Question

If `f` is a one-to-one function such that `f(2)=9`, what is `f^-1(9)`?

Answer 1

`f^(-1)(9) = f^(-1)(f(2)) = 2`

Explanation:

If `f` is a one-to-one function, then its inverse function, `f^(-1)`, is well-defined.

What does the inverse do ? Exactly what it is called.
Suppose, for example :

`f : RR \rightarrow RR`
`x \mapsto f(x) = y`

Then `f^(-1)` do the opposite/reverse :

`f^-1 : RR \rightarrow RR`
`y \mapsto f^(-1)(y) = x`

Thus, if `f(x) = y`, then `f^(-1)(f(x)) = f^(-1)(y) = x`.

Therefore, if `f(2) = 9`, you apply `f^(-1)` to both sides and you get :
`f^(-1)(f(2)) = f^(-1)(9) = 2`.