What's the difference between a function g(x) and a composite function g(f(x))?

Question

Let $f: A \to B$ And $g: B \to C$

$(gof)(x)$ is written as $g(f(x)$, does this mean that it's domain is $f(x)$ , and if it's true then how is it different from the function g whose domain and codomain are the same?

Answer 1

Let $f:A \rightarrow B$ and $g:B \rightarrow C$

Lets walk through $g \circ f = g(f)$ step by step. We start in the parentheses, so the composite function begins with the domain of $f$, which is $A$. $f$ maps elements in $A$ to $B$. Then, $g$ takes over and maps elements in $B$ to $C$. So, $g \circ f = g(f)$ maps from $A$ to $C$.

Answer 2

Since $f: A \to B$ and $g: B \to C$ we get $g \circ f:A\to C$. Since $(g \circ f)(x)=g(f(x))$ the function needs to be passed a parameter in $A$ and returns one from$C$. For example let $a\in A$ and $f(a)=b\in B$ and $g(b)=c\in C$. Then $(g \circ f)(a)=g(f(a))=g(b)=c$