To determine which relation is a function, we need to ensure that each input (x-value) maps to exactly one output (y-value). Let's analyze each of the given choices one by one:
### Choice A:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
4 & -7 \\
\hline
5 & -3 \\
\hline
5 & -2 \\
\hline
6 & 3 \\
\hline
7 & 6 \\
\hline
\end{array}
\][/tex]
In Choice A, the x-value 5 is paired with two different y-values (-3 and -2). This means that a single input (5) maps to multiple outputs, so this relation is not a function.
### Choice B:
[tex]\[
\{(-2,0),(-1,1),(0,2),(1,1),(2,0)\}
\][/tex]
In Choice B, each unique x-value is paired with exactly one y-value:
- -2 maps to 0
- -1 maps to 1
- 0 maps to 2
- 1 maps to 1
- 2 maps to 0
Since each x-value maps to exactly one y-value, this relation is indeed a function.
### Choice C:
Choice C is empty and does not contain any pairs. Therefore, it cannot be considered a function.
### Choice D:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 7 \\
\hline
\end{array}
\][/tex]
In Choice D, there is only one pair:
- 1 maps to 7
Since there are no other pairs, this relation trivially maps one x-value to one y-value, so it is considered a function.
After considering all the options, the relations that qualify as functions are Choices B and D. However, if we are asked to determine which of the given relations is also a function, we find that:
- Choice B comprehensively meets the criteria for a function.
Thus, the correct answer is:
[tex]\[
\boxed{\text{2 (Choice B)}}
\][/tex]
