Answer :
To determine which table represents a linear function, let's examine the data points in each table.
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 8 \\ \hline \end{array} \][/tex]
To check if this represents a linear function, we look at the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex] values:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 2 ([tex]\(\Delta y = 1\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 2 to 4 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 4 to 8 ([tex]\(\Delta y = 4\)[/tex])
The changes in [tex]\( y \)[/tex] values are not consistent (1, 2, 4), thus Table 1 does not represent a linear function.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 2 & 3 \\ \hline 3 & 6 \\ \hline \end{array} \][/tex]
Checking the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex]:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 0 to 1 ([tex]\(\Delta y = 1\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 3 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 3 to 6 ([tex]\(\Delta y = 3\)[/tex])
The changes in [tex]\( y \)[/tex] values are not consistent (1, 2, 3), thus Table 2 does not represent a linear function.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 2 & 0 \\ \hline 3 & 1 \\ \hline \end{array} \][/tex]
Checking the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex]:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 0 to 1 ([tex]\(\Delta y = 1\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 0 ([tex]\(\Delta y = -1\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 0 to 1 ([tex]\(\Delta y = 1\)[/tex])
The changes in [tex]\( y \)[/tex] values are not consistent (1, -1, 1), thus Table 3 does not represent a linear function.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 5 \\ \hline 3 & 7 \\ \hline \end{array} \][/tex]
Checking the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex]:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 3 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 3 to 5 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 5 to 7 ([tex]\(\Delta y = 2\)[/tex])
The changes in [tex]\( y \)[/tex] values are consistent (2, 2, 2), thus Table 4 represents a linear function.
### Conclusion:
Only Table 4 represents a linear function.
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 8 \\ \hline \end{array} \][/tex]
To check if this represents a linear function, we look at the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex] values:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 2 ([tex]\(\Delta y = 1\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 2 to 4 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 4 to 8 ([tex]\(\Delta y = 4\)[/tex])
The changes in [tex]\( y \)[/tex] values are not consistent (1, 2, 4), thus Table 1 does not represent a linear function.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 2 & 3 \\ \hline 3 & 6 \\ \hline \end{array} \][/tex]
Checking the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex]:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 0 to 1 ([tex]\(\Delta y = 1\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 3 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 3 to 6 ([tex]\(\Delta y = 3\)[/tex])
The changes in [tex]\( y \)[/tex] values are not consistent (1, 2, 3), thus Table 2 does not represent a linear function.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 2 & 0 \\ \hline 3 & 1 \\ \hline \end{array} \][/tex]
Checking the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex]:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 0 to 1 ([tex]\(\Delta y = 1\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 0 ([tex]\(\Delta y = -1\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 0 to 1 ([tex]\(\Delta y = 1\)[/tex])
The changes in [tex]\( y \)[/tex] values are not consistent (1, -1, 1), thus Table 3 does not represent a linear function.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 5 \\ \hline 3 & 7 \\ \hline \end{array} \][/tex]
Checking the changes in [tex]\( y \)[/tex] values compared to changes in [tex]\( x \)[/tex]:
- From [tex]\( x = 0 \to 1 \)[/tex]: [tex]\( y \)[/tex] changes from 1 to 3 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 1 \to 2 \)[/tex]: [tex]\( y \)[/tex] changes from 3 to 5 ([tex]\(\Delta y = 2\)[/tex])
- From [tex]\( x = 2 \to 3 \)[/tex]: [tex]\( y \)[/tex] changes from 5 to 7 ([tex]\(\Delta y = 2\)[/tex])
The changes in [tex]\( y \)[/tex] values are consistent (2, 2, 2), thus Table 4 represents a linear function.
### Conclusion:
Only Table 4 represents a linear function.
