We let $X$ some finite set of real numbers. I wanted to write the maximum element of X as $\max \{x\in X\}=f(\sum_{x\in X} g(x))$ .
the second part of the question is related to the case where X has only two elements. Then we could use that $max\{x,y\}=(x+y-|x-y|)/2$.
Is there a way to write $|x-y|$ in the form given above: $f(g(x)+g(y))$?
If you can write the maximum as a linear combinations of functions of the above form that is also interesting.
Apologies that don't know which tag this should be under. Please edit if you know a good tag.
If such functions exist, then they would have to be highly exotic.
Suppose that for some functions $f, g$, it is true that $$|x - y| = f(g(x) + g(y))$$ for all $x, y$. Then in particular, it would be true for $x = y$, and so for all $x$, $$f(2g(x)) = 0$$ This is a very restrictive condition. Let $N := \{2g(x) \mid x\in \Bbb R\}$. Then $f(N) = \{0\}$. But since $|x| = f(g(x) + g(0))$, $g$ must take on uncountably many values, and so $N$ must be uncountable as well. That is, $f(x)$ is $0$ for uncountably many values of $x$.
Further, if $u, v \in N$ with $u \ne v$, then there $x,y$ with $g(x) = \frac u2, g(y) = \frac v2$, and so $f\left(\frac{u+v}2\right) = f(g(x) + g(y)) = |x - y| \ne 0$. Therefore between every two values of $N$ is a value where $f$ is not $0$ (and thus is not in $N$). Hence, $N$ is nowhere-dense, and $f$ is not continuous at any point of $\overline N \setminus N$.
This precludes $f$ being any nice function. nothing elementary. Nothing useful for actual calculation.