Sure, let's analyze both the table of values and the given points to determine whether each represents a function.
### Analyzing the Table of Values:
To determine if a table represents a function, we need to ensure that each input value ([tex]\(x\)[/tex]) maps to exactly one output value ([tex]\(y\)[/tex]). In other words, no [tex]\(x\)[/tex] value should repeat with different [tex]\(y\)[/tex] values.
#### Table:
[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]
We will examine the [tex]\(x\)[/tex] values here to see if any of them repeat:
- [tex]\(x = -10\)[/tex]
- [tex]\(x = -5\)[/tex]
- [tex]\(x = 0\)[/tex]
- [tex]\(x = 5\)[/tex]
- [tex]\(x = 10\)[/tex]
Since each [tex]\(x\)[/tex] value appears only once, the table does indeed represent a function.
### Analyzing the List of Points:
Similar to the table analysis, for the list of points to represent a function, each [tex]\(x\)[/tex] value should map to exactly one [tex]\(y\)[/tex] value.
#### Points:
[tex]\[
(4, 5), (6, -2), (-5, 0), (6, 1)
\][/tex]
We will examine the [tex]\(x\)[/tex] values here to see if any of them repeat:
- [tex]\(x = 4\)[/tex]
- [tex]\(x = 6\)[/tex]
- [tex]\(x = -5\)[/tex]
- [tex]\(x = 6\)[/tex]
We can see that [tex]\(x = 6\)[/tex] appears twice with different [tex]\(y\)[/tex] values ([tex]\(-2\)[/tex] and [tex]\(1\)[/tex]). This means that the list of points does not represent a function.
### Conclusion:
From the analysis:
- The table of values represents a function because there are no duplicate [tex]\(x\)[/tex] values.
- The list of points does not represent a function because there is a duplicate [tex]\(x\)[/tex] value ([tex]\(x = 6\)[/tex]) with different [tex]\(y\)[/tex] values.
Therefore, the correct answer to the question is that neither the table nor the points represent a function, since the list of points violates the condition of having unique [tex]\(x\)[/tex] values. But since we are to determine which of them represents a function, only the table holds. The final determination:
- The given points [tex]\((4, 5), (6, -2), (-5, 0), (6, 1)\)[/tex] do not form a function.
Thus, considering the most inclusive answer in summary:
- False: Neither structure strictly holds the criteria under all conditions as true representations of a strict function ensemble.
