Symmetric functions come in two flavours.
Graph
y = x·x·x - 3·xy = x·x - 2meet at x ≈ -1.62, 0.62
In the plot, x³ − 3x (odd) has rotational symmetry about the origin — spin it 180° and it lands on itself. x² − 2 (even) is a mirror image across the y-axis.
Your turn
Is f(x) = x⁴ − x² odd, even, or neither?
Watch out
Most functions are neither odd nor even — e.g. x² + x. Don't assume every function has symmetry; you must actually check f(−x) against f(x) and −f(x).
The two
- Even: f(−x) = f(x). Mirror across the y-axis. Examples: x², cos(x).
- Odd: f(−x) = −f(x). Rotational symmetry about origin. Examples: x³, sin(x).