To determine the percent rate of increase for the exponential function [tex]\( f(x) = 2.5(1.3)^x \)[/tex], we'll follow these steps:
1. Identify the growth factor:
The function [tex]\( f(x) = 2.5(1.3)^x \)[/tex] has the form [tex]\( f(x) = a(b)^x \)[/tex], where [tex]\( a \)[/tex] is the initial value and [tex]\( b \)[/tex] is the base of the exponent. Here, the base [tex]\( b \)[/tex] is 1.3.
2. Determine the growth factor's meaning:
The base of the exponent, 1.3, represents the growth factor. It tells us how much the function is multiplied by for each increase of 1 in [tex]\( x \)[/tex]. Since 1.3 is greater than 1, it indicates growth.
3. Calculate the percent rate of increase:
To convert the growth factor to a percentage, we subtract 1 from the growth factor and then multiply by 100. This is because 1 represents no change (100% of the original value), so any value above 1 indicates a percentage increase.
[tex]\[
\text{Percent Increase} = (1.3 - 1) \times 100
\][/tex]
4. Simplify the expression:
[tex]\[
\text{Percent Increase} = 0.3 \times 100 = 30
\][/tex]
Therefore, the percent rate of increase for the exponential function [tex]\( f(x) = 2.5(1.3)^x \)[/tex] is [tex]\( \boxed{30\%} \)[/tex].
Thus, the correct answer is:
D. [tex]\( 30 \% \)[/tex]
