## Functions: Even and Odd Functions

Even functions
and odd functions are functions which satisfy particular symmetry
relations, with respect to additive inverses.

Let

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*in the domain of*f*:*f* (*x*)
= *f* (−*x*)

Geometrically, an even function is symmetric with
respect to the *y*-axis, meaning that its graph
remains unchanged after reflection about the *y*-axis.
An example of an even function, *f*(*x*)
= *x*^{2},
is illustrated below:

Odd Functions:

Let

*f*(

*x*) be a real-valued function of a real variable. Then

*f*is

**odd**if the following equation holds for all

*x*in the domain of

*f*:

− *f* (*x*)
= *f* (−*x*)

Geometrically, an
odd function is symmetric with respect to the origin, meaning that its
graph remains unchanged after rotation of 180 degrees about the origin.
An example of an even
function, *f* (*x*)
= *x*^{3},
is illustrated below:

Properties Relating to Odd and
Even Functions