To determine which representation from the given data sets represents a function, we need to assess whether each set of points satisfies the definition of a function. Recall that a function is a relation where each input (or \( x \)-value) is associated with exactly one output (or \( y \)-value).
Let's examine each set of points separately:
1.
[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-5 & 10 \\
\hline
-3 & 5 \\
\hline
-3 & 4 \\
\hline
0 & 0 \\
\hline
5 & -10 \\
\hline
\end{tabular}
\][/tex]
Here, the \( x \)-value -3 is associated with two different \( y \)-values (5 and 4). This means that for \( x = -3 \), there are multiple outputs, which violates the definition of a function. Therefore, this set of points does not represent a function.
2.
[tex]\[
\{(-8, -2), (-4, 1), (0, -2), (2, 3), (4, -4)\}
\][/tex]
In this set, each \( x \)-value corresponds to exactly one \( y \)-value. The \( x \)-values are: -8, -4, 0, 2, and 4, and all are unique with unique \( y \)-values associated with them. Hence, this set represents a function.
3.
[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-2 & -3 \\
\hline
-1 & -2 \\
\hline
0 & -1 \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
\end{tabular}
\][/tex]
Here, the \( x \)-value 0 has two different \( y \)-values (-1 and 0). This violates the definition of a function, as there cannot be two different \( y \)-values for the same \( x \)-value. Thus, this set does not represent a function.
4.
[tex]\[
\{(-12, 4), (-6, 10), (-4, 15), (-8, 18), (-12, 24)\}
\][/tex]
In this set, the \( x \)-value -12 is associated with two different \( y \)-values (4 and 24). This means that for \( x = -12 \), there are multiple outputs, which violates the definition of a function. Therefore, this set does not represent a function.
After evaluating all four sets, we conclude that only the second set of points represents a function.
