Even, odd and periodic functions

For even/odd tests the domain must be symmetric about \(0\). An even function satisfies \(\mathrm{f}(-x)=\mathrm{f}(x)\) for every permitted \(x\); an odd function satisfies \(\mathrm{f}(-x)=-\mathrm{f}(x)\).

The zero function is both even and odd. Testing a few points can disprove a claimed identity but cannot prove it for every input.

A periodic function repeats: \(\mathrm{f}(x+T)=\mathrm{f}(x)\) throughout its domain, which is preserved by the period. The least positive period, when it exists, is the fundamental period. For \(\sin(3x)\), it is \(\dfrac{2\pi}{3}\).

Using periodicity

If a function has period \(T\), evaluate an input outside its defining interval by adding or subtracting an integer multiple of \(T\). For period \(4\), \(\mathrm f(27)=\mathrm f(3)\). Choose the representative that lies in the stated base interval, respecting open and closed endpoints.

To extend a graph, translate its complete base pattern by \(T\) repeatedly. Carry endpoint inclusion and any jump with the pattern; do not join separate pieces across a discontinuity.

Evenness, oddness and periodicity are different tests. A repeated shape need not be even, and an even polynomial need not be periodic. Prove an identity over the domain; a few matching plotted points are only a check.