To determine whether the given table represents a linear or nonlinear function, we need to examine the relationship between the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values by calculating the rate of change, which is essentially the slope between consecutive points.
The table is as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & \frac{1}{9} \\
\hline
-1 & \frac{1}{3} \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
\end{array}
\][/tex]
To find the rate of change between each pair of points:
1. Calculate the change in [tex]\(y\)[/tex] values divided by the change in [tex]\(x\)[/tex] values for each pair of consecutive points.
[tex]\[
\Delta y = y_{n+1} - y_n
\][/tex]
[tex]\[
\Delta x = x_{n+1} - x_n
\][/tex]
[tex]\[
\text{rate of change} = \frac{\Delta y}{\Delta x}
\][/tex]
Let's compute this step-by-step for each pair of points:
### Between [tex]\(x = -2\)[/tex] and [tex]\(x = -1\)[/tex]:
[tex]\[
\Delta y = \frac{1}{3} - \frac{1}{9} = \frac{3}{9} - \frac{1}{9} = \frac{2}{9}
\][/tex]
[tex]\[
\Delta x = -1 - (-2) = 1
\][/tex]
[tex]\[
\text{rate of change} = \frac{2/9}{1} = \frac{2}{9}
\][/tex]
### Between [tex]\(x = -1\)[/tex] and [tex]\(x = 0\)[/tex]:
[tex]\[
\Delta y = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\][/tex]
[tex]\[
\Delta x = 0 - (-1) = 1
\][/tex]
[tex]\[
\text{rate of change} = \frac{2/3}{1} = \frac{2}{3}
\][/tex]
### Between [tex]\(x = 0\)[/tex] and [tex]\(x = 1\)[/tex]:
[tex]\[
\Delta y = 3 - 1 = 2
\][/tex]
[tex]\[
\Delta x = 1 - 0 = 1
\][/tex]
[tex]\[
\text{rate of change} = \frac{2}{1} = 2
\][/tex]
### Between [tex]\(x = 1\)[/tex] and [tex]\(x = 2\)[/tex]:
[tex]\[
\Delta y = 9 - 3 = 6
\][/tex]
[tex]\[
\Delta x = 2 - 1 = 1
\][/tex]
[tex]\[
\text{rate of change} = \frac{6}{1} = 6
\][/tex]
As seen from these calculations, the rate of change is not constant.
Since the rate of change is not constant, the given table does not represent a linear function. Therefore, the correct statement is:
The table represents a nonlinear function because the rate of change is not constant.
