To find the average rate of change of the function over the interval [tex]\( 1 \leq x \leq 3 \)[/tex], we can use the following steps:
1. Identify the function values at the endpoints of the interval:
- The value of the function at [tex]\( x = 1 \)[/tex] is [tex]\( f(1) = 8 \)[/tex].
- The value of the function at [tex]\( x = 3 \)[/tex] is [tex]\( f(3) = 32 \)[/tex].
2. Use the formula for the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Here, [tex]\( x_1 = 1 \)[/tex], [tex]\( x_2 = 3 \)[/tex], [tex]\( f(x_1) = 8 \)[/tex], and [tex]\( f(x_2) = 32 \)[/tex].
3. Plug the values into the formula:
[tex]\[
\text{Average rate of change} = \frac{32 - 8}{3 - 1}
\][/tex]
4. Simplify the expression:
[tex]\[
\text{Average rate of change} = \frac{24}{2} = 12
\][/tex]
Thus, the average rate of change of the function over the interval [tex]\( 1 \leq x \leq 3 \)[/tex] is [tex]\( 12 \)[/tex].
