The homogeneous functions can be defined in the following general format:

Let us consider a function f over a field F such that f : X -> Y with n number of

arguments, i.e. (x₁ ,x₂ , ... ,x_n ) and there be a positive integer n and some factor k >

0, then f will be  called a homogeneous function of degree n, if the following condition

is satisfied -

f(kx₁ ,kx₂ , ... ,kx_n ) = kⁿ f(x₁ ,x₂ , ... ,x_n )

for each (x₁ ,x₂ , ... ,x_n ) belonging to the domain of function f.

Ex: Let f(x, y) = x² - 2y² and k be a positive integer.

Now, f(kx, ky) = (kx)² - 2(ky)²

                      = k² x² - 2k² y²

                      = k² (x² - 2y² )

                      = k² f(x, y)

So, the given function is homogenous function of degree 2