Symmetric Functions

We will now look at some important properties of functions, namely, symmetry in functions.

# Even Functions

 Definition: A function $f$ is said to be Even if for every $x$ in the domain of $f$, $f(x) = f(-x)$.

A prime example of an even function is the function $f(x) = x^2$ graphed below:

We can verify that this is an even function by evaluating $f$ at $-x$:

(1)
\begin{align} f(x) = x^2 \\ f(-x) = (-x)^2 \\ f(-x) = x^2 \end{align}

Therefore, $f(x) = f(-x)$ and $f$ is an even function. We can also verify this numerically. Clearly $f(2) = f(-2) = 4$, and $f(3) = f(-3) = 9$, etc…

Geometrically, even functions have graphs that are symmetric about the $y$-axis.

# Odd Functions

 Definition: A function $f$ is said to be Odd if for every $x$ in the domain of $f$, $f(-x) = -f(x)$.

One great example of an odd function is $f(x) = x^3$.

Once again, we can verify this function is odd through our definition:

(2)
\begin{align} f(-x) = (-x)^3 \\ f(-x) = -(x^3) \\ -f(x) = -(x^3) \end{align}

Therefore, $f(-x) = -f(x)$, so our function $f$ is odd.

Geometrically, odd functions have graphs that are symmetric about the origin by a rotation of $180^{\circ}$.

 Note: While we have described both even and odd functions above, it is important to note that some functions are neither!