Mr. Integral
Math notes/Pre-Calculus/Functions Review

Odd Functions — Cheat sheet

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Key method

f(−x)=−f(x).f(-x)=-f(x).

If zero belongs to the domain, oddness forces f(0)=0f(0)=0. That necessary condition alone does not prove a function is odd.

Example

For f(x)=x3−2xf(x)=x^3-2x, f(−x)=−x3+2x=−(x3−2x)f(-x)=-x^3+2x=-(x^3-2x). The reciprocal 1/x1/x is also odd on its symmetric domain excluding zero. In contrast, x3+1x^3+1 fails because the constant does not change sign.

Avoid this mistake

The zero function is both even and odd on any symmetric domain. Most functions are neither.