To answer this question, we need to determine and compare the average rates of change for both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] over the interval [tex]\([3, 5]\)[/tex].
### Average Rate of Change of [tex]\( f \)[/tex]
The table provides the values of the function [tex]\( f \)[/tex] at specific points. We have the following data for the interval [tex]\([3, 5]\)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & 3 & 4 & 5 \\
\hline
f(x) & 59 & 63 & 71 \\
\hline
\end{array}
\][/tex]
To determine the average rate of change of [tex]\( f \)[/tex] over the interval [tex]\([3, 5]\)[/tex], use the formula:
[tex]\[
\text{Average rate of change of } f = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
where [tex]\( x_1 = 3 \)[/tex] and [tex]\( x_2 = 5 \)[/tex].
Thus,
[tex]\[
\text{Average rate of change of } f = \frac{f(5) - f(3)}{5 - 3} = \frac{71 - 59}{5 - 3} = \frac{12}{2} = 6.0
\][/tex]
### Average Rate of Change of [tex]\( g \)[/tex]
Function [tex]\( g \)[/tex] is an exponential function passing through the points [tex]\((3, -53)\)[/tex] and [tex]\((5, -41)\)[/tex].
To determine the average rate of change of [tex]\( g \)[/tex] over the interval [tex]\([3, 5]\)[/tex], use the same formula:
[tex]\[
\text{Average rate of change of } g = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\][/tex]
where [tex]\( x_1 = 3 \)[/tex], [tex]\( y_1 = g(3) = -53 \)[/tex], [tex]\( x_2 = 5 \)[/tex], and [tex]\( y_2 = g(5) = -41 \)[/tex].
Thus,
[tex]\[
\text{Average rate of change of } g = \frac{g(5) - g(3)}{5 - 3} = \frac{-41 - (-53)}{5 - 3} = \frac{12}{2} = 6.0
\][/tex]
### Comparison and Conclusion
The average rate of change of [tex]\( f \)[/tex] is 6.0, and the average rate of change of [tex]\( g \)[/tex] is also 6.0.
Therefore, the correct statement is:
[tex]\[
\boxed{\text{D. The average rate of change of } f \text{ is the same as the average rate of change of } g.}
\][/tex]
