Symmetry
Classify functions as even, odd, or neither using algebraic tests.
Objective
Classify functions as even, odd, or neither using algebraic tests.
What it means
A function is even if f(-x) = f(x) (symmetric about the y-axis). It is odd if f(-x) = -f(x) (symmetric about the origin). If neither equation holds, the function has no such symmetry.
See it
Use this model while you read. Move it around to see what the math is doing.
This rectangle has 2 lines of symmetry.
Key terms
- Even function
- f(-x) = f(x); graph symmetric about y-axis
- Odd function
- f(-x) = -f(x); graph symmetric about the origin
Rules to remember
- Substitute -x for x and simplify to compare with f(x).
- If all terms have even exponents (and constants), the function is likely even.
- If all terms have odd exponents, the function is likely odd.
Step by step
- 1Compute f(-x)Replace every x with -x and simplify.
- 2Compare to f(x) and -f(x)Check which one matches.
- 3State the classificationEven, odd, or neither.
Common mistakes
- Assuming any polynomial with exponents is automatically even or odd.
- Forgetting to check every term.
Tips
- Even exponents everywhere → even function.
- Odd exponents everywhere (no constant) → odd function.