A group of functions that guarantees the existence of a specific function

Question

Let $f$ be a function that has continuous derivative in $\mathrm{R}$. Then, I want to find a set of functions such guarantees the existence of the function $g$ such that $$g(f(x))=(f'(x))^2$$ Or in other words, I want to find $f$ such that the points $(f(x), (f'(x))^2)$ (defined parameterically) will form a function, namely $g$. There are two cases that I am thinking.

1. When f is monotone, then $f^{-1}$ exists, and therefore $g$ can be defined explicitly, $g(x)=(f'(f^{-1}(x)))^2$ by putting in $f^{-1}(x)$ instead of $x$. This part is pretty obvious.
2. Now this is the part that I am having a hard time to prove mathematically. Assume that the function is not monotone. In other words, there exists $a, b$ such that $f(a)=f(b)$. Then, $g(f(a))=g(f(b))$, and we get $(f'(a))^2=(f'(b))^2$. In other words, if two points have the same function value, then the absolute value of the derivative of each points must be the same. Doodling a little about function satisfying this, and I found out that functions with both point reflection and line reflection will satisfy the condition(such as $\sin x, \cos x$, and so on). I have a little proposition about this, but I am not sure about the exact proof.

(This is part of the proof that I was trying to do.)

Draw a line $y=c$, and let $a<b \in R$ such that $f(a)=f(b)=c$, and $b-a$ is minimized. (I am actually not sure if such minimum exists for all functions, which I assume not, but if it is) If $f'(a)=f'(b)$(Assume this positive for convenience), then by continuity, there exists $\epsilon_1, \epsilon_2>0$ such that $f(a+\epsilon_1)>0$ and $f(b-\epsilon_2)<0$, and by IVT, there must exist $t \in [a, b]$ such that $f(t)=c$. Then, $t-a<b-a$, so this is contradiction to minimality of $b-a$. Therefore, we conclude that $f'(a)=-f'(b)$ for adjacent roots of $f(x)=c$. Therefore, for all of the roots of $f(x)=c$, denote it as $..., a_{-1}, a_0, a_1, ...$, then we get $f'(a_n)=-f'(a_{n+1})$ for all $n$.

I am currently guessing that such functions satisfying this must be a function with line reflection, however I am having a hard time trying to prove it.