Answer :
To determine which table represents a function, we need to check if every [tex]\( x \)[/tex] value (input) maps to exactly one [tex]\( y \)[/tex] value (output). Let's consider each table.
### 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 see that each [tex]\( x \)[/tex] value corresponds to one unique [tex]\( y \)[/tex] value:
- [tex]\( x = -3 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
- [tex]\( x = 8 \)[/tex] maps to [tex]\( y = 1 \)[/tex]
Since no [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table 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, the [tex]\( x = -5 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values:
- [tex]\( x = -5 \)[/tex] maps to [tex]\( y = -5 \)[/tex]
- [tex]\( x = -5 \)[/tex] maps to [tex]\( y = 5 \)[/tex]
Because an [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table 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, the [tex]\( x = -2 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values:
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = 2 \)[/tex]
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
Because an [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table 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, the [tex]\( x = -4 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values:
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 2 \)[/tex]
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
Because an [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table does not represent a function.
### Conclusion
Among the tables presented:
- Table 1 represents a function.
- Table 2, Table 3, and Table 4 do not represent functions.
### 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 see that each [tex]\( x \)[/tex] value corresponds to one unique [tex]\( y \)[/tex] value:
- [tex]\( x = -3 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
- [tex]\( x = 8 \)[/tex] maps to [tex]\( y = 1 \)[/tex]
Since no [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table 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, the [tex]\( x = -5 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values:
- [tex]\( x = -5 \)[/tex] maps to [tex]\( y = -5 \)[/tex]
- [tex]\( x = -5 \)[/tex] maps to [tex]\( y = 5 \)[/tex]
Because an [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table 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, the [tex]\( x = -2 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values:
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = 2 \)[/tex]
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
Because an [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table 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, the [tex]\( x = -4 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values:
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 2 \)[/tex]
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
Because an [tex]\( x \)[/tex] value is repeated with a different [tex]\( y \)[/tex] value, this table does not represent a function.
### Conclusion
Among the tables presented:
- Table 1 represents a function.
- Table 2, Table 3, and Table 4 do not represent functions.
