Range of two equal functions

I know two functions $f$ and $g$ are equal if

(1) their domains are equal

(2) their co-domains are equal

(3) $f(x) = g(x)$

I want to ask if two functions are equal is it necessary that they will have equal range too?

Answer

Take $f : A \rightarrow B \wedge g : A \rightarrow B.$ If these functions are equal, then $\forall x \in A. f(x) = g(x).$ Consider their ranges $f(A)$ and $g(A).$ If their ranges are not equal, $f(A) \neq g(A),$ which implies that there exists an element in one of their ranges that is not in the other. Without loss of generality. Say there exists some $y \in f(A)$ such that $y \notin g(A).$ This means that
$$\exists x \in A. f(x) = y \wedge \forall x \in A. g(x) \neq y.$$
Fix this $x \in A$ such that $f(x) = y.$ Because $f$ and $g$ are equal, then
$$f(x) = g(x) = y \implies g(x) = y,$$
which is a contradiction of the statement that $\forall x\in A. g(x) \neq y.$ Thus, the ranges of $f$ and $g$ must be equal.

Your suggested functions are not equal, since

$$fof(3) = 1 \neq 2 = f(3).$$