The domain of every function f(x) is the set of x-values that are 'plugged' into the function f. It then follows that the domain of f(u) is the set of u-values plugged into the function f. Make the substitution u=g(x). The domain of g(x) determines the set of u-values that are plugged into f(x).
In short
Domain of g(x) –(g)-> Range of g(x) = Domain of f(u) –(f)-> Range of f(u) = Range of f(g(x))
Thus the domain of f(g(x)) = set of x-values that are plugged into the fg function = set of x-values that are plugged into the g function = domain of g(x) = x > -2 (for real values of sqrt(2x+4), 2x+4>0 \Rightarrow x > -2