Associativity of function composition?

Question

I am working on an exercises that should show that automorphism under composition satisfy a group definition, there are basically four things I need to prove:

1.closure, 2.associativity, 3.inverse, 4.identity

but I am stuck on number 2, usually for elements in a group ie: $$a,b,c \in G$$ if can show associativity by proving: $$(a \circ b)\circ c = a \circ (b \circ c)$$

but when element of the group are functions....what does it even mean?
I know when "$\circ$" means composition, we have $$a \circ b \circ c (g) = a(b(c(g)))$$ but what is $$(a \circ b) \circ c (g) = $$ and how do I prove$$(a \circ b)\circ c(g) = a \circ (b \circ c)(g)$$

Answer 1

From the definition of composition we have: $$ (g \circ f)(x)=g(f(x)) \quad \forall x \in X $$ so: $$ [h \circ (g\circ f)](x)=h((g \circ f)(x))=h(g(f(x))) $$ and

$$ [(h \circ g)\circ f)](x)=(h\circ g) (f(x))=h(g(f(x))) $$

Answer 2

Things may be clarified if you adjust your parentheses a bit. Suppose $x$ is an element of the domain of $h$. On one hand, we have

$$((f \circ g) \circ h)(x) = (f \circ g)(h(x))= f(g(h(x))),$$

and on the other hand,

$$(f \circ (g \circ h))(x) = f((g \circ h)(x)) = f(g(h(x))).$$