# Function Composition, Definition?

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 $$G$$

I'm not sure what you mean by "how the argument ($$x$$) is distributed to $$f$$ below". Each of the steps you show is simply applying the definition of $$\circ$$:
$$(F \circ G)(x) = F(G(x))$$
For example, 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 $$G$$.