The domain of a function consists of all the possible input values (usually $x$-values) that are valid for the function. Based on the set of points you've provided:

$(-3, -7),\ (8, -2),\ (0, 9),\ (-1, -7)$

The domain is simply the set of all the $x$-coordinates from the points:

$\{-3, 8, 0, -1\}$

So, the domain of this function is $\{-3, 8, 0, -1\}$.

Would you like to explore any specific part of this? Here are some related questions:

1. What is the range of this function?
2. Can this function be considered a one-to-one function?
3. How do you determine the domain of a function from a graph?
4. What if a function has an undefined value, how does that affect its domain?
5. What is the domain for a function with an equation, rather than just points?

Tip: The domain includes all $x$-values, and the range includes all $y$-values.