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

##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##.