Even and Odd Functions

Pre-Calculus · Axiom Academy

Two kinds of symmetry — a mirror across the y-axis, a half-turn about the origin — that you can spot from the formula alone. 1. Even: a Mirror Across the y-Axis A function is even when replacing x with -x changes nothing: the input -a lands at exactly the same height as the input a . Geometrically, the left half of the graph is the mirror image of the right half across the y-axis . for every x in the domain — the height at -x equals the height at x . 2. Odd: a Half-Turn About the Origin A function is odd when replacing x with -x flips the sign of the output: the input -a lands at the height -f(a) . Geometrically, rotating the graph about the origin leaves it unchanged — the origin sits exactly halfway between each point and its partner. the height at -x is the negative of the height at x . Why the connector passes through the origin The midpoint of and is — the origin, for every a . That is precisely what a rotation about the origin means. Familiar odd functions: x , x^3 , , and . 3. The Test: Substitute -x and Compare You don't need the graph to decide. Compute f(-x) by replacing every x with -x , simplify, and compare the result against f(x) and against -f(x) . Exactly one of three things happens. Every sign came back the same. Example: f(x) = x^4 - 2x^2 + 1 gives f(-x) = x^4 - 2x^2 + 1 . Every term flipped sign. Example: f(x) = x^3 gives f(-x) = -x^3 = -f(x) . It matches f(x) on some terms and -f(x) on others. Example: f(x) = x^2 + x gives f(-x) = x^2 - x .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.