I've lost track of how many times I've said this, but: by showing that the definition holds.

The definition of "inverse function" is:

is the inverse function to g(x) if and only if and .

If g(x)= 1- 1/(x-1) and , then [tex]g^{-1}(g(x))= g^{-1}(1- 1/(x-1))= 1+ 1/(1-[1-1/(x-1)]). Show that that is equal to x and the do it the other way.