Let's explore how the function [tex]\( f(x) = 1.5^x \)[/tex] changes from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex].
1. Evaluate the function at [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 1.5^2 = (1.5 \times 1.5) = 2.25
\][/tex]
2. Evaluate the function at [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 1.5^3 = (1.5 \times 1.5 \times 1.5) = 3.375
\][/tex]
3. Calculate the absolute change from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[
\Delta f = f(3) - f(2) = 3.375 - 2.25 = 1.125
\][/tex]
4. Calculate the percentage change:
[tex]\[
\text{Percentage change} = \left( \frac{\Delta f}{f(2)} \right) \times 100 = \left( \frac{1.125}{2.25} \right) \times 100
\][/tex]
5. Simplify the fraction:
[tex]\[
\frac{1.125}{2.25} = \frac{1}{2} = 0.5
\][/tex]
6. Convert this ratio into a percentage:
[tex]\[
0.5 \times 100 = 50\%
\][/tex]
Therefore, [tex]\( f(x) \)[/tex] increases by [tex]\( 50\% \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]. Hence, the correct answer is:
[tex]\( f(x) \)[/tex] increases by [tex]\( 50\% \)[/tex] (choice b).
