# Functions

A **function** \(f\) from a set \(D\) to a set \(Y\) is a rule that assigns a unique value \(f(x)\) in \(Y\) to each \(x\) in \(D\).

The set \(D\) of all possible input values is called the **domain** of the function. The set of all output values of \(f(x)\) as \(x\) varies throughout \(D\) is called the **range** of the function. The range might not include every element in the set \(Y\).

### Even and odd functions

A function \(y = f(x)\) is an

**even function of \(x\)** if \(f(-x) = f(x)\)

**odd function of \(x\)** if \(f(-x) = -f(x)\)

for every \(x\) in the function's domain.

The graph of an even function is **symmetric about the y-axis**.

The graph of an odd function is **symmetric about the origin**.

*\(f(x) = 0\) is even and odd function*

### Linear functions

A function of the form \(f(x) = mx + b\), where \(m\) and \(b\) are fixed constants, is called a **linear function**.

### Power functions

A function of the form \(f(x) = x^a\), where \(a\) is a constant is called a **power function**.

### Exponential functions

A function of the form \(f(x) = a^x\), where \(a > 0\) and \(a \ne 1\), is called an **exponential function** (with base a). All exponential functions have domain \((-\infty, \infty)\) and range \((0, \infty)\).