To determine if a relation is also a function, we need to check if each [tex]\( x \)[/tex]-value maps to exactly one [tex]\( y \)[/tex]-value. In other words, a relation is a function if and only if no [tex]\( x \)[/tex]-value is repeated with different [tex]\( y \)[/tex]-values.
Let's examine the given relation step-by-step:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
4 & -7 \\
\hline
5 & -3 \\
\hline
5 & -2 \\
\hline
6 & 3 \\
\hline
7 & 6 \\
\hline
\end{tabular}
\][/tex]
1. The [tex]\( x \)[/tex]-value [tex]\( 4 \)[/tex] maps to the [tex]\( y \)[/tex]-value [tex]\( -7 \)[/tex].
2. The [tex]\( x \)[/tex]-value [tex]\( 5 \)[/tex] maps to the [tex]\( y \)[/tex]-value [tex]\( -3 \)[/tex].
3. The [tex]\( x \)[/tex]-value [tex]\( 5 \)[/tex] also maps to the [tex]\( y \)[/tex]-value [tex]\( -2 \)[/tex].
4. The [tex]\( x \)[/tex]-value [tex]\( 6 \)[/tex] maps to the [tex]\( y \)[/tex]-value [tex]\( 3 \)[/tex].
5. The [tex]\( x \)[/tex]-value [tex]\( 7 \)[/tex] maps to the [tex]\( y \)[/tex]-value [tex]\( 6 \)[/tex].
Notice that the [tex]\( x \)[/tex]-value [tex]\( 5 \)[/tex] maps to two different [tex]\( y \)[/tex]-values: [tex]\( -3 \)[/tex] and [tex]\( -2 \)[/tex]. This means that for [tex]\( x = 5 \)[/tex], there are two corresponding [tex]\( y \)[/tex]-values, which violates the definition of a function.
Therefore, this relation is not a function.
The final conclusion is that the relation is not a function, so the correct choice is:
B.
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
4 & -7 \\
\hline
5 & -3 \\
\hline
5 & -2 \\
\hline
6 & 3 \\
\hline
7 & 6 \\
\hline
\end{tabular}
\][/tex]
is not a function.
