Inverses of Functions

Function inverses are often functions in their own right, which “undo” what the original function did. As an analogy, imagine that tying your shoe is a function. The inverse of this function would, of course, be untying that shoe. Now, we could have reversed the order of those two actions, and they would still have an inverse relationship. In other words, if you imagine that untying your shoe is a function, then tying it would be the inverse.

In order to determine if a function has an inverse that is also a function, we need to check if it passes the horizontal line test. This test is exactly the same as the vertical line test, except you check to see if any horizontal lines ever intersect the graph in more than one place. For more information, look at the module Vertical and Horizontal Line Tests.

As a more mathematical example, say we have the function . Its graph passes the vertical line test, so we know it is a function. Its graph passes the horizontal line test, so we know this function has an inverse that is also a function. Now, what might that inverse be? If the function just adds 5 to whatever value you give it, what would be the inverse? What would “undo” adding 5? Naturally, subtracting 5 “undoes” adding 5. So, we can see that the inverse function must be .

In summary, if , then the inverse of this function, denoted , is
.

(Notation: the -1 superscript is different for functions than it is for numbers. While , it is virtually never the case that

But we need a more rigorous way to find inverses. The easiest way to do this is to interchange x and y. If you think about it, if a function has an ordered pair (a, b), then its inverse has the ordered pair (b, a). In other words, the “x” and “y” coordinates just trade places.

Example 1

Find the inverse of:

Solution 1

First, we set it up as , and then we switch the x and y, obtaining . Now, solve for y

Therefore, the inverse of is .

Example 2

Find the inverse of .

Solution 2

First, we set it up as , and then switch variables, so that we have. Then solve for y:

Therefore, for , we know that .

Example 3

Find the inverse of

Solution 3

We switch variables: , and then we solve for y:

Therefore, for , we know that .

(Note: there is a visual relationship between the graph of a function and the graph of its inverse. If a function has an inverse, the graph of the inverse can be obtained by reflecting the graph of the function about the line . Here are a couple of graphs which have both the function and its inverse, one in green and the other in red, with the line indicated by a dashed line.)