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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. Which function is even?

  1. Worked Example: First identify exactly what the question is asking: Which function is even?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Even powers keep the same value when x is replaced by -x.
  4. Every term in x^4 - 3x^2 has even power.
  5. So f(-x) = f(x).
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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

Practice problems

1. Which function is even?

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which function is even?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Even powers keep the same value when x is replaced by -x.
  4. Every term in x^4 - 3x^2 has even power.
  5. So f(-x) = f(x).
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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

2. Which function is even? (variation 2)

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which function is even?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Even powers keep the same value when x is replaced by -x.
  4. Every term in x^4 - 3x^2 has even power.
  5. So f(-x) = f(x).
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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

3. Which function is odd?

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Which function is odd?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Odd functions satisfy f(-x) = -f(x).
  4. Both x^3 and x change sign.
  5. So x^3 - x 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: f(x) = x^3 - x

4. Which function is odd? (variation 2)

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Which function is odd?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Odd functions satisfy f(-x) = -f(x).
  4. Both x^3 and x change sign.
  5. So x^3 - x 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: f(x) = x^3 - x

5. An even function has graph symmetry across the:

Choices: y-axis · x-axis · origin · line y = x

Show solution
  1. Core Practice: First identify exactly what the question is asking: An even function has graph symmetry across the:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Even means f(-x) = f(x).
  4. Points left and right of the y-axis have matching y-values.
  5. That is y-axis symmetry.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y-axis

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