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To determine which table represents a linear function, we need to examine each table and check if the rate of change (or the slope) between the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values is constant. A function is linear if there is a constant difference in [tex]\(y\)[/tex] values when [tex]\(x\)[/tex] values increase by a constant amount.
### Analyzing Each Table
Table 1: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{2} \\ \hline 2 & 1 \\ \hline 3 & 1 \frac{1}{2} \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(1 - \frac{1}{2} = \frac{1}{2}\)[/tex], [tex]\(1 \frac{1}{2} - 1 = \frac{1}{2}\)[/tex], [tex]\(2 - 1 \frac{1}{2} = \frac{1}{2}\)[/tex]
The differences (ratios) are consistent ([tex]\(\frac{1}{2}/1 = \frac{1}{2}\)[/tex]), so this table represents a linear function.
Table 2: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 1 \\ \hline 2 & \frac{1}{2} \\ \hline 3 & \frac{1}{3} \\ \hline 4 & \frac{1}{4} \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(\frac{1}{2} - 1 = -\frac{1}{2}\)[/tex], [tex]\(\frac{1}{3} - \frac{1}{2} = -\frac{1}{6}\)[/tex], [tex]\(\frac{1}{4} - \frac{1}{3} = -\frac{1}{12}\)[/tex]
The differences are not consistent, so this table does not represent a linear function.
Table 3: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 7 \\ \hline 2 & 9 \\ \hline 3 & 13 \\ \hline 4 & 21 \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(9 - 7 = 2\)[/tex], [tex]\(13 - 9 = 4\)[/tex], [tex]\(21 - 13 = 8\)[/tex]
The differences are not consistent, so this table does not represent a linear function.
Table 4: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 0 \\ \hline 2 & 6 \\ \hline 3 & 16 \\ \hline 4 & 30 \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(6 - 0 = 6\)[/tex], [tex]\(16 - 6 = 10\)[/tex], [tex]\(30 - 16 = 14\)[/tex]
The differences are not consistent, so this table does not represent a linear function.
### Conclusion
Based on the examination above, only the first table represents a linear function.
### Analyzing Each Table
Table 1: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{2} \\ \hline 2 & 1 \\ \hline 3 & 1 \frac{1}{2} \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(1 - \frac{1}{2} = \frac{1}{2}\)[/tex], [tex]\(1 \frac{1}{2} - 1 = \frac{1}{2}\)[/tex], [tex]\(2 - 1 \frac{1}{2} = \frac{1}{2}\)[/tex]
The differences (ratios) are consistent ([tex]\(\frac{1}{2}/1 = \frac{1}{2}\)[/tex]), so this table represents a linear function.
Table 2: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 1 \\ \hline 2 & \frac{1}{2} \\ \hline 3 & \frac{1}{3} \\ \hline 4 & \frac{1}{4} \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(\frac{1}{2} - 1 = -\frac{1}{2}\)[/tex], [tex]\(\frac{1}{3} - \frac{1}{2} = -\frac{1}{6}\)[/tex], [tex]\(\frac{1}{4} - \frac{1}{3} = -\frac{1}{12}\)[/tex]
The differences are not consistent, so this table does not represent a linear function.
Table 3: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 7 \\ \hline 2 & 9 \\ \hline 3 & 13 \\ \hline 4 & 21 \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(9 - 7 = 2\)[/tex], [tex]\(13 - 9 = 4\)[/tex], [tex]\(21 - 13 = 8\)[/tex]
The differences are not consistent, so this table does not represent a linear function.
Table 4: [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 0 \\ \hline 2 & 6 \\ \hline 3 & 16 \\ \hline 4 & 30 \\ \hline \end{array} \][/tex]
Differences in [tex]\(x\)[/tex]: [tex]\(2-1=1\)[/tex], [tex]\(3-2=1\)[/tex], [tex]\(4-3=1\)[/tex]
Differences in [tex]\(y\)[/tex]: [tex]\(6 - 0 = 6\)[/tex], [tex]\(16 - 6 = 10\)[/tex], [tex]\(30 - 16 = 14\)[/tex]
The differences are not consistent, so this table does not represent a linear function.
### Conclusion
Based on the examination above, only the first table represents a linear function.
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