{"id":723,"date":"2025-07-14T21:32:38","date_gmt":"2025-07-14T21:32:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=723"},"modified":"2026-01-08T22:48:44","modified_gmt":"2026-01-08T22:48:44","slug":"inverse-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-functions-learn-it-4\/","title":{"raw":"Inverse Functions: Learn It 4","rendered":"Inverse Functions: Learn It 4"},"content":{"raw":"

Finding Inverse Functions and Their Graphs<\/h2>\r\nNow that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function [latex]f\\left(x\\right)={x}^{2}[\/latex] restricted to the domain [latex]\\left[0,\\infty \\right)[\/latex], on which this function is one-to-one, and graph it as below.\r\n\r\n[caption id=\"attachment_4927\" align=\"aligncenter\" width=\"336\"]\"Graph Quadratic function with domain restricted to [0, \u221e).[\/caption]Restricting the domain<\/strong> to [latex]\\left[0,\\infty \\right)[\/latex] makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.\r\n\r\nWe already know that the inverse of the toolkit quadratic function is the square root function, that is, [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}[\/latex]. What happens if we graph both [latex]f\\text{ }[\/latex] and [latex]{f}^{-1}[\/latex] on the same set of axes, using the [latex]x\\text{-}[\/latex] axis for the input to both [latex]f\\text{ and }{f}^{-1}?[\/latex]\r\n\r\nWe notice a distinct relationship: The graph of [latex]{f}^{-1}\\left(x\\right)[\/latex] is the graph of [latex]f\\left(x\\right)[\/latex] reflected about the diagonal line [latex]y=x[\/latex], which we will call the identity line, shown below.\r\n\r\n[caption id=\"attachment_4925\" align=\"aligncenter\" width=\"331\"]\"Graph Square and square-root functions on the non-negative domain.[\/caption]\r\n\r\nThis relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.\r\n\r\n
Given the graph of [latex]f\\left(x\\right)[\/latex], sketch a graph of [latex]{f}^{-1}\\left(x\\right)[\/latex].\"Graph\r\n[reveal-answer q=\"694066\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"694066\"]This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of [latex]\\left(0,\\infty \\right)[\/latex] and range of [latex]\\left(-\\infty ,\\infty \\right)[\/latex], so the inverse will have a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] and range of [latex]\\left(0,\\infty \\right)[\/latex].\r\n[latex]\\\\[\/latex]\r\nIf we reflect this graph over the line [latex]y=x[\/latex], the point [latex]\\left(1,0\\right)[\/latex] reflects to [latex]\\left(0,1\\right)[\/latex] and the point [latex]\\left(4,2\\right)[\/latex] reflects to [latex]\\left(2,4\\right)[\/latex]. Sketching the inverse on the same axes as the original graph gives us\u00a0the result in the graph below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The function and its inverse, showing reflection about the identity line[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
Is there any function that is equal to its own inverse?\r\n<\/strong>\r\n\r\n
\r\n\r\nYes. If [latex]f={f}^{-1}[\/latex], then [latex]f\\left(f\\left(x\\right)\\right)=x[\/latex], and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because\r\n

[latex]\\frac{1}{\\frac{1}{x}}=x[\/latex]<\/p>\r\nAny function [latex]f\\left(x\\right)=c-x[\/latex], where [latex]c[\/latex] is a constant, is also equal to its own inverse.\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]293887[\/ohm_question]<\/section>
[ohm_question hide_question_numbers=1]317949[\/ohm_question]<\/section><\/section>","rendered":"

Finding Inverse Functions and Their Graphs<\/h2>\n

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function [latex]f\\left(x\\right)={x}^{2}[\/latex] restricted to the domain [latex]\\left[0,\\infty \\right)[\/latex], on which this function is one-to-one, and graph it as below.<\/p>\n

\"Graph
Quadratic function with domain restricted to [0, \u221e).<\/figcaption><\/figure>\n

Restricting the domain<\/strong> to [latex]\\left[0,\\infty \\right)[\/latex] makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.<\/p>\n

We already know that the inverse of the toolkit quadratic function is the square root function, that is, [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}[\/latex]. What happens if we graph both [latex]f\\text{ }[\/latex] and [latex]{f}^{-1}[\/latex] on the same set of axes, using the [latex]x\\text{-}[\/latex] axis for the input to both [latex]f\\text{ and }{f}^{-1}?[\/latex]<\/p>\n

We notice a distinct relationship: The graph of [latex]{f}^{-1}\\left(x\\right)[\/latex] is the graph of [latex]f\\left(x\\right)[\/latex] reflected about the diagonal line [latex]y=x[\/latex], which we will call the identity line, shown below.<\/p>\n

\"Graph
Square and square-root functions on the non-negative domain.<\/figcaption><\/figure>\n

This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.<\/p>\n

Given the graph of [latex]f\\left(x\\right)[\/latex], sketch a graph of [latex]{f}^{-1}\\left(x\\right)[\/latex].\"Graph<\/p>\n