To determine which of the given sets of data represents a function, we need to check if for each input value ([tex]\(x\)[/tex]), there is exactly one corresponding output value ([tex]\(y\)[/tex]).
### Explanation:
A relation is a function if and only if no [tex]\(x\)[/tex]-value is repeated with different [tex]\(y\)[/tex]-values.
### Given Data:
1. Table Data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-10 & 84 \\
\hline
-5 & 31.5 \\
\hline
0 & 4 \\
\hline
5 & 1.5 \\
\hline
10 & 24 \\
\hline
\end{array}
\][/tex]
2. Set of Pairs:
[tex]\[
(4, 5), (6, -2), (-5, 0), (6, 1)
\][/tex]
### Step-by-Step Solution:
1. Check the Table Data:
- The [tex]\(x\)[/tex]-values are: [tex]\(-10, -5, 0, 5, 10\)[/tex]
- Check if there are any repeating [tex]\(x\)[/tex]-values:
- Each [tex]\(x\)[/tex]-value is unique.
- Since no [tex]\(x\)[/tex]-value is repeated, this set of data represents a function.
2. Check the Set of Pairs:
- The pairs are: [tex]\((4, 5), (6, -2), (-5, 0), (6, 1)\)[/tex]
- The [tex]\(x\)[/tex]-values are: [tex]\(4, 6, -5, 6\)[/tex]
- Observe that the [tex]\(x\)[/tex]-value [tex]\(6\)[/tex] is repeated with different corresponding [tex]\(y\)[/tex]-values ([tex]\(-2\)[/tex] and [tex]\(1\)[/tex]).
- Since the [tex]\(x\)[/tex]-value [tex]\(6\)[/tex] corresponds to two different [tex]\(y\)[/tex]-values, this set of data does not represent a function.
### Conclusion:
- The table data represents a function.
- The set of pairs [tex]\((4, 5), (6, -2), (-5, 0), (6, 1)\)[/tex] does not represent a function.
Therefore, the results are:
- For the table data, it represents a function: True
- For the set of pairs, it does not represent a function: False
