# Functional equation

A **functional equation**, roughly speaking, is an equation in which some of the unknowns to be solved for are functions. For example, the following are functional equations:

## Contents

## Introductory Topics

### The Inverse of a Function

The inverse of a function is a function that "undoes" a function. For an example, consider the function: . The function has the property that . In this case, is called the **(right) inverse function**. (Similarly, a function so that is called the **left inverse function**. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the **inverse function**.) Often the inverse of a function is denoted by .

## Intermediate Topics

### Cyclic Functions

A cyclic function is a function that has the property that:

A classic example of such a function is because . Cyclic functions can significantly help in solving functional identities. Consider this problem:

Find such that . In this functional equation, let and let . This yields two new equations:

Now, if we multiply the first equation by 3 and the second equation by 4, and add the two equations, we have:

So, clearly,