Even and Odd Functions
The symmetry classification f(−x) = f(x) for even and f(−x) = −f(x) for odd, and the payoff it gives for free when integrating over a symmetric interval.
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The symmetry classification f(−x) = f(x) for even and f(−x) = −f(x) for odd, and the payoff it gives for free when integrating over a symmetric interval.
Numbers are even or odd; so are some functions. The classification rests on symmetry — how a function behaves when the sign of its input is flipped — and it is not merely descriptive. Knowing which kind you have can remove half the work from an integral.
Most functions are neither. mixes an even term with an odd one and so satisfies no condition.
Classify .
The method is always the same: substitute , simplify, and see which of the two forms you land on.
Classify .
This is the ordinary case. Even and odd functions are the exceptions, which is precisely why they are worth naming.
Over an interval symmetric about zero, an even function contributes the same area on both sides — so compute one side and double it.
An odd function contributes a positive area on one side and an equal negative area on the other. They cancel exactly, and the integral is zero without any calculation at all.