This section introduces the analytic viewpoint of invertability, as well as one-to-one functions.

Inverse Function - The Analytic View

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.

How to solve for inverse analytically

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.

1 : In order to analytically solve for an inverse function you can...
Switch the and variable rolls, and solve for the new independent variable. Change the variable to another letter. Change the variable to another letter. Use the horizontal line test to determine if an inverse exists.