### Learning Objectives

Determine the conditions for when a function has an inverse.Use the horizontal line test to recognize when a function is one-to-one.Find the inverse of a given function.Draw the graph of an inverse function.Evaluate inverse trigonometric functions.

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An inverse function reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it. In this section, we define an inverse function formally and state the necessary conditions for an inverse function to exist. We examine how to find an inverse function and study the relationship between the graph of a function and the graph of its inverse. Then we apply these ideas to define and discuss properties of the inverse trigonometric functions.

Existence of an Inverse Function

We begin with an example. Given a function and an output , we are often interested in finding what value or values were mapped to by . For example, consider the function . Since any output , we can solve this equation for to find that the input is . This equation defines as a function of . Denoting this function as , and writing , we see that for any in the domain of . Thus, this new function, , “undid” what the original function did. A function with this property is called the inverse function of the original function.

Given a function with domain and range , its inverse function (if it exists) is the function with domain and range such that if . In other words, for a function and its inverse ,

for all in , and for all in .

Note that is read as “f inverse.” Here, the -1 is not used as an exponent and .(Figure) shows the relationship between the domain and range of and the domain and range of .

Figure 1. Given a function and its inverse if and only if . The range of becomes the domain of and the domain of becomes the range of .

Recall that a function has exactly one output for each input. Therefore, to define an inverse function, we need to map each input to exactly one output. For example, let’s try to find the inverse function for . Solving the equation for , we arrive at the equation . This equation does not describe as a function of because there are two solutions to this equation for every 0" title="Rendered by QuickLaTeX.com" height="16" width="42" style="vertical-align: -4px;" />. The problem with trying to find an inverse function for is that two inputs are sent to the same output for each output 0" title="Rendered by QuickLaTeX.com" height="16" width="42" style="vertical-align: -4px;" />. The function discussed earlier did not have this problem. For that function, each input was sent to a different output. A function that sends each input to a different output is called a one-to-one function.

We say a is a one-to-one function if when .

One way to determine whether a function is one-to-one is by looking at its graph. If a function is one-to-one, then no two inputs can be sent to the same output. Therefore, if we draw a horizontal line anywhere in the -plane, according to the horizontal line test, it cannot intersect the graph more than once. We note that the horizontal line test is different from the vertical line test. The vertical line test determines whether a graph is the graph of a function. The horizontal line test determines whether a function is one-to-one ((Figure)).

### Rule: Horizontal Line Test

A function is one-to-one if and only if every horizontal line intersects the graph of no more than once.

Figure 2. (a) The function is not one-to-one because it fails the horizontal line test. (b) The function is one-to-one because it passes the horizontal line test.

For each of the following functions, use the horizontal line test to determine whether it is one-to-one.

SolutionSince the horizontal line for any integer intersects the graph more than once, this function is not one-to-one.Since every horizontal line intersects the graph once (at most), this function is one-to-one.

Is the function graphed in the following image one-to-one?

Solution

No.

Hint

Use the horizontal line test.

Finding a Function’s Inverse

We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of to elements in the range of . The inverse function maps each element from the range of back to its corresponding element from the domain of . Therefore, to find the inverse function of a one-to-one function , given any in the range of , we need to determine which in the domain of satisfies . Since is one-to-one, there is exactly one such value . We can find that value by solving the equation for . Doing so, we are able to write as a function of where the domain of this function is the range of and the range of this new function is the domain of . Consequently, this function is the inverse of , and we write . Since we typically use the variable to denote the independent variable and to denote the dependent variable, we often interchange the roles of and , and write . Representing the inverse function in this way is also helpful later when we graph a function and its inverse on the same axes.

### Problem-Solving Strategy: Finding an Inverse Function

Solve the equation for .Interchange the variables and and write .

Find the inverse for the function . State the domain and range of the inverse function. Verify that .

Solution

Follow the steps outlined in the strategy.

Step 1. If , then and .

Step 2. Rewrite as and let .

Therefore, .

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Since the domain of is , the range of is . Since the range of is , the domain of is .