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Answer :
To determine which tables represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to understand that a function assigns exactly one output [tex]\( y \)[/tex] for each input [tex]\( x \)[/tex]. This means that for the table to represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], every [tex]\( x \)[/tex] value must connect to only one [tex]\( y \)[/tex] value.
Let's examine each of the given tables:
1. Table 1:
[tex]\[
\begin{array}{c|c|c|c}
x & 1 & 9 & 9 \\
\hline
y & 5 & 6 & 14 \\
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex]-value of 9 corresponds to two different [tex]\( y \)[/tex]-values (6 and 14). This means that Table 1 does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] because a single input [tex]\( x \)[/tex] has multiple outputs.
2. Table 2:
[tex]\[
\begin{array}{c|c|c|c}
x & 1 & 9 & 12 \\
\hline
y & 5 & 6 & 6 \\
\end{array}
\][/tex]
- In this table, every [tex]\( x \)[/tex]-value (1, 9, 12) corresponds to exactly one [tex]\( y \)[/tex]-value (5 for 1, 6 for 9 and 6 for 12). Thus, Table 2 does represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
3. Table 3:
[tex]\[
\begin{array}{c|c|c|c}
x & 1 & 9 & 12 \\
\hline
y & 5 & 6 & 14 \\
\end{array}
\][/tex]
- Here, each [tex]\( x \)[/tex]-value has a unique corresponding [tex]\( y \)[/tex]-value (5 for 1, 6 for 9, and 14 for 12). Therefore, Table 3 also represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Based on this analysis, the tables that represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] are Table 2 and Table 3.
Let's examine each of the given tables:
1. Table 1:
[tex]\[
\begin{array}{c|c|c|c}
x & 1 & 9 & 9 \\
\hline
y & 5 & 6 & 14 \\
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex]-value of 9 corresponds to two different [tex]\( y \)[/tex]-values (6 and 14). This means that Table 1 does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] because a single input [tex]\( x \)[/tex] has multiple outputs.
2. Table 2:
[tex]\[
\begin{array}{c|c|c|c}
x & 1 & 9 & 12 \\
\hline
y & 5 & 6 & 6 \\
\end{array}
\][/tex]
- In this table, every [tex]\( x \)[/tex]-value (1, 9, 12) corresponds to exactly one [tex]\( y \)[/tex]-value (5 for 1, 6 for 9 and 6 for 12). Thus, Table 2 does represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
3. Table 3:
[tex]\[
\begin{array}{c|c|c|c}
x & 1 & 9 & 12 \\
\hline
y & 5 & 6 & 14 \\
\end{array}
\][/tex]
- Here, each [tex]\( x \)[/tex]-value has a unique corresponding [tex]\( y \)[/tex]-value (5 for 1, 6 for 9, and 14 for 12). Therefore, Table 3 also represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Based on this analysis, the tables that represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] are Table 2 and Table 3.
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