Question

The domain of f(x) is the set of all real values except 7, and the domain of g(x) is the set of all real values except –3. What describes the domain of (gºf)(x)?

Answer 1

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) = f(x) ∈ Dom(g)

In your case:
Dom (g°f) = { x ∈ R - {7} | f(x) ∈ R - {-3} }