# Homogeneous Functions – Homogeneous check to the function x in the power of y – Exercise 7048

Exercise

Determine if the following function:

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

Is homogeneous.

The function is not homogeneous

Solution

We look at the function:

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

By definition, a function is homogeneous of degree n if and only if the following holds:

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

For a parameter t.

Therefore,

$$f(tx,ty)=$$

We plug in our function and get

$$={(tx)}^{ty}=$$

We open brackets and get

$$=t^{ty}x^{ty}=$$

$$=t^{ty}{(x^y)}^t=$$

And we got the following:

$$\neq t^nf(x,y)$$

For any t and any n.

In short, we got the following:

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

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

Have a question? Found a mistake? – Write a comment below!
Was it helpful? You can buy me a cup of coffee here, which will make me very happy and will help me upload more solutions!

Share with Friends