This is something that's not really crucial to my understanding, but it's bothered me for a while.

Say you have a function $\displaystyle f(x)$. Why is it that the inverse of that function is called $\displaystyle f^{-1}(x)$? I mean, if we want to square the function, it's my understanding that we still say $\displaystyle (f(x))(f(x)) = f^2(x)$, so logically, shouldn't $\displaystyle f^{-1}(x)$ be equivalent to $\displaystyle \frac{1}{f(x)}$? And yet it seems that $\displaystyle f^{-1}(x)$ represents the inverse of $\displaystyle f(x)$ while $\displaystyle \frac{1}{f(x)}$ represents the reciprocal of x.

It's really trig functions that got to me originally. Now that I'm in university I've found out that there is another way to denote it. i.e. the inverse of $\displaystyle \sin (x)$ can be written as $\displaystyle \arcsin (x)$, which I prefer to use. But still, if I were to use $\displaystyle \sin^{-1} (x)$ to mean $\displaystyle \frac{1}{\sin (x)}$ or equivalently $\displaystyle \csc (x)$, most people would read it to mean the inverse function of sine x.

Is there any reason for this, or advantage of the "to the power of negative one" notation? To me it just seems confusing, and conflicts with those times when you're actually indicating powers of the function.