To determine which statement correctly compares the two functions on the interval [tex]\([0, 4]\)[/tex], let us analyze both functions.
### Function [tex]\( f \)[/tex]
Function [tex]\( f \)[/tex] is an exponential decay function with an initial value of 64, decreasing by 50% each unit increase in [tex]\( x \)[/tex]. This can be expressed as:
[tex]\[ f(x) = 64 \cdot (0.5)^x \][/tex]
We need to determine the values of [tex]\( f \)[/tex] and its average rate of decrease over the interval [tex]\([0, 4]\)[/tex]:
- [tex]\( f(0) \)[/tex] (initial value) = 64
- [tex]\( f(4) \)[/tex] (value at [tex]\( x = 4 \)[/tex]) = 64 \cdot (0.5)^4 = 64 \cdot 0.0625 = 4
Now, the average rate of decrease for function [tex]\( f \)[/tex] over the interval [tex]\([0, 4]\)[/tex] can be calculated as:
[tex]\[ \text{Average rate of } f = \frac{f(4) - f(0)}{4 - 0} = \frac{4 - 64}{4} = \frac{-60}{4} = -15 \][/tex]
### Function [tex]\( g \)[/tex]
The values of [tex]\( g \)[/tex] are given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 & 4 \\
\hline
g(x) & 75 & 43 & 27 & 19 & 15 \\
\hline
\end{array} \][/tex]
We need to determine the initial and final values of [tex]\( g \)[/tex] and its average rate of decrease over the interval [tex]\([0, 4]\)[/tex]:
- [tex]\( g(0) \)[/tex] (initial value) = 75
- [tex]\( g(4) \)[/tex] (value at [tex]\( x = 4 \)[/tex]) = 15
Now, the average rate of decrease for function [tex]\( g \)[/tex] over the interval [tex]\([0, 4]\)[/tex] can be calculated as:
[tex]\[ \text{Average rate of } g = \frac{g(4) - g(0)}{4 - 0} = \frac{15 - 75}{4} = \frac{-60}{4} = -15 \][/tex]
### Comparison
We observed that both functions are decreasing over the interval [tex]\([0, 4]\)[/tex] with the same average rate of decrease:
- [tex]\( f \)[/tex] decreases at an average rate of [tex]\(-15\)[/tex]
- [tex]\( g \)[/tex] decreases at an average rate of [tex]\(-15\)[/tex]
Hence, the correct statement is:
C. Both functions are decreasing at the same average rate on that interval.
