Even and Odd Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Even and Odd Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An even function satisfies f(-x) = f(x) (symmetric about y-axis); an odd function satisfies f(-x) = -f(x) (rotational symmetry about origin).

Even means mirror across y-axis; odd means rotational symmetry through the origin.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: To test: compute f(-x) and simplify. If f(-x) = f(x), even; if f(-x) = -f(x), odd; if neither, the function is neither even nor odd.

Common stuck point: Students test only one value pair and generalize too quickly.

Sense of Study hint: Substitute -x symbolically and compare to original and negative original.

Worked Examples

Example 1

easy
Determine whether f(x) = x^4 - 3x^2 + 2 is even, odd, or neither.

Solution

  1. 1
    Compute f(-x): (-x)^4 - 3(-x)^2 + 2 = x^4 - 3x^2 + 2.
  2. 2
    Compare: f(-x) = x^4 - 3x^2 + 2 = f(x).
  3. 3
    Since f(-x) = f(x) for all x, the function is even.

Answer

\text{Even}
A function is even if f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Polynomials with only even powers of x (including constant terms, which are x^0) are always even functions.

Example 2

medium
Determine whether g(x) = \frac{x}{x^2 + 1} is even, odd, or neither.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Is h(x) = x^3 + x^2 even, odd, or neither?

Example 2

hard
Prove that the product of two odd functions is an even function.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function notationreflecting functionsalgebraic symmetry