# 3.4 Determining Even and Odd Functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions $f\left(x\right)={x}^{2}$ or $f\left(x\right)=|x|$ will result in the original graph. We say that these types of graphs are symmetric about the y-axis. Functions whose graphs are symmetric about the y-axis are called even functions.

If the graphs of$\text{}f\left(x\right)={x}^{3}\text{}$or$\text{}f\left(x\right)=\frac{1}{x}\text{}$ were reflected over both axes, the result would be the original graph, as shown in Figure 3-11.

We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function.

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, $\text{}f\left(x\right)={2}^{x}\text{}$ is neither even nor odd. Also, the only function that is both even and odd is the constant function $\text{}f\left(x\right)=0$.

## Even and Odd Functions

A function is called an even function if for every input $\text{}x$

$f\left(x\right)=f\left(-x\right)$

The graph of an even function is symmetric about the $y\text{-}$ axis.

A function is called an odd function if for every input $\text{}x$

$f\left(x\right)=-f\left(-x\right)$

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

## How To

Given the formula for a function, determine if the function is even, odd, or neither.

1. Determine whether the function satisfies $\text{}f\left(x\right)=f\left(-x\right)\text{}$. If it does, it is even.
2. Determine whether the function satisfies $\text{}f\left(x\right)=-f\left(-x\right)\text{}$. If it does, it is odd.
3. If the function does not satisfy either rule, it is neither even nor odd.

## Example 1: Determining whether a Function is Even, Odd, or Neither

1. Is the function $\text{}f\left(x\right)={x}^{3}+2x\text{}$ even, odd, or neither?

### Analysis

Consider the graph of $\text{}f\text{}$ in Figure 3-12. Notice that the graph is symmetric about the origin. For every point $\text{}\left(x,y\right)\text{}$ on the graph, the corresponding point $\text{}\left(-x,-y\right)\text{}$ is also on the graph. For example, (1, 3) is on the graph of $\text{}f\text{}$, and the corresponding point $\left(-1,-3\right)$ is also on the graph.

2. Is the function $\text{}f\left(s\right)={s}^{4}+3{s}^{2}+7\text{}$ even, odd, or neither?