To find the multiplicative rate of change of the given exponential function, we will analyze the ratio of consecutive [tex]\(y\)[/tex]-values at distinct [tex]\(x\)[/tex]-values. The rate of change should be constant for an exponential function.
First, let's extract the values from the table:
- When [tex]\(x = 1\)[/tex], [tex]\(y = 2\)[/tex]
- When [tex]\(x = 2\)[/tex], [tex]\(y = \frac{2}{5}\)[/tex]
- When [tex]\(x = 3\)[/tex], [tex]\(y = \frac{2}{25}\)[/tex]
- When [tex]\(x = 4\)[/tex], [tex]\(y = \frac{2}{125}\)[/tex]
Now let's determine the multiplicative rate of change by dividing each [tex]\(y\)[/tex]-value by its preceding [tex]\(y\)[/tex]-value:
1. From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]:
[tex]\[
\text{Rate} = \frac{\frac{2}{5}}{2} = \frac{2}{5} \times \frac{1}{2} = \frac{2}{10} = 0.2
\][/tex]
2. From [tex]\(x = 2\)[/tex] to [tex]\(x = 3\)[/tex]:
[tex]\[
\text{Rate} = \frac{\frac{2}{25}}{\frac{2}{5}} = \frac{2}{25} \times \frac{5}{2} = \frac{10}{50} = 0.2
\][/tex]
3. From [tex]\(x = 3\)[/tex] to [tex]\(x = 4\)[/tex]:
[tex]\[
\text{Rate} = \frac{\frac{2}{125}}{\frac{2}{25}} = \frac{2}{125} \times \frac{25}{2} = \frac{50}{250} = 0.2
\][/tex]
Since all the calculated rates are equal, the multiplicative rate of change of the function is:
[tex]\[
0.2
\][/tex]
This corresponds to [tex]\(\frac{1}{5}\)[/tex], because:
[tex]\[
0.2 = \frac{1}{5}
\][/tex]
Therefore, the multiplicative rate of change of the function is [tex]\(\boxed{\frac{1}{5}}\)[/tex].
