ON INVERSE FUNCTIONS

With Restricted Domains

You can always find the inverse of a one-to-one function without restricting the domain of the function. Recall that a function is a rule that links an element in the domain to just one number in the range. You could have points (3, 7), (8, 7) and (14,7) on the graph of a function. A one-to-one function adds the requirement that each element in the range is linked to just one number in the domain. In this case, the above three points would not be points on the graph of a one-to-one function because 7 links to different numbers in the domain.

You can identify a one-to-one function from its graph by using the Horizontal Line Test. Mentally scan the graph with a horizontal line; if the line intersects the graph in more than one place, it is not the graph of a one-to-one function. If the function is not one-to-one, then its inverse will not be unique, and the inverse function must be unique. The domain of the original function must be restricted so that its inverse will be unique.

This section will show you how to restrict the domain and then find a unique inverse on that domain. Study the graph of a function that is not one-to-one and choose a part of the graph that is one-to-one. We will find the inverse for just that part of the graph.

The following is an example of finding the inverse of a function that is not one-to-one.

Example 4: How would you restrict the domain of the function so that a unique inverse exists?

Solution:

Recall that for a function to have an inverse, the function must be a one-to-one function. From the graph below you can see that there are three ways to restrict the domain so that the portion of the graph in that domain has a unique inverse. Recall that the inverse will be different for each of the domains.

It appears that the function is increasing on the interval , decreasing on the interval , and increasing on the interval . Functions that are increasing are one-to-one functions, and functions that are decreasing are one-to-one functions.

Therefore, if we restrict our domain to one of these three sets, we can find the inverse function with respect to that set.

This ends the review of inverse functions. Work the problems below, or go back to the start to again review several of the topics.

Work the following problems and check the answers and solutions.

1. Find the inverse of the function . Answer.

2. Find the inverse of the function . Answer.

3. Find the inverse of the function . Answer.

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