Okay, so you have inverse functions: $ x^{2^{\frac{1}{2}}} = x$
But you also have negative functions $ x^{2^{-2}} \neq x$
But when I look for inverse functions, they are usually defined like this:
$$ f^{-1} $$
However when I write $sin^{-1}(x)$, I am usually talking about $csc(x)$, which is not the inverse, which would be $asin(x)$.
Is there some better terminology for negative functions? What is now the actual inverse function?
There's some finicky stuff that goes on with terminology.
We have $\sin^2(x)=(\sin(x))^2$, but $\sin^{-1}(x)$ is just inverse sin (or asin, arcsin, etc.)
If you want to say $\frac{1}{\sin x}$, the standard is to just use $csc(x).$