Arithmetic on Even and Odd Functions — Cheat sheet

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Key method

Even + even is even; odd + odd is odd. Products obey: even × even is even, odd × odd is even, and even × odd is odd. A sum of a nonzero even part and a nonzero odd part is generally neither.

Example

The function $x\sin x$ is even: both $x$ and $\sin x$ are odd, so their sign changes cancel. The function $x\cos x$ is odd. For $f(x)=x^2+x$ , its even part is $(f(x)+f(-x))/2=x^2$ and its odd part is $(f(x)-f(-x))/2=x$ .

Avoid this mistake

An odd power in a numerator does not determine the parity of an entire rational expression; inspect the denominator too.