To determine the domain of the set of points \(\{(1,2),(-5,7),(4,1),(0,0)\}\), we need to identify all the first elements (or x-coordinates) in these ordered pairs.
Let's list the points and their x-coordinates:
- Point \((1,2)\) has an x-coordinate of \(1\).
- Point \((-5,7)\) has an x-coordinate of \(-5\).
- Point \((4,1)\) has an x-coordinate of \(4\).
- Point \((0,0)\) has an x-coordinate of \(0\).
So, the x-coordinates from the given set of points are \(\{1, -5, 4, 0\}\).
The domain of the set of points is therefore \(\{-5, 0, 1, 4\}\).
Now, let's match this with the given choices:
- \( (-\infty, \infty) \) implies all real numbers, which is not our list.
- \(\{-5, 0, 1, 4\}\) matches our domain exactly.
- \(\{0, 1, 2, 7\}\) does not match our domain.
- The choice \(4\) is ambiguous and does not relate to the domain of the points.
Given our analysis, the correct answer is:
\(\{-5, 0, 1, 4\}\).
Therefore, the numerical result is:
[tex]\[ \boxed{2} \][/tex]
