You've got the domain of the function correct.

The trick to finding inverse functions is that we are going to write the same function, except that we are going to interchange the roles of x and y. Here, then, we are going to write:

x = 1/(y - 1)

and solve for y. The new y will be the inverse function.

x(y - 1) = 1

xy - x = 1

xy = x + 1

y = (x + 1)/x = 1 + 1/x (whichever form you prefer)

So the inverse function to f(x) = 1/(x - 1) is g(x) = 1 + 1/x.

You can work out the domain and range of g(x) yourself, it looks like you know how to do this.

f(g(x)) = f(1+1/x) = 1/([1+1/x] - 1) = 1/(1/x) = x (Check)

g(f(x)) = g(1/(x-1)) = 1 + 1/(1/[x-1]) = 1 + x - 1 = x (Check)

-Dan