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

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##.