Graphing Inverse Functions
Let’s consider the relationship between the graph of a function and the graph of its inverse.
Consider the graph of shown in Figure 9(a) and a point on the graph.

Since , then . Therefore, when we graph , the point is on the graph. As a result, the graph of is a reflection of the graph of about the line .
How to: Graph the Inverse of a Function
- Plot the Function: Graph the original function and plot a few key points.
- Reflect Over Line: Reflect these points over the line to find the corresponding points on .
- Draw the Inverse: Connect these reflected points to graph the inverse function.
- Check: Ensure that each point on the original function corresponds to the point on the inverse function.
- Line of Symmetry: The line should act as a line of symmetry between the function and its inverse.
For the graph of in the following image, sketch a graph of by sketching the line and using symmetry. Identify the domain and range of .

Restricting Domains
As we have seen, does not have an inverse function because it is not one-to-one. However, we can choose a subset of the domain of such that the function is one-to-one. This subset is called a restricted domain.
By restricting the domain of , we can define a new function such that the domain of is the restricted domain of and for all in the domain of . Then we can define an inverse function for on that domain.
For example, since is one-to-one on the interval , we can define a new function such that the domain of is and for all in its domain. Since is a one-to-one function, it has an inverse function, given by the formula .
On the other hand, the function is also one-to-one on the domain . Therefore, we could also define a new function such that the domain of is and for all in the domain of . Then is a one-to-one function and must also have an inverse. Its inverse is given by the formula (Figure 13).

restricted domain
Some functions don’t have inverses over their full domains because they’re not one-to-one. By restricting the domain, we ensure the function is one-to-one. Once the domain is restricted, we can define an inverse.
Consider the function .
- Sketch the graph of and use the horizontal line test to show that is not one-to-one.
- Show that is one-to-one on the restricted domain . Determine the domain and range for the inverse of on this restricted domain and find a formula for .