Answer :
To determine which table represents a function, we need to verify that each \( x \) value maps to exactly one \( y \) value. In other words, there should be no repeated \( x \) values with different \( y \) values.
Let's analyze 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]
Analysis:
- \( x = -3 \) maps to \( y = -1 \).
- \( x = 0 \) maps to \( y = 0 \).
- \( x = -2 \) maps to \( y = -1 \).
- \( x = 8 \) maps to \( y = 1 \).
There are no repeated \( x \) values with different \( y \) values. 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]
Analysis:
- \( x = -5 \) maps to \( y = -5 \).
- \( x = 0 \) maps to \( y = 0 \).
- \( x = -5 \) maps to \( y = 5 \) (this is a repetition with a different \( y \) value from the first row).
- \( x = 6 \) maps to \( y = -6 \).
Since \( x = -5 \) maps to both \( y = -5 \) and \( y = 5 \), 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]
Analysis:
- \( x = -4 \) maps to \( y = 8 \).
- \( x = -2 \) maps to \( y = 2 \).
- \( x = -2 \) maps to \( y = 4 \) (this is a repetition with a different \( y \) value from the second row).
- \( x = 0 \) maps to \( y = 2 \).
Since \( x = -2 \) maps to both \( y = 2 \) and \( y = 4 \), 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]
Analysis:
- \( x = -4 \) maps to \( y = 2 \).
- \( x = 3 \) maps to \( y = 5 \).
- \( x = 1 \) maps to \( y = 3 \).
- \( x = -4 \) maps to \( y = 0 \) (this is a repetition with a different \( y \) value from the first row).
Since \( x = -4 \) maps to both \( y = 2 \) and \( y = 0 \), Table 4 does not represent a function.
### Conclusion
Table 1 is the only table where each \( x \) value maps to exactly one \( y \) value, thus Table 1 represents a function. The results for each table are as follows:
- Table 1: Function
- Table 2: Not a function
- Table 3: Not a function
- Table 4: Not a function
So, the correct answer is that Table 1 represents a function.
Let's analyze 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]
Analysis:
- \( x = -3 \) maps to \( y = -1 \).
- \( x = 0 \) maps to \( y = 0 \).
- \( x = -2 \) maps to \( y = -1 \).
- \( x = 8 \) maps to \( y = 1 \).
There are no repeated \( x \) values with different \( y \) values. 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]
Analysis:
- \( x = -5 \) maps to \( y = -5 \).
- \( x = 0 \) maps to \( y = 0 \).
- \( x = -5 \) maps to \( y = 5 \) (this is a repetition with a different \( y \) value from the first row).
- \( x = 6 \) maps to \( y = -6 \).
Since \( x = -5 \) maps to both \( y = -5 \) and \( y = 5 \), 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]
Analysis:
- \( x = -4 \) maps to \( y = 8 \).
- \( x = -2 \) maps to \( y = 2 \).
- \( x = -2 \) maps to \( y = 4 \) (this is a repetition with a different \( y \) value from the second row).
- \( x = 0 \) maps to \( y = 2 \).
Since \( x = -2 \) maps to both \( y = 2 \) and \( y = 4 \), 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]
Analysis:
- \( x = -4 \) maps to \( y = 2 \).
- \( x = 3 \) maps to \( y = 5 \).
- \( x = 1 \) maps to \( y = 3 \).
- \( x = -4 \) maps to \( y = 0 \) (this is a repetition with a different \( y \) value from the first row).
Since \( x = -4 \) maps to both \( y = 2 \) and \( y = 0 \), Table 4 does not represent a function.
### Conclusion
Table 1 is the only table where each \( x \) value maps to exactly one \( y \) value, thus Table 1 represents a function. The results for each table are as follows:
- Table 1: Function
- Table 2: Not a function
- Table 3: Not a function
- Table 4: Not a function
So, the correct answer is that Table 1 represents a function.
