When working with functions and looking at their graphs, we might notice some interesting patterns or behaviors. For example, a function like the sine function appears to repeat itself over and over again, and the quadratic function defined by appears to be symmetric about the -axis.

In this section, we’ll discuss new vocabulary we can use to describe these behaviors as well as how to show analytically that a function has a certain behavior.

We’ll also discuss the important concept of inverse functions, which can provide a way to “undo” functions.

Odd and even functions

Consider the two functions, and , whose graphs are shown below.

Note that the graph of seems to be symmetric about the origin, meaning that when we rotate the graph a half-turn, we get the same graph. Also, the graph of seems to be symmetric about the -axis, meaning that when we flip the graph across the -axis, we get the same graph.

Let’s first consider the case of by looking at a few test points.

We see that , , , and are points on the graph which indicates is an odd function. We can check this for all point, showing that is on the graph whenever is. In other words, we need to show satisfies the equation whenever does. Substituting into the equation gives

This shows that is an odd function.

Consider the function defined by . We’ll try to prove that is symmetric about the -axis.
a.
Assume is a generic point on the graph of , so . What point is symmetric to about the -axis?
b.
Show your answer to part a is on the graph of whenever is. Conclude that is symmetric about the -axis.

Notice that to test an equation’s graph for symmetry about the origin, we replaced and with and , respectively. Doing this substitution in the equation results in . Solving the latter equation for gives . In order for this equation to be equivalent to the original equation we need , or, equivalently, . In the exploration, you checked whether the graph of an equation was symmetric about the -axis by replacing with and checking to see if an equivalent equation results. If we are graphing the equation , substituting for results in the equation . In order for this equation to be equivalent to the original equation we need . This leads us to the definition of an even function and an odd function.

A function is even if and only if its graph is symmetric about the -axis. A function is odd if and only if its graph is symmetric about the origin.

Completing the Graph of Even or Odd Functions

If we know a function is either even or odd, we can determine one half of the graph or function values if we are giving the other half. This is because we know that:

  • If a function is even and the point is on the function graph, then the point is also on the function graph.
  • If a function is odd and the point is on the function graph, then the point is also on the function graph.

Try the two examples below and then compare your work with the solution.