Even and Odd Functions

What is an Even Function?
Definition:
A function f(x) is even if f(-x) = f(x) for all x in its domain.
Graphical Interpretation:
The graph of an even function is symmetric with respect to the y-axis. This means that if you reflect the graph across the y-axis, it remains unchanged. This means that plugging in a negative x-value gives you the same y-value as plugging in the positive x-value.
Examples:
f(x) = x²
f(x) = cos(x)
f(x) = |x| (absolute value of x)

What is an Odd Function?
Definition:
A function f(x) is odd if f(-x) = -f(x) for all x in its domain.
Graphical Interpretation:
The graph of an odd function is symmetric with respect to the origin. This means that if you rotate the graph 180 degrees about the origin, it remains unchanged.
Examples:
f(x) = x³
f(x) = sin(x)
f(x) = x

Neither:
If a function does not satisfy either of the above conditions, it is neither even nor odd.

How to determine if a function is even, odd, or neither?

The following diagrams show how to distinguish between even and odd functions. Scroll down the page for more examples and solutions.

 

Graphical Test

• The graph of an even function is symmetric with respect to the y-axis.
• The graph of an odd function is symmetric with respect to the origin.

Algebraic Test

• If f(-x) is exactly the same as the original function f(x), then the function is even.
• If f(-x) is exactly the negative of the original function f(x), then the function is odd.
• If neither of these conditions is met, then the function is neither even nor odd.

Determining if a Function is Even, Odd or Neither

• Show Video Lesson

Even and Odd Functions
How to determine if a function is even, odd, or neither?

Examples of Even and Odd Functions
How to algebraically determine whether a function is even, odd or neither.
Determine if a function is even, odd, or neither. Mostly algebraic with two basic graphic examples.

Recognizing Odd and Even Functions

• Show Video Lesson

Check out our most popular games!
Fraction Concoction	Fact Family Game	Number Bond Garden
Online Addition Subtraction Game	Penguin Solitaire	Sawayama Solitaire
Ark Solitaire	Eldritch Invasion	Shenzhen Solitaire

Check out more Solitaire games here.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.