Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = xⁿ is an even function if n is an even integer, and it is an odd function if n is an odd integer.

Definition and examples

The concept of evenness or oddness is defined for functions whose domain and image both have an additive inverse. This includes additive groups, all rings, all fields, and all vector spaces. Thus, for example, a real-valued function of a real variable could be even or odd, as could a complex-valued function of a vector variable, and so on.

The examples are real-valued functions of a real variable, to illustrate the symmetry of their graphs.

Even functions

Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x and -x in the domain of f: