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Even and Odd Functions

A free Precalculus lesson from the “Transformations and Combinations of Functions” unit, with a worked example and practice problems including step-by-step solutions.

Even functions have y-axis symmetry; odd functions have origin symmetry. Algebraically, compare f(-x) to f(x) and -f(x). This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Transformations let students predict how a model changes when a situation is shifted, scaled, reflected, or combined.

Worked example

Problem. Classify f(x) = x^4 - 3x^2 as even, odd, or neither.

  1. Replace every x with -x.
  2. Even powers are unchanged: (-x)^4 = x^4 and (-x)^2 = x^2.
  3. f(-x) = f(x), so the function is even.

Answer: even

Practice problems

1. Classify f(x) = x^4 - 3x^2 as even, odd, or neither.

Choices: even · odd · neither · both even and odd

Show solution
  1. Replace every x with -x.
  2. Even powers are unchanged: (-x)^4 = x^4 and (-x)^2 = x^2.
  3. f(-x) = f(x), so the function is even.

Answer: even

2. Classify f(x) = x^3 - 3x as even, odd, or neither.

Choices: odd · even · neither · both even and odd

Show solution
  1. Warm-up: First identify exactly what the question is asking: Classify f(x) = x^3 - 3x as even, odd, or neither.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Replace every x with -x.
  4. Odd powers flip sign: (-x)^3 = -x^3 and -x becomes +x.
  5. f(-x) = -f(x), so the function is odd.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: odd

3. Classify f(x) = x^2 + 2x as even, odd, or neither.

Choices: neither · even · odd · both even and odd

Show solution
  1. This mixes an even-power term (x^2) with an odd-power term (x).
  2. f(-x) = x^2 - 2x, which equals neither f(x) nor -f(x).
  3. So the function is neither even nor odd.

Answer: neither

4. Which function is even?

Choices: f(x) = x^4 - 3x^2 · f(x) = x^3 - 1x · f(x) = x + 2 · f(x) = x^2 + 2x

Show solution
  1. An even function contains only even-power terms.
  2. x^4 - 3x^2 has all even powers, so f(-x) = f(x).
  3. The other choices contain an odd-power term.

Answer: f(x) = x^4 - 3x^2

5. Which function is odd?

Choices: f(x) = x^3 - 2x · f(x) = x^2 + 3 · f(x) = x^4 - 4x^2 · f(x) = 3

Show solution
  1. An odd function contains only odd-power terms.
  2. x^3 - 2x has all odd powers, so f(-x) = -f(x).
  3. The other choices contain an even-power or constant term.

Answer: f(x) = x^3 - 2x

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