Even and odd function are functions which satisfy some symmetry relation.

Geometrically, the graph of an even function is symmetric with respect to the y-axis. Therefore, let

be a real-valued function with a real variable. Then

*f* is even if the following equation holds for all

*x* in the domain of

*f*.

.

Geometrically, the graph of an odd function has rotational symmetry with respect to the origin; i.e., the graph remains unchanged after rotation of 180 degrees about the origin. Therefore, Therefore, let

be a real-valued function with a real variable. Then

*f* is even if the following equation holds for all

*x* in the domain of

*f*.

.

A function is neither even nor odd if it fails to hold for the above equations.

,

, therefore

and

so

is neither even nor odd.

,

, therefore

and

so

is also neither even nor odd.