To determine the multiplicative rate of change of the function based on the given table, we need to calculate the ratio of successive [tex]\( y \)[/tex] values.
Here is the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & \frac{3}{2} \\
\hline
2 & \frac{9}{8} \\
\hline
3 & \frac{27}{32} \\
\hline
4 & \frac{81}{128} \\
\hline
\end{array}
\][/tex]
We will calculate the ratios of each consecutive pair of [tex]\( y \)[/tex] values:
1. Calculate the ratio between [tex]\( y_2 \)[/tex] and [tex]\( y_1 \)[/tex]:
[tex]\[
\frac{y_2}{y_1} = \frac{\frac{9}{8}}{\frac{3}{2}} = \frac{9}{8} \times \frac{2}{3} = \frac{18}{24} = \frac{3}{4}
\][/tex]
2. Calculate the ratio between [tex]\( y_3 \)[/tex] and [tex]\( y_2 \)[/tex]:
[tex]\[
\frac{y_3}{y_2} = \frac{\frac{27}{32}}{\frac{9}{8}} = \frac{27}{32} \times \frac{8}{9} = \frac{216}{288} = \frac{3}{4}
\][/tex]
3. Calculate the ratio between [tex]\( y_4 \)[/tex] and [tex]\( y_3 \)[/tex]:
[tex]\[
\frac{y_4}{y_3} = \frac{\frac{81}{128}}{\frac{27}{32}} = \frac{81}{128} \times \frac{32}{27} = \frac{2592}{3456} = \frac{3}{4}
\][/tex]
Observing these ratios, we see that they are all the same. Therefore, the multiplicative rate of change is consistent.
Thus, the multiplicative rate of change for this function is [tex]\(\frac{3}{4}\)[/tex].
The correct answer is:
[tex]\(\frac{3}{4}\)[/tex]
