Even, Odd, or Neither Functions

Definitions

• Even function: $f(-x) = f(x)$ for all $x$. The graph has y-axis symmetry (mirror image across the y-axis). Examples: $x^2$, $x^4$, $\cos x$, $|x|$.
• Odd function: $f(-x) = -f(x)$ for all $x$. The graph has origin symmetry (180° rotation about the origin). Examples: $x^3$, $x$, $\sin x$.
• Neither: Most functions are neither even nor odd. 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 $-f(x)$.

Step-by-step: Test $f(x) = x^3 + x$

Step 1: $f(-x) = (-x)^3 + (-x) = -x^3 - x$

Step 2: Compare with $-f(x) = -(x^3 + x) = -x^3 - x$.

Since $f(-x) = -f(x)$, the function is odd. ✓

Geometric check: $f(2) = 8 + 2 = 10$ and $f(-2) = -8 - 2 = -10 = -f(2)$. The point $(2, 10)$ and $(-2, -10)$ are reflections through the origin.

Step-by-step: Test $g(x) = x^4 + x^2$

Step 1: $g(-x) = (-x)^4 + (-x)^2 = x^4 + x^2$

Step 2: $g(-x) = g(x)$ → the function is even. ✓

Geometric check: $g(3) = 81 + 9 = 90$ and $g(-3) = 81 + 9 = 90 = g(3)$. Same y-value at $x = 3$ and $x = -3$.

Step-by-step: Test $h(x) = x^3 + x^2$

Step 1: $h(-x) = (-x)^3 + (-x)^2 = -x^3 + x^2$

Step 2: Is $h(-x) = h(x) = x^3 + x^2$? No ($-x^3 + x^2 \neq x^3 + x^2$). Is $h(-x) = -h(x) = -x^3 - x^2$? No ($-x^3 + x^2 \neq -x^3 - x^2$). Neither even nor odd.

Quick pattern

• All terms have even exponents (including constants) → even function.
• All terms have odd exponents → odd function.
• Mix of even and odd exponents → neither.

Try it in the visualization

Select a function and see $f(x)$ (cyan) and $f(-x)$ (pink dashed) graphed together. For even functions, they overlap perfectly. For odd functions, $f(-x)$ matches $-f(x)$. Test at specific points to see the numerical relationship.