Can i apply a function in a functional equation?

Question

I have posted a similar exercise before but i want to generalize. Suppose the following equation: $f(x) = g(x) $ , where $Df = Dg$ and $f(A) = g(A) $ , with $f(A) \subseteq Df $ . Moreover $f$ is injective ($ 1-1 $ ). The given equation might be true for all $x$ , for $1$ $x$ or not true at all, that is not given. Is the following implication always true? : $ f(x) = g(x) \iff f(f(x)) = f(g(x)) $

Answer 1

Only if the targets of f and g are in the domain of f...:

f a function : $\forall x, y\in D_f: x=y \implies f(x)=f(y)$

f injective : $\forall x, y\in D_f: f(x) =f(y) \implies x=y$

Now by assuming that the targets of f and g are in $D_f$: $\forall x , y\in D_f:f(x),g(y)\in D_f\ \ \ $

by injectivity : $f(f(x))=f(g(y)) \implies f(x)=g(y)$

since f is a function: $f(x)=g(x)\implies f(f(x))=f(g(x))$