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Select all of the following tables which represent [tex] y [/tex] as a function of [tex] x [/tex].

1.
[tex]
\[
\begin{array}{|c|c|c|c|}
\hline
x & 1 & 9 & 15 \\
\hline
y & 2 & 6 & 11 \\
\hline
\end{array}
\]
[/tex]

2.
[tex]
\[
\begin{array}{|c|c|c|c|}
\hline
x & 1 & 9 & 9 \\
\hline
y & 2 & 6 & 11 \\
\hline
\end{array}
\]
[/tex]

3.
[tex]
\[
\begin{array}{|c|c|c|c|}
\hline
x & 1 & 9 & 15 \\
\hline
y & 2 & 6 & 6 \\
\hline
\end{array}
\]
[/tex]

Answer :

To determine which tables represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to understand the basic definition of a function. A function assigns exactly one output [tex]\( y \)[/tex] for each input [tex]\( x \)[/tex].

Let's examine each table one by one:

1. First Table:
- [tex]\( x \)[/tex]: 1, 9, 15
- [tex]\( y \)[/tex]: 2, 6, 11

In this table, each [tex]\( x \)[/tex] value is associated with a unique [tex]\( y \)[/tex] value. For [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex]; for [tex]\( x = 9 \)[/tex], [tex]\( y = 6 \)[/tex]; and for [tex]\( x = 15 \)[/tex], [tex]\( y = 11 \)[/tex]. Hence, [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex] in this table.

2. Second Table:
- [tex]\( x \)[/tex]: 1, 9, 9
- [tex]\( y \)[/tex]: 2, 6, 11

Here, the value [tex]\( x = 9 \)[/tex] is associated with two different [tex]\( y \)[/tex] values: 6 and 11. This means that for [tex]\( x = 9 \)[/tex], there are two possible outputs, which violates the rule that each input must correspond to exactly one output in a function. Therefore, [tex]\( y \)[/tex] is not a function of [tex]\( x \)[/tex] in this table.

3. Third Table:
- [tex]\( x \)[/tex]: 1, 9, 15
- [tex]\( y \)[/tex]: 2, 6, 6

In this table, each [tex]\( x \)[/tex] value has exactly one [tex]\( y \)[/tex] value. For [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex]; for [tex]\( x = 9 \)[/tex], [tex]\( y = 6 \)[/tex]; and for [tex]\( x = 15 \)[/tex], [tex]\( y = 6 \)[/tex]. Even though the [tex]\( y \)[/tex] values for different [tex]\( x \)[/tex] values can be the same, what matters is the consistency of the output for each unique [tex]\( x \)[/tex]. So, [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex] in this table.

Based on this analysis, the first and third tables represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]. Therefore, the correct answer is tables 1 and 3.

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