Why is $f(x)^{-1}$ used to denote the inverse of a function, and not its reciprocal? Function notation says that any operations applied to a variable inside the parenthesis are applied to the variable before it enters the function, and anything applied to the function as a whole is applied to the entire function or to the result of the function.  So you could say that...
If $f(x)=x^2$,
then $f(3x-2)=(3x-2)^2$, and $3f(x)-3=3x^{2}-3$.
I understand that $f^{-1}(x)$ is used to denote the inverse of a function so...
$$f^{-1}(x)=\pm\sqrt{x}.$$
But putting something to the $-1$ power gives you its reciprocal like...
$$x^{-1}=\frac{x^{-1}}{1}=\frac{1}{x^1}= \frac{1}{can}$$
So would it not make sense that...
$$f^{-1}(x)=\frac{1}{x^2}.$$
My question is why $f^{-1}(x)$ is considered the inverse of $f(x)$?  Was this notation chosen randomly, or is there some logical explanation for why it is that way?

That notation, in my opinion, is not normal.
We always let $f^{-1}(x)$ or $f^{-1}$ denote the inverse of a function.
