Well, how is f-1 defined?
f^{-1} = \{ (y, x) : (x, y) \in f \}. \text{ So } f^{-1}(y) = x \Leftrightarrow f(x) = y.
Now just take y = f(x) to see that f-1(f(x)) = x.
Edit: Doh, you changed your questions :( Well I'll leave it there in case it helps anyone else.
As for your new question, I'm...