To determine whether the given relation is a function, we need to check if each [tex]\( x \)[/tex]-value is associated with exactly one [tex]\( y \)[/tex]-value.
Let's look at the relation given:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
\hline
1 & -3 \\
\hline
2 & 1 \\
\hline
3 & -2 \\
\hline
\end{array}
\][/tex]
We'll analyze each [tex]\( x \)[/tex]-value and see if it's associated with more than one [tex]\( y \)[/tex]-value:
1. The [tex]\( x \)[/tex]-value [tex]\( 1 \)[/tex] is associated with [tex]\( y \)[/tex]-values [tex]\(-2\)[/tex] and [tex]\(-3\)[/tex]. This means that the [tex]\( x \)[/tex]-value [tex]\( 1 \)[/tex] has more than one corresponding [tex]\( y \)[/tex]-value.
2. The [tex]\( x \)[/tex]-value [tex]\( 2 \)[/tex] is associated with the [tex]\( y \)[/tex]-value [tex]\( 1 \)[/tex].
3. The [tex]\( x \)[/tex]-value [tex]\( 3 \)[/tex] is associated with the [tex]\( y \)[/tex]-value [tex]\(-2\)[/tex].
A relation is a function if and only if each [tex]\( x \)[/tex]-value has exactly one corresponding [tex]\( y \)[/tex]-value.
In this case, since the [tex]\( x \)[/tex]-value [tex]\( 1 \)[/tex] has two different [tex]\( y \)[/tex]-values ([tex]\(-2\)[/tex] and [tex]\(-3\)[/tex]), the relation is not a function.
Therefore, the answer to whether the given relation is a function is:
No
