This section introduces the analytic viewpoint of invertability, as well as one-to-one functions.
The geometric view is insightful to understanding what the inverse means, but it doesn’t really help us explicitly determine what the inverse of a function is. To do this, we use the analytic view.
Before we give a technique for explicitly obtaining an inverse, it is very important to know how to check if a function actually is an inverse analytically. This is because the process we have for obtaining an inverse can (and does) often fail, but it fails in a way that may not be clear without trying to verify if your result is a legitimate inverse. This means whenever you solve for an inverse of a function, you should always check to ensure it is an inverse according to the following definition.
To do this we must show first that ; Next we must show that ; Thus, since we have shown that and we can conclude that , ie that is the inverse function of .
Remember from our geometric view, that the inverse function is the function that reverses the roles of and . In essence, the inverse function is switching the roles for the input and output variable. So to find a function that does this, we ‘merely’ switch the independent and dependent variables, then solve for the independent variable again. Consider our previous example, but this time we will determine the inverse function.
To find the inverse function we will first switch the input and output variable. Since there is no explicit output variable, we will assign one by setting , thus we switch the location of the and variables to go from to .
Next we want to solve our new equality for . To do this we need to cube root both sides, which gives: So our proposed inverse function is . Keep in mind this is only a proposed inverse until we prove it is an inverse by showing that and (which we did in the previous example). Once we have shown that it is indeed the inverse we can conclude that and we’re done.