 # Even and Odd Functions: Definition, Test

Share on

Even and odd functions have special symmetries about the origin or y-axis.

• A function is even if it is symmetric about the vertical y-axis; if this is the case, f(-x) = x for every x in the domain.
• A function is odd if it is symmetric about the origin. In this case, f(-x) = x for every x in the domain.

## Examples of Even and Odd Functions

The parabola f(x) = x2 is an example of an even function. As you can see in the image below, it is symmetric about the vertical y axis. The function f(x) = x3, on the other hand, is an example of an odd function. It is symmetric about the origin. Since the definition of even and odd are not mutually exclusive, it is possible to have a function that is both even and odd. For such a function, both f(-x) = f(x) and f(-x)= -f(x) are true. Substituting those equations into each other, we find that, for these functions, f(x) = -f(x). There is only one way that can be true: if f(x) = 0. So the function f(x) = 0 is the one and only function that is both even and odd.

## Properties of Odd and Even Functions

Here are some basic properties of odd and even functions that are worth remembering:

• The sum of two odd functions is odd, and the sum of two even functions is even (note that this is not the same as the rule for even and odd integers)
• The sum of an even and an odd function is neither even nor odd unless one or both of them is actually equal to zero.
• The difference of two even functions is even; likewise, the difference of two odd functions is odd.
• The product of two even functions is another even function, and the product of two odd functions is an even function also.
• The product of an even function and an odd function is an odd function.

## Test for Even and Odd Functions

This simple test states:

A function y – f(x) is even if f(-x) = f(x)
A function y – f(x) is odd if f(-x) = f(x)

Example question #1: Is the function f(x) = x3 – x odd or even?

Solution:
Step 1: Replace any “x” in the function with “-x”:
f(-x) = (-x)3 – (-x)

Step 2: Solve with algebra.The goal is to get the function in a form that is either the original function, x3 – x, or its negative -(x3 – x),

• f(-x) = (-x)3 – (-x) =
• = -x3 + x

The result, -x3 + x, is the negative of the function so this function is odd.

Example question #2: Is the function f(x) = 2 + cos(x) odd or even?

Solution:
Step 1: Replace any “x” in the function with “-x”:
f(-x) = 2 + cos(-x)

Step 2: Solve with algebra.The goal is to get the function in a form that is either the original function, x3 – x, or its negative -(x3 – x),

• f(-x) = 2 + cos(x)*

The result, 2 + cos(x) is the original function, so this is an even function.
One of the Trigonometric identities for negatives states that -cos(x) = cos(x).