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

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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

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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