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To determine which of the given items is not a function, we need to analyze each option to see if they all comply with the definition of a function. A function is defined as a relation where each input [tex]\( x \)[/tex] is related to exactly one output [tex]\( y \)[/tex].
### Option 1: The table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 3 \\ \hline 2 & 4 \\ \hline 2 & 5 \\ \hline 3 & 9 \\ \hline \end{tabular} \][/tex]
In this table, we can see that the input [tex]\( x = 2 \)[/tex] is associated with two different outputs ([tex]\( y = 4 \)[/tex] and [tex]\( y = 5 \)[/tex]). Therefore, this table does not meet the definition of a function because an input [tex]\( x \)[/tex] should map to only one output [tex]\( y \)[/tex].
### Option 2: The table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline -2 & 0 \\ \hline 1 & 3 \\ \hline 2 & 4 \\ \hline 3 & 7 \\ \hline \end{tabular} \][/tex]
In this table, each input [tex]\( x \)[/tex] is associated with exactly one output [tex]\( y \)[/tex]:
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = 1 \)[/tex] maps to [tex]\( y = 3 \)[/tex]
- [tex]\( x = 2 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 3 \)[/tex] maps to [tex]\( y = 7 \)[/tex]
Since no input is repeated with different outputs, this table represents a function.
### Option 3: The equation [tex]\( y = 3x^2 - 6x + 4 \)[/tex]
This equation is a quadratic function. For each input [tex]\( x \)[/tex], it will produce exactly one output [tex]\( y \)[/tex]. For any value of [tex]\( x \)[/tex] there will be only one corresponding value of [tex]\( y \)[/tex]. As such, this equation satisfies the definition of a function.
### Option 4: The set of points [tex]\(\{(3, 4), (6, 5), (7, 9), (9, 15)\}\)[/tex]
In this set of points, each [tex]\( x \)[/tex] value appears exactly once:
- [tex]\( x = 3 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 6 \)[/tex] maps to [tex]\( y = 5 \)[/tex]
- [tex]\( x = 7 \)[/tex] maps to [tex]\( y = 9 \)[/tex]
- [tex]\( x = 9 \)[/tex] maps to [tex]\( y = 15 \)[/tex]
Since each [tex]\( x \)[/tex] is associated with exactly one [tex]\( y \)[/tex], this set of points represents a function.
### Conclusion
From our analysis, the first table is the one that does not meet the criteria for being a function because the input [tex]\( x = 2 \)[/tex] corresponds to two different outputs ([tex]\( y = 4 \)[/tex] and [tex]\( y = 5 \)[/tex]). Therefore, the option that is not a function is:
[tex]\[ \boxed{1} \][/tex]
### Option 1: The table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 3 \\ \hline 2 & 4 \\ \hline 2 & 5 \\ \hline 3 & 9 \\ \hline \end{tabular} \][/tex]
In this table, we can see that the input [tex]\( x = 2 \)[/tex] is associated with two different outputs ([tex]\( y = 4 \)[/tex] and [tex]\( y = 5 \)[/tex]). Therefore, this table does not meet the definition of a function because an input [tex]\( x \)[/tex] should map to only one output [tex]\( y \)[/tex].
### Option 2: The table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline -2 & 0 \\ \hline 1 & 3 \\ \hline 2 & 4 \\ \hline 3 & 7 \\ \hline \end{tabular} \][/tex]
In this table, each input [tex]\( x \)[/tex] is associated with exactly one output [tex]\( y \)[/tex]:
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = 1 \)[/tex] maps to [tex]\( y = 3 \)[/tex]
- [tex]\( x = 2 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 3 \)[/tex] maps to [tex]\( y = 7 \)[/tex]
Since no input is repeated with different outputs, this table represents a function.
### Option 3: The equation [tex]\( y = 3x^2 - 6x + 4 \)[/tex]
This equation is a quadratic function. For each input [tex]\( x \)[/tex], it will produce exactly one output [tex]\( y \)[/tex]. For any value of [tex]\( x \)[/tex] there will be only one corresponding value of [tex]\( y \)[/tex]. As such, this equation satisfies the definition of a function.
### Option 4: The set of points [tex]\(\{(3, 4), (6, 5), (7, 9), (9, 15)\}\)[/tex]
In this set of points, each [tex]\( x \)[/tex] value appears exactly once:
- [tex]\( x = 3 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 6 \)[/tex] maps to [tex]\( y = 5 \)[/tex]
- [tex]\( x = 7 \)[/tex] maps to [tex]\( y = 9 \)[/tex]
- [tex]\( x = 9 \)[/tex] maps to [tex]\( y = 15 \)[/tex]
Since each [tex]\( x \)[/tex] is associated with exactly one [tex]\( y \)[/tex], this set of points represents a function.
### Conclusion
From our analysis, the first table is the one that does not meet the criteria for being a function because the input [tex]\( x = 2 \)[/tex] corresponds to two different outputs ([tex]\( y = 4 \)[/tex] and [tex]\( y = 5 \)[/tex]). Therefore, the option that is not a function is:
[tex]\[ \boxed{1} \][/tex]
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