I understand intuition of the following formula and function composition, however could use guidance understanding formalisms on how the argument $(x)$ is distributed to $f$ below:
$f\colon X\to Y$ and $g\colon Y$ to Z and $h\colon Z\to W$
$x \in X$
$ h\circ(g\circ f)(x)=h(g\circ f(x))= h(g(f(x))=h\circ g(f(x))=(h\circ g)\circ f(x). $
Is there a rule or identity such that the ordered list of arguments $(x)$ is used as an argument to $f$?
The definition of $\circ$:
$$(F \circ G)(x) = F(G(x))$$
The first step $h \circ (g \circ f)(x) = h((g \circ f)(x))$ applies the definition by putting in $h$ for $F$, and $g \circ f$ for $G$