To create a function from the given table, each input value [tex]\( x \)[/tex] must be associated with exactly one output value [tex]\( y \)[/tex]. Therefore, we must ensure that no [tex]\( x \)[/tex]-value repeats in our table.
Let's analyze the given pairs:
- [tex]\( x = 6 \)[/tex] is paired with [tex]\( y = 6 \)[/tex]
- [tex]\( x = 3 \)[/tex] is paired with [tex]\( y = 8 \)[/tex]
- [tex]\( x = 9 \)[/tex] is paired with [tex]\( y = 12 \)[/tex]
- [tex]\( x = 7 \)[/tex] is paired with [tex]\( y = 8 \)[/tex]
- [tex]\( x = 3 \)[/tex] is paired with [tex]\( y = 6 \)[/tex] (This is an issue because [tex]\( 3 \)[/tex] appears twice with different [tex]\( y \)[/tex]-values)
To correct this and ensure that every [tex]\( x \)[/tex] value is unique, we need to remove one of the pairs where [tex]\( x = 3 \)[/tex] and replace it with a new pair that is not already in the table.
We’ll remove the last pair [tex]\((3, 6)\)[/tex] and replace it with a pair with a new [tex]\( x \)[/tex]-value which isn't 6, 3, 9, or 7, and any [tex]\( y \)[/tex]-value. Let's choose [tex]\( x = 8 \)[/tex] and to keep consistency, let's use a new [tex]\( y \)[/tex]-value, for instance, [tex]\( y = 10 \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
6 & 6 \\
\hline
3 & 8 \\
\hline
9 & 12 \\
\hline
7 & 8 \\
\hline
8 & 10 \\
\hline
\end{array}
\][/tex]
We now have a table where each [tex]\( x \)[/tex] value appears only once, ensuring it represents a function.
