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I have read on many sites(per example this MSE' answer) that a $gof$ composite function is "well defined" or "is possible" when:

$f: A \to B$, $g: B \to C$, $gof:A\to C$

What this mean? is definition or what?

Why the domain of $gof$ is $A$? it is not supposed to be the intersection of $x$ in $Dom(f)$ and $f(x)$ in $Dom(g)$? The only reason I see for the definition of $gof$ is that enters $A$ values(because is the domain of the first function $f$) and get $C$ possible values(because is the codomain of the last function)

Why the domain of $g$ is $B$? i think that the domain of $g$ in a composition will be $B$ if the sets $Range(f)$ and $Dom(g)$ are equal, but $Range(f)$ can be subset of $Dom(g)$

ESCM
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You're forgetting that, in the composition, $f$ is applied first. Thus, $f$ takes all of the elements of $A$ and sends them to a subset of $B$ called $\text{range}(f)$, and then $g$ (since the composition is $g(f(x))$) sends all of the elements of $\text{range}(f) \subseteq B$ to a subset of $C$.

Individually $f,g$ are separate functions, so $g$ could have a domain bigger than $\text{range}(f)$ if desired, since $g \circ f$ only is sampling from a subset of the domain of $g$ (the subset $f$ maps to) and not the entire domain.

PrincessEev
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    I understand what you say in the first paragraph. However, why is the domain of g the codomain of f? For example, to demonstrate this, they use that "definition" or whatever. – ESCM Apr 19 '20 at 01:30
  • or, they are only specific conditions for prove that? – ESCM Apr 19 '20 at 01:31
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    Well, suppose otherwise, that the domain of $g$ is not the codomain of $f$. How would you suggest $g \circ f$ handle elements $f(x)$ which are in the range of $f$ and thus in $B$, but not in the domain of $g$? – PrincessEev Apr 19 '20 at 01:59
  • i know that the domain of $g$ must contain the range of $f$ but not necessary be equal – ESCM Apr 19 '20 at 03:26
  • Per example range of $f(x) = x^2$ is $[0, \infty)$ and domain of $g(x) = x^3 - 2/x$ is $\mathbb{R} - {0}$, so the range of $f$ is different of domain of $g$. – ESCM Apr 19 '20 at 03:28
  • Help me, please. – ESCM Apr 19 '20 at 03:40