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 ) = x n {displaystyle f(x)=x^{n}} is an even function if n {displaystyle n} is an even integer, and it is an odd function if n {displaystyle n} is an odd integer. f ( x ) = f ( − x ) {displaystyle f(x)=f(-x)}     (Eq.1) − f ( x ) = f ( − x ) {displaystyle -f(x)=f(-x)}     (Eq.2) f e ( x ) = f ( x ) + f ( − x ) 2 {displaystyle f_{ ext{e}}(x)={frac {f(x)+f(-x)}{2}}}     (Eq.3) f o ( x ) = f ( x ) − f ( − x ) 2 {displaystyle f_{ ext{o}}(x)={frac {f(x)-f(-x)}{2}}}     (Eq.4) 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 ) = x n {displaystyle f(x)=x^{n}} is an even function if n {displaystyle n} is an even integer, and it is an odd function if n {displaystyle n} is an odd integer. Evenness or oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have an additive inverse. This includes additive groups, all rings, all fields, and all vector spaces. Thus, for example, a real function, as could a complex-valued function of a vector variable, and so on. The given examples are real functions, to illustrate the symmetry of their graphs. Let f ( x ) {displaystyle f(x)} be a real-valued function of a real variable. Then f {displaystyle f} is even if the following equation holds for all x {displaystyle x} and − x {displaystyle -x} in the domain of f {displaystyle f} ::p. 11 or equivalently if the following equation holds for all x {displaystyle x} and − x {displaystyle -x} in the domain of f {displaystyle f} : Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.