1. ## Inverse function notation

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

Say you have a function $f(x)$. Why is it that the inverse of that function is called $f^{-1}(x)$? I mean, if we want to square the function, it's my understanding that we still say $(f(x))(f(x)) = f^2(x)$, so logically, shouldn't $f^{-1}(x)$ be equivalent to $\frac{1}{f(x)}$? And yet it seems that $f^{-1}(x)$ represents the inverse of $f(x)$ while $\frac{1}{f(x)}$ 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 $\sin (x)$ can be written as $\arcsin (x)$, which I prefer to use. But still, if I were to use $\sin^{-1} (x)$ to mean $\frac{1}{\sin (x)}$ or equivalently $\csc (x)$, 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.

2. The notation $f^{-1}(x)$ is pure accident of history.
It is so ingrained in mathematics, we have been able to replace it.
Several years ago some of us tried to use $\overleftarrow f (y)$.
That did not have many takers.

But for the inverse trig functions we can use $\arcsin(x)$ for $\sin^{-1}(x)$.

3. I believe the $f^{-1}$ notation comes from the composition of a function with its inverse, which yields the Identity function ...

$f \circ f^{-1} = I$

and is analogous to the multiplicative inverse ...

$x \cdot x^{-1} = 1$

no real advantage, and yes, it can be confusing. you just have to be clear in its meaning and usage. confusion can be reduced as follows ...

$\sin^{-1}{x}$ is the inverse sine function

$(\sin{x})^{-1}$ is the reciprocal of the sine function, which can be more clear when $\csc{x}$ is used.