In the last example from the previous section we looked at the two functions and and saw that

and as noted in that section this means that there is a nice relationship between these two functions. Let�s see just what that relationship is. Consider the following evaluations.

In the first case we plugged into and got a value of -5. We then turned around and plugged into and got a value of -1, the number that we started off with.

In the second case we did something similar. Here we plugged into and got a value of , we turned around and plugged this into and got a value of 2, which is again the number that we started with.

Note that we really are doing some function composition here. The first case is really,

and the second case is really,

Note as well that these both agree with the formula for the compositions that we found in the previous section. We get back out of the function evaluation the number that we originally plugged into the composition.

So, just what is going on here? In some way we can think of these two functions as undoing what the other did to a number. In the first case we plugged into and then plugged the result from this function evaluation back into and in some way undid what had done to and gave us back the original x that we started with.

Function pairs that exhibit this behavior are called inverse functions. Before formally defining inverse functions and the notation that we�re going to use for them we need to get a definition out of the way.

A function is called one-to-one if no two values of x produce the same y. Mathematically this is the same as saying,

So, a function is one-to-one if whenever we plug different values into the function we get different function values.

Sometimes it is easier to understand this definition if we see a function that isn�t one-to-one. Let�s take a look at a function that isn�t one-to-one. The function is not one-to-one because both and . In other words there are two different values of x that produce the same value of y. Note that we can turn into a one-to-one function if we restrict ourselves to . This can sometimes be done with functions.