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To determine which table represents a function, we need to understand that a function associates each input [tex]\( x \)[/tex] with exactly one output [tex]\( y \)[/tex]. In other words, no [tex]\( x \)[/tex] value should map to multiple [tex]\( y \)[/tex] values.
Let's examine each table step by step:
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & -1 \\ \hline 0 & 0 \\ \hline -2 & -1 \\ \hline 8 & 1 \\ \hline \end{array} \][/tex]
In Table 1, we have the following pairs: [tex]\((-3, -1)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((-2, -1)\)[/tex], [tex]\((8, 1)\)[/tex]. None of the [tex]\( x \)[/tex] values are repeated, so each [tex]\( x \)[/tex] maps to exactly one [tex]\( y \)[/tex]. Therefore, Table 1 represents a function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & -5 \\ \hline 0 & 0 \\ \hline -5 & 5 \\ \hline 6 & -6 \\ \hline \end{array} \][/tex]
In Table 2, we have the pairs: [tex]\((-5, -5)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((-5, 5)\)[/tex], [tex]\((6, -6)\)[/tex]. The [tex]\( x \)[/tex] value [tex]\(-5\)[/tex] is repeated and it maps to two different [tex]\( y \)[/tex] values ([tex]\(-5\)[/tex] and [tex]\( 5 \)[/tex]). Therefore, Table 2 does not represent a function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & 8 \\ \hline -2 & 2 \\ \hline -2 & 4 \\ \hline 0 & 2 \\ \hline \end{array} \][/tex]
In Table 3, we have the pairs: [tex]\((-4, 8)\)[/tex], [tex]\((-2, 2)\)[/tex], [tex]\((-2, 4)\)[/tex], [tex]\((0, 2)\)[/tex]. The [tex]\( x \)[/tex] value [tex]\(-2\)[/tex] is repeated and it maps to two different [tex]\( y \)[/tex] values ([tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex]). Therefore, Table 3 does not represent a function.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & 2 \\ \hline 3 & 5 \\ \hline 1 & 3 \\ \hline -4 & 0 \\ \hline \end{array} \][/tex]
In Table 4, we have the pairs: [tex]\((-4, 2)\)[/tex], [tex]\((3, 5)\)[/tex], [tex]\((1, 3)\)[/tex], [tex]\((-4, 0)\)[/tex]. The [tex]\( x \)[/tex] value [tex]\(-4\)[/tex] is repeated and it maps to two different [tex]\( y \)[/tex] values ([tex]\( 2 \)[/tex] and [tex]\( 0 \)[/tex]). Therefore, Table 4 does not represent a function.
Considering the above analysis, only Table 1 represents a function. Thus, the first table is the one that represents a function.
[tex]\[ \boxed{0} \][/tex]
Let's examine each table step by step:
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & -1 \\ \hline 0 & 0 \\ \hline -2 & -1 \\ \hline 8 & 1 \\ \hline \end{array} \][/tex]
In Table 1, we have the following pairs: [tex]\((-3, -1)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((-2, -1)\)[/tex], [tex]\((8, 1)\)[/tex]. None of the [tex]\( x \)[/tex] values are repeated, so each [tex]\( x \)[/tex] maps to exactly one [tex]\( y \)[/tex]. Therefore, Table 1 represents a function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & -5 \\ \hline 0 & 0 \\ \hline -5 & 5 \\ \hline 6 & -6 \\ \hline \end{array} \][/tex]
In Table 2, we have the pairs: [tex]\((-5, -5)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((-5, 5)\)[/tex], [tex]\((6, -6)\)[/tex]. The [tex]\( x \)[/tex] value [tex]\(-5\)[/tex] is repeated and it maps to two different [tex]\( y \)[/tex] values ([tex]\(-5\)[/tex] and [tex]\( 5 \)[/tex]). Therefore, Table 2 does not represent a function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & 8 \\ \hline -2 & 2 \\ \hline -2 & 4 \\ \hline 0 & 2 \\ \hline \end{array} \][/tex]
In Table 3, we have the pairs: [tex]\((-4, 8)\)[/tex], [tex]\((-2, 2)\)[/tex], [tex]\((-2, 4)\)[/tex], [tex]\((0, 2)\)[/tex]. The [tex]\( x \)[/tex] value [tex]\(-2\)[/tex] is repeated and it maps to two different [tex]\( y \)[/tex] values ([tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex]). Therefore, Table 3 does not represent a function.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & 2 \\ \hline 3 & 5 \\ \hline 1 & 3 \\ \hline -4 & 0 \\ \hline \end{array} \][/tex]
In Table 4, we have the pairs: [tex]\((-4, 2)\)[/tex], [tex]\((3, 5)\)[/tex], [tex]\((1, 3)\)[/tex], [tex]\((-4, 0)\)[/tex]. The [tex]\( x \)[/tex] value [tex]\(-4\)[/tex] is repeated and it maps to two different [tex]\( y \)[/tex] values ([tex]\( 2 \)[/tex] and [tex]\( 0 \)[/tex]). Therefore, Table 4 does not represent a function.
Considering the above analysis, only Table 1 represents a function. Thus, the first table is the one that represents a function.
[tex]\[ \boxed{0} \][/tex]
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