Symmetry of functions

Operations for functions: New functions from old

Symmetry of functions

Symmetry of functions is more of a geometrical concept than an algebraic concept. A function can be symmetric about a vertical line or about a point. However, a function cannot be symmetric about the $x$ -axis (or any other horizontal line of the type $y=a$ ), where $a$ is a constant), since anything that is mirrored around an horizontal line will not pass the vertical line test (and for that reason cannot be a function by definition). Two special cases are even and odd functions.

Symmetry of a graph

A graph is said to be symmetric with respect to a vertical line $\ell$ if, for each point of the graph, its image under reflection about $\ell$ is also a point of the graph. The line we will mostly be looking at is the $y$ -axis.

A graph is said to be symmetric with respect to a point $P$ if, for each point of the graph, its image under the point reflection about $P$ is also a point of the graph. The point we will mostly be looking at is the origin.

A circle is symmetric with respect to its center and with respect to each line through the center.

A line is symmetric with respect to each point on the line.

Even function

A function $f$ is said to be even if its graph is symmetric with respect to the $y$ -axis. In other words: if

$f(-x)=f(x)$ for each point $x$ in the domain of $f$ .

For instance, the function $f(x)=x^2$ is even:

Odd function

A function $f$ is called an odd function if its graph is symmetric with respect to the origin. It has the property

$f(-x)=-f(x)$

For instance, the function $f(x)=x^3$ is odd:

Is the real function $f(x)=x^7$ even, odd or neither of these?

odd

This can be seen in the following graph of the function $f$ :

New example