Homogeneous Function

A homogeneous function is a function which multiplicative scaling behavior.

Example

The function $f(x,y)$ can be expressed in a different way be defining:

• two real constants $k,n\in \mathbb{R}$
• $x^{\prime }=kx$
• $y^{\prime }=ky$

The function $f$ is homogeneous if we can write

$$f(x^{\prime },y^{\prime })=f(kx,ky)=k^{n}f(x,y)$$

Human Interpretation

An easy way to tell if a function is homogeneous is to see if the function is equal to zero.

$f(x)=5x$ is non-homogeneous

$f(x)=0$ is homogeneous

Degree

$$f(kx,ky)=k^{n}f(x,y)$$

is called a homogeneous function of degree $n$.