The function g°f(x) can be written as g(f(x)), which means that you take a random x, put it into f(x) which gives you a value (let's call it y). You then put y into g(y) and you find the result.
Therefore the Domain you are looking for in words is every x that can be put into f(x) and that gives you a value y that can be put in g(y).
Mathematically:
Dom (g°f) = {x ∈ Dom(f) | f(x) ∈ Dom(g)}
In your case:
Dom (g°f) = { x ∈ R - {7} | f(x) ∈ R - {-3} }
