Finding the domain and Range of a composite functions

Question

if f(x) and g(x) are well defined functions and f∘g(x) exists, is the following generalization true for all scenarios (i.e. when domain of either or both f(x) and g(x) is restricted):

Domain of f∘g(x) = Domain of g(x) ∩ domain of f∘g(x)

Range of f∘g(x) = Range of f(x) ∩ range of f∘g(x)

If the above is true, how do we derive the identities?

Edit:

I want to find the domain and range of the composite function \begin{equation}f \circ g(x)\end{equation}

where

\begin{equation} f(x) = x^2-4, x\in R, x > 8 \end{equation}

\begin{equation} g(x) = 2x-2,x\in R, x > 6 \end{equation}

$\begin{equation} D_{fg} = D_{g} \cap D_{f\circ g(x)} \end{equation}$ = $\begin{equation} (x>8) \cap (x\in R) \end{equation}$ = $\begin{equation} x>8 \end{equation}$

$\begin{equation} R_{fg} = R_{f} \cap R_{f\circ g(x)} \end{equation}$ = $\begin{equation} (y>60)\cap (y\geq -4) \end{equation}$ = $\begin{equation} y>60 \end{equation}$

1. is my approach correct?

2. If so, can this be generalized for all well defined composite functions?

3. if I want to prove that "Domain of f∘g(x) = Domain of g(x) ∩ domain of f∘g(x) and Range of f∘g(x) = Range of f(x) ∩ range of f∘g(x)", how do I approach it?