Answer :
To determine whether each table represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to check if each input (or [tex]\( x \)[/tex] value) has exactly one output (or [tex]\( y \)[/tex] value). Remember, a function cannot have any [tex]\( x \)[/tex] value that corresponds to more than one [tex]\( y \)[/tex] value.
Analyzing each table:
1. Table 1:
[tex]\[
\begin{array}{|r|r|r|r|}
\hline
x & 5 & 6 & 6 \\
\hline
y & 1 & 9 & 13 \\
\hline
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex] value of 6 corresponds to two different [tex]\( y \)[/tex] values (9 and 13).
- This means for [tex]\( x = 6 \)[/tex], there are two outputs, which violates the rule of a function.
- Therefore, this table does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
2. Table 2:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & 5 & 6 & 15 \\
\hline
y & 1 & 9 & 9 \\
\hline
\end{array}
\][/tex]
- Here, each [tex]\( x \)[/tex] value (5, 6, and 15) has a single, unique [tex]\( y \)[/tex] value corresponding to it.
- There are no repetitions of [tex]\( x \)[/tex] with different [tex]\( y \)[/tex] values.
- Therefore, this table does represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
3. Table 3:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & 5 & 6 & 15 \\
\hline
y & 1 & 9 & 13 \\
\hline
\end{array}
\][/tex]
- Similarly, in this table, each [tex]\( x \)[/tex] value (5, 6, and 15) corresponds to exactly one [tex]\( y \)[/tex] value.
- No [tex]\( x \)[/tex] value is paired with multiple [tex]\( y \)[/tex] values.
- Thus, this table does represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Conclusion:
- Table 1: Does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
- Table 2: Represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
- Table 3: Represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
The correct selection of tables representing [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] is: Table 2 and Table 3.
Analyzing each table:
1. Table 1:
[tex]\[
\begin{array}{|r|r|r|r|}
\hline
x & 5 & 6 & 6 \\
\hline
y & 1 & 9 & 13 \\
\hline
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex] value of 6 corresponds to two different [tex]\( y \)[/tex] values (9 and 13).
- This means for [tex]\( x = 6 \)[/tex], there are two outputs, which violates the rule of a function.
- Therefore, this table does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
2. Table 2:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & 5 & 6 & 15 \\
\hline
y & 1 & 9 & 9 \\
\hline
\end{array}
\][/tex]
- Here, each [tex]\( x \)[/tex] value (5, 6, and 15) has a single, unique [tex]\( y \)[/tex] value corresponding to it.
- There are no repetitions of [tex]\( x \)[/tex] with different [tex]\( y \)[/tex] values.
- Therefore, this table does represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
3. Table 3:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & 5 & 6 & 15 \\
\hline
y & 1 & 9 & 13 \\
\hline
\end{array}
\][/tex]
- Similarly, in this table, each [tex]\( x \)[/tex] value (5, 6, and 15) corresponds to exactly one [tex]\( y \)[/tex] value.
- No [tex]\( x \)[/tex] value is paired with multiple [tex]\( y \)[/tex] values.
- Thus, this table does represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Conclusion:
- Table 1: Does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
- Table 2: Represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
- Table 3: Represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
The correct selection of tables representing [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] is: Table 2 and Table 3.
