Homogeneous Function

The function f(x,y) is a homogeneous function of degree $\alpha$ if the following is true:

$f(tx,ty) = t^\alpha f(x,y)$

Example:

$f(x,y) = x^2 + y^2$

For this equation, $f(tx,ty) = t^2x^2 + t^2y^2 = t^2(x^2 + y^2) = t^2 f(x,y)$

A similar example:

$f(x,y) = x^3 + y^3$

$f(tx,ty) = t^3x^3 + t^3y^3 = t^3(x^3 + y^3) = t^3 f(x,y)$

After seeing these examples for 2 and 3, you should be able to generalize this.

Let’s look at one more example:

$f(x,y) = x^2 + xy + x^2$

$f(tx,ty) = t^2x^2 + (tx)(ty) + t^2y^2 = t^2x^2 + t^2xy + t^2y^2 =t^2(x^2 + xy + y^2)$

Below is another example for which the number of x’s and y’s is the same for every term:

$x^4 + x^3y + x^2y^2 +xy^3 + y^4$