How to Determine if a Function is Odd or Even

An even function is a function, which has a graph with symmetry about the y-axis. On the other hand, the odd function has a graph with rotational symmetry of 180° about the origin. A function is even when f(-x) = f(x) and it is odd when f(-x) = f(x).

Here, we will learn how to determine whether a function is even or odd, both graphically and algebraically. We will look at some examples to practice the concepts.

ALGEBRA
Formulas for even and odd functions

Relevant for

Learning to determine if a function is even or odd.

See methods

ALGEBRA
Formulas for even and odd functions

Relevant for

Learning to determine if a function is even or odd.

See methods

How to determine if a function is even or odd graphically

Since the graphs of even and odd functions have unique characteristics, we can determine whether a function is even or odd using its graph.

Even Function: The graph of an even function is symmetric about the y-axis. This means that if we were to fold the function on the y-axis, we would get two equal parts of the graph.

For example, the function $latex f(x)=x^2$ is an even function because it has symmetry on the y-axis as shown in the following graph:

Graph of quadratic function

Odd Function: The graph of an odd function has rotational symmetry of 180° about the origin. This means that if we rotate the graph by 180°, the graph remains unchanged.

For example, the function $latex f(x)=x^3$ is an odd function because its graph does not change when we rotate it by 180° about the origin:

Graph of odd function x cubed

How to determine if a function is even or odd algebraically

When we don’t have a graph of the function, we can determine if a function is even or odd algebraically. For this, we consider the following.

Even Function: A function is even if $latex f(-x)=f(x)$ for all values of x that belong to the domain of the function.

For example, the function $latex f(x)=x^2$ is even, since:

$latex f(-x)=(-x)^2=x^2=f(x)$

Odd Function: A function is odd if $latex f(-x)=-f(x)$ for all values of x that belong to the domain of the function.

For example, the function $latex f(x)=x^3$ is odd, since:

$latex f(-x)=(-x)^3=-x^3=-f(x)$

In short, if we have a function f such that $latex f(-x)=f(x)$, the function is even. If we have a function f such that $latex f(-x)=-f(x)$, the function is odd


Determining if a function is even or odd – Examples with answers

Graphical and algebraic methods are used to solve the following odd and even function examples. Try to solve the problems yourself before looking at the solution.

EXAMPLE 1

Prove that the function $latex f(x)=x^2-2$ is even.

The graph of the function $latex f(x)=x^2-2$ is equal to the graph of $latex f(x)=x^2$ translated 2 units down:

Graph of quadratic function 3

Clearly, we see that the graph of $latex f(x)=x^2-2$ is symmetric about the y-axis. This means that the function is even.

EXAMPLE 2

Prove that the function $latex f(x)=2x^4+x^2-1$ is an even function.

Proving this graphically would be more difficult. Therefore, let’s prove algebraically.

If the function is even, we must have $latex f(-x)=f(x)$. Thus, we check:

$latex f(-x)=2(-x)^2+(-x)^2-1$

$latex f(-x)=2x^2+x^2-1$

We see that we got the original function, so we have proved that the function is even.

EXAMPLE 3

Prove that the function $latex f(x)=4x^3-x$ is an odd function.

Odd functions are generally difficult to demonstrate graphically. Therefore, let’s prove algebraically.

If a function is odd, we must have $latex f(-x)=-f(x)$. Then, we check this:

$latex f(-x)=4(-x)^3-(-x)$

$latex f(-x)=-4-x^3+x$

The expression on the right-hand side is equivalent to having $latex -f(x)$. This means that the function is odd.

EXAMPLE 4

Determine whether the function $latex f(x)=3x^2-|x|$ is odd or even.

To prove whether a function is even or odd, we can start by performing the even function test. If the function fails that test, we can proceed with the odd function test.

For a function to be even, we must have $latex f(-x)=f(x)$. Checking, we have:

$latex f(-x)=3(-x)^2-|-x|$

$latex f(-x)=3x^2-|x|$

We see that the expression we got is equal to $latex f(x)$, so the function is even.

Since the function is even, we no longer have to test for odd functions.

EXAMPLE 5

Determine whether the function $latex f(x)=\frac{1}{x}+x$ is odd or even.

Again, we can start with the even functions test and then carry out the test for odd functions.

For a function to be even, we must have $latex f(-x)=f(x)$. Checking, we have:

$latex f(-x)=\frac{1}{-x}+(-x)$

$latex f(-x)=-\frac{1}{x}-x$

We see that the expression we got is not equal to $latex f(x)$, so the function is not even. Thus, we proceed with the test for odd functions.

For a function to be odd, we must have $latex f(-x)=-f(x)$. Looking at the expression we got above, we see that the resulting expression is indeed equal to $latex -f(x)$, so the function is odd.

EXAMPLE 6

Determine whether the function $latex f(x)=(x^3-5)^2$ is even, odd, or neither.

We start with the test for even functions. Therefore, we check if $latex f(-x)=f(x)$:

$latex f(-x)=((-x)^3-5)^2$

$latex f(-x)=(-x^3-5)^2$

The expression we got is not equal to $latex f(x)$. Thus, we proceed with the test for odd functions.

To check if the function is odd, we must have $latex f(-x)=-f(x)$. However, the function is also not odd, since the expression obtained above is not equal to $latex -f(x)$.

Note: $latex -f(x)$ would be equal to $latex -(x^3-5)^2$.


Determining if a function is even or odd – Practice problems

Use the odd and even function tests to solve the following practice problems. You can use the examples with answers shown above as a guide.

Determine whether the function $latex f(x)=2x^2-1$ is even, odd or neither.

Choose an answer





Determine whether the function $latex f(x)=x^2-x$ is even, odd, or neither.

Choose an answer





Determine whether the function $latex f(x)=1+|x|$ is even, odd, or neither.

Choose an answer





Determine whether the function $latex f(x)=x^5-1$ is even, odd, or neither.

Choose an answer





Determine whether the function $latex f(x)\frac{1}{1+x^2}$ is even, odd, or neither.

Choose an answer






See also

Interested in learning more about algebraic functions? Take a look at these pages:

Learn mathematics with our additional resources in different topics

LEARN MORE