Recipe

Classify a function as even, odd, or neither

Use when: you're given a formula for f and need to know whether f(-x) equals f(x), equals -f(x), or matches neither

Goal: a settled even, odd, or neither label, backed by an identity that holds for every x, not a check at one value

Taught in Periodic functions and even/odd symmetry, revisited

1. 1

   Look at whether f is built entirely as a sum of pure powers of x, meaning a polynomial with nothing else mixed in.

   Example

   f(x) = x^4+2x^2: both powers (4 and 2) are even, so the shortcut predicts even. h(x)=x^3+x^2: the powers 3 and 2 are mixed, so the shortcut already says neither. g(x)=x\cos x: \cos x isn't a pure power of x, so the shortcut doesn't apply, and Step 2 is needed.

   • every term has an even power, including a bare constant — the shortcut predicts even; go to Step 3 to confirm.
   • every term has an odd power, with no constant term — the shortcut predicts odd; go to Step 3 to confirm.
   • the powers are mixed, some even and some odd — the shortcut already says neither; go to Step 3 to confirm.
   • f isn't a sum of pure powers of x at all, such as \sin x, |x|, a product, or a piecewise rule — the shortcut doesn't apply; continue to Step 2.

   Why: the power pattern is a genuine shortcut only for a sum of pure powers of x; it says nothing about a product or a transcendental function like cosine.

2. 2

   Replace every x in the formula with -x, then simplify with known identities, such as cosine being even, until nothing further reduces.

   Example

   g(-x) = (-x)\cos(-x) = (-x)(\cos x) = -x\cos x.

   Why: f(-x) is the one quantity that actually decides even or odd; no shortcut replaces computing it once f isn't a sum of pure powers.

3. 3

   Match what Step 2 produced against the original f(x) and against its negative, -f(x); whichever one it equals settles the label.

   Example

   g(-x) = -x\cos x = -g(x) exactly, so g is odd. Checking f(x)=x^4+2x^2 directly confirms Step 1's shortcut: f(-x) = (-x)^4+2(-x)^2 = x^4+2x^2 = f(x).

   • the result matches f(x) exactly — f is even.
   • the result matches -f(x) exactly — f is odd.
   • the result matches neither — f is neither even nor odd.

   Why: even and odd are properties of the whole function, so only a match holding for every x counts, not a coincidence at one value.