\n$$y = x + 5$$<br>\r\n$$x = y + 5$$<br>\r\n$$y = x \\\\;-\\\\; 5$$<br>\r\nThus, $$f^{-1}(x) = x\\\\;-\\\\;5$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$f^{-1}(x) = x\\\\;-\\\\;5$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$f(x) = x + 5$$<br>\r\n$$y = x + 5$$<br>\r\n$$x = y + 5$$<br>\r\n$$y = x \\\\;-\\\\; 5$$<br>\r\nThus, $$f^{-1}(x) = x\\\\;-\\\\;5$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$f(x) = x + 5$$<br>\r\n$$y = x + 5$$<br>\r\n$$x = y + 5$$<br>\r\n$$y = x \\\\;-\\\\; 5$$<br>\r\nThus, $$f^{-1}(x) = x\\\\;-\\\\;5$$\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"15-frequently-asked-questions-on-inverse-function\">Frequently Asked Questions On Inverse Function<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-948fc825-2afb-4861-bbbe-6881fe2ff35e\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-948fc825-2afb-4861-bbbe-6881fe2ff35e\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-948fc825-2afb-4861-bbbe-6881fe2ff35e\"><strong>What are inverse operations?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-948fc825-2afb-4861-bbbe-6881fe2ff35e\">\n\n<p class=\"eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/inverse-operations\">Inverse operations<\/a> are opposite operations that reverse or cancel the action of one another.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/addition\">Addition<\/a> and <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/subtraction\/subtract\">subtraction<\/a> are inverse operations.<a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/multiplication\/multiplication\">Multiplication<\/a> and <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/division\/division\">division<\/a> are inverse operations.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-948fc825-2afb-4861-bbbe-6881fe2ff35e\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-948fc825-2afb-4861-bbbe-6881fe2ff35e\"><strong>What is a bijective function?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-948fc825-2afb-4861-bbbe-6881fe2ff35e\">\n\n<p class=\"eplus-wrapper\">Bijective function is a one-one and onto function.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-948fc825-2afb-4861-bbbe-6881fe2ff35e\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-948fc825-2afb-4861-bbbe-6881fe2ff35e\"><strong>Is <\/strong>$f^{-1}(x)$<strong> same as <\/strong>$f(x)^{-1}$<strong>?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-948fc825-2afb-4861-bbbe-6881fe2ff35e\">\n\n<p class=\"eplus-wrapper\">No. $f^{-1}(x)$ is the inverse of the function f.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$f(x)^{-1} = \\frac{1}{f(x)} =$ Reciprocal of $f(x)$<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is an Inverse Function? The inverse function $f^{-1}$ undoes the action performed by the function f.&nbsp; We read $f^{-1}$ as \u201cf inverse.\u201d If $f^{-1}$ is an inverse of the function f, then f is an inverse function of $f^{-1}$. Thus, we can say that&nbsp; f and $f^{-1}$ reverse each other. A function from set &#8230; <a title=\"Inverse Function &#8211; Definition, Types, Examples, Facts, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/inverse-function\" aria-label=\"More on Inverse Function &#8211; Definition, Types, Examples, Facts, FAQs\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-29339","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=29339"}],"version-history":[{"count":13,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29339\/revisions"}],"predecessor-version":[{"id":35814,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29339\/revisions\/35814"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=29339"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=29339"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=29339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}