Function composition and inverse

Question

Consider f : ℝ \ {1} → ℝ \ {1} given by f(x) = x/(x-1)

I need to find:

1) f ◦ f ◦ f and

2) the inverse function f^-1(x)

So far I have:

1) f(f(x/(x-1)) = f(x) = x/(x-1) which is suspicious to me so I was wondering if I messed that up.

2) If I understand correctly, the inverse will be x=y/(y-1) solved for y?

Answer 1

You're correct that the inverse can be found by solving for $y$ as you wrote.

But your answer to part 1 suggests that $f^2$ composed with $f(x)$, gives the identity function $g(x) = x$. That suggests an easy way to identify $f^{-1}$.