To determine the rate of change of the function described in the table, we need to calculate the rate of change between each consecutive pair of points.
The formula for the rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Let's apply this formula to each consecutive pair of points provided in the table.
1. Between [tex]\( (-1, \frac{1}{10}) \)[/tex] and [tex]\( (0, \frac{1}{2}) \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{\frac{1}{2} - \frac{1}{10}}{0 - (-1)} = \frac{\frac{1}{2} - \frac{1}{10}}{1} = \frac{5/10 - 1/10}{1} = \frac{4/10}{1} = \frac{2/5} = 0.4
\][/tex]
2. Between [tex]\( (0, \frac{1}{2}) \)[/tex] and [tex]\( (1, \frac{5}{2}) \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{\frac{5}{2} - \frac{1}{2}}{1 - 0} = \frac{\frac{5}{2} - \frac{1}{2}}{1} = \frac{4/2}{1} = 2
\][/tex]
3. Between [tex]\( (1, \frac{5}{2}) \)[/tex] and [tex]\( (2, \frac{25}{2}) \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{\frac{25}{2} - \frac{5}{2}}{2 - 1} = \frac{\frac{25}{2} - \frac{5}{2}}{1} = \frac{20/2}{1} = 10
\][/tex]
4. Between [tex]\( (2, \frac{25}{2}) \)[/tex] and [tex]\( (3, \frac{125}{2}) \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{\frac{125}{2} - \frac{25}{2}}{3 - 2} = \frac{\frac{125}{2} - \frac{25}{2}}{1} = \frac{100/2}{1} = 50
\][/tex]
The calculated rates of change are:
[tex]\[
0.4, \quad 2, \quad 10, \quad 50
\][/tex]
Therefore, the rate of change values can be matched with the given options, but none of the exact provided options ([tex]\(\frac{12}{5}\)[/tex], 5, \(\frac{25}{2})\, and 25) match any of these calculated values.
Thus, the correct answer is none of the given options.
