Even and odd functions

An even function satisfies f(−x)=f(x), while an odd function satisfies f(−x)=−f(x) wherever both sides are defined. These identities express symmetry about the y-axis or the origin, respectively.

Function: A relation assigning each input in its domain exactly one output. Parity classifies functions by how their outputs change when inputs are negated.
Negation (mathematics): The operation that replaces a number or expression with its additive inverse. Parity tests compare a function's value at x with its value at the negated input.
Definite integral: A number representing accumulated signed area or total change over an interval. On symmetric intervals, odd contributions cancel and even contributions double.
Neither even nor odd function: A function that satisfies neither the even-function nor the odd-function identity on its domain. Most functions lack either symmetry, even when their graphs look roughly balanced.
Domain of a function: The set of inputs for which a function is defined. An even or odd function must have a domain closed under negation.
Polynomial: An expression formed from constants and nonnegative integer powers of variables. A polynomial is even or odd according to the powers that have nonzero coefficients.
Taylor series: A power series representing a function through its derivatives at a point. Even and odd functions have Taylor expansions containing only matching powers.
Periodic function: A function whose values repeat after shifts by a fixed nonzero period. Periodicity concerns translation, unlike parity's behavior under input negation.
Reflection symmetry: A symmetry in which a figure remains unchanged after reflection across a line or plane. Evenness is reflection symmetry across the vertical axis of a graph.
Sum of functions: The function formed by adding the outputs of two functions at each input. Sums preserve parity when both summands have the same parity.

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Sine and cosineRelated: Cosine is even, while sine is odd, reflecting unit-circle symmetries.