Question

calculate the average rate of change for the function ( f(x) = 3^{x-1} ) on the interval 1, 3.
options: 9, 2, 4, 2.5

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 1 \), \( b = 3 \), and \( f(x)=3^{x - 1} \).

Step2: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1)=3^{1 - 1}=3^{0}=1 \)

Step3: Calculate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\( f(3)=3^{3 - 1}=3^{2}=9 \)

Step4: Apply the average rate of change formula

Using the formula \( \frac{f(3)-f(1)}{3 - 1} \), substitute the values of \( f(3) \) and \( f(1) \):
\( \frac{9 - 1}{3 - 1}=\frac{8}{2}=4 \)

Answer:

4