# Function Repository Resource:

# FunctionParity

Determine the parity of a function (whether it is even or odd) with respect to one or more variables

Contributed by: Wolfram|Alpha Math Team
 ResourceFunction["FunctionParity"][expr,x] gives an integer indicating the parity of expr with respect to x, or Undefined.

## Details and Options

FunctionParity returns one of the following results:
 -1 expr is an odd function of x 1 expr is an even function of x 0 expr is both an even and an odd function of x none of the above is true
A function is considered even if f(x)=f(-x) for all x and is considered odd if -f(x)=f(-x).

## Examples

### Basic Examples

Find the parity of a basic power function:

 In:= Out= Test another power function:

 In:= Out= Show that a constant function has even parity:

 In:= Out= Find the parity of a rational function:

 In:= Out= Find the parity of the absolute value function:

 In:= Out= Find the parity of a trigonometric function:

 In:= Out= Find the parity of a Gaussian function:

 In:= Out= Test a signed Gaussian function:

 In:= Out= Find the parity of the hyperbolic tangent function:

 In:= Out= Find the parity of the error function:

 In:= Out= Find the parity of a Fresnel integral:

 In:= Out= Find the parity of a shifted Fresnel integral:

 In:= Out= ### Scope

Show that an implicitly defined circle is even in both x and y:

 In:= Out= Find the parity of a function of two variables:

 In:= Out= Find the parity of a function of three variables:

 In:= Out= ### Properties and Relations

FunctionParity returns Undefined for functions that are neither even nor odd:

 In:= Out= The constant function f(x)=0 is both even and odd:

 In:= Out= 