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.

Final Answer

Show final answer

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.

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