Even and Odd Functions: Symmetry

Even and odd functions: symmetry tests

Even: $f(-x) = f(x)$ for all $x$ → y-axis symmetry. The graph is a mirror image across the y-axis. Examples: $x^2$, $x^4$, $\cos x$, $|x|$.

Odd: $f(-x) = -f(x)$ for all $x$ → origin symmetry (180° rotation). Examples: $x^3$, $x$, $\sin x$.

Neither: Most functions have no special symmetry. Example: $x^2 + x$.

How to test algebraically

Step 1: Compute $f(-x)$ by replacing every $x$ with $(-x)$.

Step 2: Compare $f(-x)$ with $f(x)$ and $-f(x)$.

Example: $f(x) = x^3$. $f(-x) = (-x)^3 = -x^3 = -f(x)$ → odd ✓.

Quick pattern for polynomials

All terms have even exponents → even. All terms have odd exponents → odd. Mix → neither.

Try it in the visualization

Select a function. $f(x)$ and $f(-x)$ are graphed — for even functions they overlap; for odd, $f(-x) = -f(x)$. Test specific points to verify. $f(-x) = -f(x)$ — symmetric about the origin (180° rotation). Neither: most functions don't have either symmetry.