Homogeneous Functions – Homogeneous check to sum of functions with powers – Exercise 7062

Exercise

Determine if the following function:

$$f(x,y)=x^2+y^2+3x$$

Is homogeneous.

The function is not homogeneous

Solution

$$f(x,y)=x^2+y^2+3x$$

Function f is called homogeneous of degree r if it satisfies the equation:

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

for all t.

$$f(tx,ty)=$$

$$=(tx)^2+(ty)^2+3(tx)=$$

$$=t^2\cdot x^2+t^2\cdot y^2+t\cdot 3x$$

We got

$$f(tx,ty)\neq t^rf(x,y)$$

Hence, by definition, the given function is not homogeneous.

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