To determine which of the given tables represents a non-linear function, we need to check the consistency of the differences in the [tex]\(y\)[/tex] values when the [tex]\(x\)[/tex] values increase by a constant amount. A linear function will exhibit a constant rate of change, i.e., the difference between consecutive [tex]\(y\)[/tex] values will be the same.
### Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
17 & 20 \\
\hline
18 & 19 \\
\hline
19 & 18 \\
\hline
20 & 17 \\
\hline
\end{array}
\][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[
\begin{align*}
19 - 20 &= -1 \\
18 - 19 &= -1 \\
17 - 18 &= -1
\end{align*}
\][/tex]
All the differences are consistent [tex]\((-1)\)[/tex], indicating this table represents a linear function.
### Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
17 & -16 \\
\hline
18 & -17 \\
\hline
19 & -18 \\
\hline
20 & -19 \\
\hline
\end{array}
\][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[
\begin{align*}
-17 - (-16) &= -1 \\
-18 - (-17) &= -1 \\
-19 - (-18) &= -1
\end{align*}
\][/tex]
All the differences are consistent [tex]\((-1)\)[/tex], indicating this table represents a linear function.
### Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
17 & 16 \\
\hline
18 & 17 \\
\hline
19 & 19 \\
\hline
20 & 20 \\
\hline
\end{array}
\][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[
\begin{align*}
17 - 16 &= 1 \\
19 - 17 &= 2 \\
20 - 19 &= 1
\end{align*}
\][/tex]
The differences [tex]\( (1, 2, 1) \)[/tex] are not consistent, indicating that this table does not represent a linear function.
### Conclusion:
By examining the differences in [tex]\(y\)[/tex] values, we determine that Table 3 does not represent a linear function. Therefore, the table representing a non-linear function is Table 3.
