1. The problem asks for the domain of the relation given by the set of ordered pairs $\{(4, -1), (-5, 2), (-9, 4), (0, 3)\}$. 2. The domain of a relation is the set of all first elements (x-values) from each ordered pair. 3. Extract the first elements from each pair: $4$, $-5$, $-9$, and $0$. 4. Therefore, the domain is $\{-9, -5, 0, 4\}$. 5. Among the given options, the correct domain is $\{-9, -5, 0, 4\}$.