To make this table show a function, each [tex]\( x \)[/tex] value must map to a unique [tex]\( y \)[/tex] value. We need to add an ordered pair [tex]\((x, y)\)[/tex] without repeating any [tex]\( x \)[/tex]-values or introducing any repeated [tex]\( y \)[/tex]-values within the table. Here is the step-by-step process to find the correct ordered pair:
1. Initial [tex]\( x, y \)[/tex] pairs in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
6 & 6 \\
\hline
3 & 8 \\
\hline
9 & 12 \\
\hline
7 & 8 \\
\hline
\end{array}
\][/tex]
2. Identify the potential [tex]\( y \)[/tex]-values: The available [tex]\( y \)[/tex]-values to choose from are 3, 6, 7, 8, 9, and 12.
3. Examine current [tex]\( y \)[/tex]-values in the table:
[tex]\[
6, 8, 12, 8
\][/tex]
We see that [tex]\( 8 \)[/tex] is repeated and needs correction.
4. Choose [tex]\( y \)[/tex] value for [tex]\( x = 7 \)[/tex]: We can reassociate [tex]\( x = 7 \)[/tex] to an unused potential [tex]\( y \)[/tex]-value that does not repeat in the table.
The unused [tex]\( y \)[/tex]-values from the list are 3, 7, and 9. Among these values:
- [tex]\( 6 \)[/tex] is already in use.
- [tex]\( 8 \)[/tex] and [tex]\( 12 \)[/tex] are already used multiples in table.
- [tex]\( 7 \)[/tex] is not currently in the table, so we can assign:
[tex]\[
7 \rightarrow 3
\][/tex]
5. Updated table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
6 & 6 \\
\hline
3 & 8 \\
\hline
9 & 12 \\
\hline
7 & 3 \\
\hline
\end{array}
\][/tex]
6. Verify unique [tex]\( y \)[/tex]-values: The final list of [tex]\( y \)[/tex]-values should be unique:
[tex]\[
\begin{array}{|c|c|}
\hline
6, 8, 12, 3 \\
\hline
\end{array}
\][/tex]
Thus, by adding the ordered pair [tex]\((7, 3)\)[/tex], we ensure that the table shows a function with unique [tex]\( y \)[/tex]-values for each [tex]\( x \)[/tex]-value.
