The main idea is that if you have some well defined function $f$, is there a function $f^{\frac{1}{2}}$ such that $f^{\frac{1}{2}}\left(f^{\frac{1}{2}}\left(x\right)\right) = f\left(x\right)$. For lack of a better name, I will call the process of finding $f^{\frac{1}{2}}$ anti-composition. My question is, are there any known techniques to anti-composing function? Is this even a problem that conventional mathematics has looked at?
For example, the problem I am working on finding the anti-composition of $f\left(x\right) = x^2+C$. I know that when $C=0$, $f^{\frac{1}{2}}\left(x\right)=x^\sqrt{2}$, because $\left(x^\sqrt{2}\right)^\sqrt{2}=x^2$. Due to this rather simple form, I assume that $f^{\frac{1}{2}}\left(x\right)=\left(x-h\right)^n+k$ for some $h$,$k$, and $n$, which may depend on $C$. However, the composition of this is $f\left(x\right)=\left(\left(x-h\right)^n+k-h\right)^n+k$. I assume that $n=\sqrt{2}$, so $x^{n^2} = x^2$. Also, I know that $h=0$, because any change in $h$ will translate the whole function, whereas $x^2+C$ definitely has no x-translation. This means $f\left(x\right)=\left(x^\sqrt{2}+k\right)^\sqrt{2}+k$. However, in order for this to equal $x^2+C$, $k$ must equal $0$ and $C$ simultaneously, which is not possible unless $C=0$.
That's all I've worked on it so far, but I really have no idea what to try next. I have an idea to use the above properties and use a point to get other points of the function and then manipulate the inputs until it looks like a function, but I have no idea how to generalize it or fill in the intervals between the points produced. My attempt at can be found at https://www.desmos.com/calculator/uhc46vpkif, but I didn't find anything useful, except for the general shape of the function.
Any help is appreciated!
