Question

If a function i homogenous is it inverse homogenous also

Answer 1

Yes it does automatically imply that f−1(αx)=αf−1(x). We can proceed as follows. Taking the inverse of both side of the original equation gives:

f−1(f(αx))=f−1(αf(x))

This evaluates to

αx=f−1(αf(x))

Now if we define y=f(x), then x=f−1(y) and

αf−1(y)=f−1(αy)