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.
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.