$\require{cancel}\newcommand\degree{^{\circ}} \newcommand\Ccancel[black]{\renewcommand\CancelColor{\color{#1}}\cancel{#2}} \newcommand{\alert}{\boldsymbol{\color{magenta}{#1}}} \newcommand{\blert}{\boldsymbol{\color{blue}{#1}}} \newcommand{\bluetext}{\color{blue}{#1}} \delimitershortfall-1sp \newcommand\abs{\left|#1\right|} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$

## SectionReflections and Even and Odd Functions

In the previous section we discussed shifting a function horizontally and veritically. Now, we will take a look at what happens as we reflect a function across the $x$-axis or $y$-axis.

Let $f(x)=(x+1)^2+1\text{,}$ and set

\begin{equation*} h(x)=f(-x)=(-x+1)^2+1. \end{equation*}

Compared to $f(x)\text{,}$ the graph of $h(x)$ is flipped, or reflected, about the $y$-axis as shown below. $x$ $f(x)$ $h(x)=f(-x)$ $-2$ $2$ $10$ $-1$ $1$ $5$ $0$ $2$ $2$ $1$ $5$ $1$ $2$ $10$ $2$

Looking at the graph and table above, we notice a relationship between the outputs of $f(x)$ and $f(-x)$ in this particular case, and as it so happens, this relationship holds in general.

Notice that if you take the column of output values for $f(x)$ and flip them upside down you get the outputs of $f(-x)\text{.}$ This makes sense as this is saying, for example,

as $h(x)=f(-x)\text{.}$ In general, we have the following definition:

Compared with the graph of $y=f(x)\text{,}$ the graph of $f(-x)$ is reflected about the $y$-axis.

There are some functions which do not change when we apply a reflection about the $y$-axis. The most common example of this is the function $y=x^2\text{.}$ Looking at a graph of this function, it is easy to see that when you reflect it across the $y$-axis it remains unchanged. These sorts of functions have a particular name:

###### Even Function

A function $f$ is called an even function if

\begin{equation*} f(x)=f(-x) \end{equation*}

for all $x$ in the domain of $f \text{.}$ In other words, a function is even if performing a reflection about the $y$-axis does not change the graph of the function.

To help remember the definition of an even function, notice that the example of an even function we gave was of $y=x^2\text{.}$ Other examples are $y= x^4 \text{,}$ $y=x^6 \text{,}$ $y=x^8\text{,}$ etc. Notice that the exponent of each of these functions is an even number.

Compared to $y=x^2\text{,}$ the graph of $h(x) = -x^2$ is flipped, or reflected, about the $x$-axis. The $y$-coordinate of each point on the graph of $y = x^2$ is replaced by its additive inverse. $x$ $y=x^2$ $h(x)=-x^2$ $-2$ $4$ $-4$ $-1$ $1$ $-1$ $0$ $0$ $0$ $1$ $1$ $-1$ $2$ $4$ $-4$

In general, we have the following definition:

Compared with the graph of $y=f(x)\text{,}$ the graph of $-f(x)$ is reflected about the $x$-axis.

We now know that a function satisfying $f(x)=f(-x)$ is an even function. While there do not exist any functions satisfying $f(x)=-f(x)\text{,}$ we do have another type of symmetry to discuss here.

###### Example257

Consider the function $f(x)=x^3 \text{.}$ If we reflect $f$ about the $y$-axis, we get the function $f(-x)$ as shown below:

If we now reflect the function $f(-x)$ about the $x$ axis, we can see from the above picture that we end up back with the original function. As an equation, this says

\begin{equation*} f(x)=-f(-x). \end{equation*}

This type of function is said to have symmetry about the origin, or called an odd function.

###### Odd Function

A function $f$ is called an odd function if

\begin{equation*} f(x)=-f(-x) \end{equation*}

for all $x$ in the domain of $f \text{.}$ In other words, a function is odd if performing a reflection about the $y$-axis and $x$-axis (doesn't matter which is performed first) does not change the graph of the function.

To help remember the definition of an odd function, we have a similar strategy as for even funcitons. Some examples of odd functions are $y=x^3\text{,}$ $y=x^5\text{,}$ $y=x^7 \text{,}$ etc. Each of these examples have exponents which are odd numbers, and they are odd functions.