Answer :
Certainly! Let's determine which of these tables represents a function. Recall that a relation (table of [tex]\((x, y)\)[/tex] pairs) is a function if every [tex]\(x\)[/tex] value corresponds to exactly one [tex]\(y\)[/tex] value. In other words, for each [tex]\(x\)[/tex] value, there should be only one [tex]\(y\)[/tex] value associated with it.
Let’s analyze each table one by one.
### Table 1
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-3 & -1 \\ \hline 0 & 0 \\ \hline-2 & -1 \\ \hline 8 & 1 \\ \hline \end{tabular} \][/tex]
- x = -3 corresponds to y = -1
- x = 0 corresponds to y = 0
- x = -2 corresponds to y = -1
- x = 8 corresponds to y = 1
All [tex]\(x\)[/tex] values are unique. Therefore, Table 1 represents a function.
### Table 2
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-5 & -5 \\ \hline 0 & 0 \\ \hline-5 & 5 \\ \hline 6 & -6 \\ \hline \end{tabular} \][/tex]
- x = -5 corresponds to y = -5
- x = 0 corresponds to y = 0
- x = -5 corresponds to y = 5
- x = 6 corresponds to y = -6
The [tex]\(x\)[/tex] value -5 corresponds to both y = -5 and y = 5. Therefore, Table 2 does not represent a function.
### Table 3
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-4 & 8 \\ \hline-2 & 2 \\ \hline-2 & 4 \\ \hline 0 & 2 \\ \hline \end{tabular} \][/tex]
- x = -4 corresponds to y = 8
- x = -2 corresponds to y = 2
- x = -2 corresponds to y = 4
- x = 0 corresponds to y = 2
The [tex]\(x\)[/tex] value -2 corresponds to both y = 2 and y = 4. Therefore, Table 3 does not represent a function.
### Table 4
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-4 & 2 \\ \hline 3 & 5 \\ \hline 1 & 3 \\ \hline-4 & 0 \\ \hline \end{tabular} \][/tex]
- x = -4 corresponds to y = 2
- x = 3 corresponds to y = 5
- x = 1 corresponds to y = 3
- x = -4 corresponds to y = 0
The [tex]\(x\)[/tex] value -4 corresponds to both y = 2 and y = 0. Therefore, Table 4 does not represent a function.
### Conclusion
Only Table 1 represents a function, as it is the only table where each [tex]\(x\)[/tex] value is associated with exactly one [tex]\(y\)[/tex] value.
Let’s analyze each table one by one.
### Table 1
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-3 & -1 \\ \hline 0 & 0 \\ \hline-2 & -1 \\ \hline 8 & 1 \\ \hline \end{tabular} \][/tex]
- x = -3 corresponds to y = -1
- x = 0 corresponds to y = 0
- x = -2 corresponds to y = -1
- x = 8 corresponds to y = 1
All [tex]\(x\)[/tex] values are unique. Therefore, Table 1 represents a function.
### Table 2
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-5 & -5 \\ \hline 0 & 0 \\ \hline-5 & 5 \\ \hline 6 & -6 \\ \hline \end{tabular} \][/tex]
- x = -5 corresponds to y = -5
- x = 0 corresponds to y = 0
- x = -5 corresponds to y = 5
- x = 6 corresponds to y = -6
The [tex]\(x\)[/tex] value -5 corresponds to both y = -5 and y = 5. Therefore, Table 2 does not represent a function.
### Table 3
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-4 & 8 \\ \hline-2 & 2 \\ \hline-2 & 4 \\ \hline 0 & 2 \\ \hline \end{tabular} \][/tex]
- x = -4 corresponds to y = 8
- x = -2 corresponds to y = 2
- x = -2 corresponds to y = 4
- x = 0 corresponds to y = 2
The [tex]\(x\)[/tex] value -2 corresponds to both y = 2 and y = 4. Therefore, Table 3 does not represent a function.
### Table 4
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-4 & 2 \\ \hline 3 & 5 \\ \hline 1 & 3 \\ \hline-4 & 0 \\ \hline \end{tabular} \][/tex]
- x = -4 corresponds to y = 2
- x = 3 corresponds to y = 5
- x = 1 corresponds to y = 3
- x = -4 corresponds to y = 0
The [tex]\(x\)[/tex] value -4 corresponds to both y = 2 and y = 0. Therefore, Table 4 does not represent a function.
### Conclusion
Only Table 1 represents a function, as it is the only table where each [tex]\(x\)[/tex] value is associated with exactly one [tex]\(y\)[/tex] value.
