Homogeneous Functions – Homogeneous check to a sum of functions with powers of parameters – Exercise 7060

Exercise

Determine if the following function:

$f(x,y)=x^m+x^{m-n}y^n$

Is homogeneous.

(m,n parameters)

Show final answer

Final Answer

$f(x,y)=x^m+x^{m-n}y^n$

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

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

for all t.

$f(tx,ty)=$

$=(tx)^m+(tx)^{m-n}\cdot (ty)^n=$

$=t^m\cdot x^m+t^{m-n}\cdot x^{m-n}\cdot t^n\cdot y^n=$

$=t^m\cdot x^m+t^m\cdot x^{m-n}\cdot y^n=$

$=t^m(x^m+x^{m-n}\cdot y^n)=$

$=t^mf(x,y)$

We got

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

Hence, by definition, the given function is homogeneous of degree m.

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