To determine whether the given relation is a function, we need to verify that each input \(x\) is associated with exactly one output \(y\). If any \(x\) value is paired with more than one \(y\) value, then the relation is not a function.
Let's analyze the given pairs one by one:
1. The first pair is \((1, -2)\):
- The input \(x = 1\) corresponds to the output \(y = -2\).
2. The second pair is \((1, -3)\):
- The input \(x = 1\) corresponds to a different output \(y = -3\).
At this point, we observe that the input \(x = 1\) has been paired with two different outputs (\(-2\) and \(-3\)). This violates the definition of a function, which requires each input to be associated with only one output.
Though the remaining pairs are:
3. The third pair is \((2, 1)\):
- The input \(x = 2\) corresponds to the output \(y = 1\).
4. The fourth pair is \((3, -2)\):
- The input \(x = 3\) corresponds to the output \(y = -2\).
These pairs do not contradict the definition of a function independently, but since we have already identified a conflict with \(x = 1\), the entire relation cannot be a function.
Therefore, the relation given in the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & -2 \\
\hline
1 & -3 \\
\hline
2 & 1 \\
\hline
3 & -2 \\
\hline
\end{tabular}
\][/tex]
is not a function.
