To determine the multiplicative rate of change of the given exponential function represented by the table with [tex]\( x \)[/tex] values and corresponding [tex]\( y \)[/tex] values, we can follow these steps:
### Step 1: Identify the y-values from the table
The given table is:
[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]
### Step 2: Determine the multiplicative rates between consecutive y-values
To find the multiplicative rate of change, we calculate the ratio of consecutive [tex]\( y \)[/tex]-values:
1. Rate between the 1st and 2nd terms:
[tex]\[
\text{Rate}_{12} = \frac{y_2}{y_1} = \frac{\frac{9}{8}}{\frac{3}{2}} = \frac{9}{8} \times \frac{2}{3} = \frac{9 \times 2}{8 \times 3} = \frac{18}{24} = \frac{3}{4}
\][/tex]
2. Rate between the 2nd and 3rd terms:
[tex]\[
\text{Rate}_{23} = \frac{y_3}{y_2} = \frac{\frac{27}{32}}{\frac{9}{8}} = \frac{27}{32} \times \frac{8}{9} = \frac{27 \times 8}{32 \times 9} = \frac{216}{288} = \frac{3}{4}
\][/tex]
3. Rate between the 3rd and 4th terms:
[tex]\[
\text{Rate}_{34} = \frac{y_4}{y_3} = \frac{\frac{81}{128}}{\frac{27}{32}} = \frac{81}{128} \times \frac{32}{27} = \frac{81 \times 32}{128 \times 27} = \frac{2592}{3456} = \frac{3}{4}
\][/tex]
### Step 3: Calculate the average multiplicative rate of change
Since this is an exponential function, the consecutive multiplicative rates should be the same. Observe that each calculated rate is [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\text{Average Rate} = \frac{\text{Rate}_{12} + \text{Rate}_{23} + \text{Rate}_{34}}{3} = \frac{\frac{3}{4} + \frac{3}{4} + \frac{3}{4}}{3} = \frac{3 \times \frac{3}{4}}{3} = \frac{3}{4}
\][/tex]
### Conclusion
The multiplicative rate of change for the given exponential function is [tex]\(\frac{3}{4}\)[/tex]. Therefore, the correct answer is:
[tex]\(\boxed{\frac{3}{4}}\)[/tex]
