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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

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.

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