# 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 . Let and in this functional equation. 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,

### Problem Examples

## Advanced Topics

### Functions and Relations

Given a set and , the Cartesian Product of these sets (denoted ) gives all ordered pairs with and . Symbolically,

A relation is a subset of . A function is a special time of relation where for every in the ordered pair , there exists a unique .

### Injectivity and Surjectivity

Consider a function be a function from the set to the set , i.e., is the domain of and is the codomain of .

The function is injective (or one-to-one) if for all in the domain , if and only if . Symbolically, \begin{equation} f(x) \ \text{is injective} \iff (\forall a,b \in \mathcal{X}, f(a)=f(b)\implies a=b). \end{equation}

The function is surjective (or onto) if for all in the codomain there exists a in the domain such that . Symbolically,
\begin{equation}
f(x) \ \text{is surjective} \iff \forall a \in \mathcal{Y},\exists b \in \mathcal{X}: f(b)=a.
\end{equation}

The function is bijective (or one-to-one and onto) if it is both injective and subjective. Symbolically,
\begin{equation}
f(x) \ \text{is bijective} \iff \forall a \in \mathcal{Y},\exists! b \in \mathcal{X}: f(b)=a.
\end{equation}

The function has an inverse function , where , if and only if it is a bijective function.