To determine the multiplicative rate of change of an exponential function based on given values, we first observe the pattern in the values of [tex]\(y\)[/tex] as [tex]\(x\)[/tex] changes. Here is the table we are given:
[tex]\[
\begin{tabular}{|c|c|}
\hline
x & y \\
\hline
1 & 2 \\
\hline
2 & \frac{2}{5} \\
\hline
3 & \frac{2}{25} \\
\hline
4 & \frac{2}{125} \\
\hline
\end{tabular}
\][/tex]
An exponential function has the form [tex]\(y = ab^x\)[/tex]. To find the multiplicative rate of change, we need to determine the base [tex]\(b\)[/tex] of the exponential function.
Given the function values:
- When [tex]\(x = 1\)[/tex], [tex]\(y = 2\)[/tex]
- When [tex]\(x = 2\)[/tex], [tex]\(y = \frac{2}{5}\)[/tex]
The multiplicative rate of change, [tex]\(b\)[/tex], can be found by considering the ratio of consecutive [tex]\(y\)[/tex]-values. Let's calculate the ratio between [tex]\(y\)[/tex]-values for [tex]\(x = 2\)[/tex] and [tex]\(x = 1\)[/tex]:
[tex]\[
\frac{\frac{2}{5}}{2}
\][/tex]
To simplify this ratio:
[tex]\[
\frac{\frac{2}{5}}{2} = \frac{2}{5} \times \frac{1}{2} = \frac{2}{5} \times \frac{1}{2} = \frac{2 \times 1}{5 \times 2} = \frac{2}{10} = \frac{1}{5}
\][/tex]
Therefore, the multiplicative rate of change of the function is [tex]\(\frac{1}{5}\)[/tex]. So, the correct answer is:
[tex]\(\frac{1}{5}\)[/tex]
