To determine which of the given tables represents a linear function, we need to examine whether the differences between consecutive \( y \)-values are consistent, indicating a constant rate of change, or slope.
### Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & \frac{1}{2} \\
\hline
2 & 1 \\
\hline
3 & \frac{3}{2} \\
\hline
4 & 2 \\
\hline
\end{array}
\][/tex]
The \( y \)-values are \(\frac{1}{2}, 1, \frac{3}{2}, 2 \).
Calculate the differences between consecutive \( y \)-values:
[tex]\[
\begin{array}{l}
1 - \frac{1}{2} = \frac{1}{2} \\
\frac{3}{2} - 1 = \frac{1}{2} \\
2 - \frac{3}{2} = \frac{1}{2} \\
\end{array}
\][/tex]
The differences are constant (\(\frac{1}{2}\)), indicating that Table 1 represents a linear function.
### Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & \frac{1}{1} \\
\hline
2 & \frac{1}{2} \\
\hline
3 & \frac{1}{3} \\
\hline
4 & \frac{1}{4} \\
\hline
\end{array}
\][/tex]
The \( y \)-values are \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \).
Calculate the differences between consecutive \( y \)-values:
[tex]\[
\begin{array}{l}
\frac{1}{2} - 1 \neq \text{constant} \\
\frac{1}{3} - \frac{1}{2} \neq \text{constant} \\
\frac{1}{4} - \frac{1}{3} \neq \text{constant}
\end{array}
\][/tex]
The differences are not consistent, indicating that Table 2 does not represent a linear function.
### Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 7 \\
\hline
2 & 9 \\
\hline
3 & 13 \\
\hline
4 & 21 \\
\hline
\end{array}
\][/tex]
The \( y \)-values are \(7, 9, 13, 21\).
Calculate the differences between consecutive \( y \)-values:
[tex]\[
\begin{array}{l}
9 - 7 = 2 \\
13 - 9 = 4 \\
21 - 13 = 8
\end{array}
\][/tex]
The differences are not constant, indicating that Table 3 does not represent a linear function.
### Table 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 0 \\
\hline
2 & 6 \\
\hline
3 & 16 \\
\hline
4 & 30 \\
\hline
\end{array}
\][/tex]
The \( y \)-values are \(0, 6, 16, 30\).
Calculate the differences between consecutive \( y \)-values:
[tex]\[
\begin{array}{l}
6 - 0 = 6 \\
16 - 6 = 10 \\
30 - 16 = 14
\end{array}
\][/tex]
The differences are not constant, indicating that Table 4 does not represent a linear function.
### Conclusion:
Upon examining all the tables, Table 1 is the only one that has consistent differences between the \( y \)-values. Therefore, Table 1 represents a linear function.
Thus, the table that represents a linear function is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & \frac{1}{2} \\
\hline
2 & 1 \\
\hline
3 & \frac{3}{2} \\
\hline
4 & 2 \\
\hline
\end{array}
\][/tex]
