To find the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 4 \leq x \leq 8 \)[/tex], we can use the formula for the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(x_{\text{end}}) - f(x_{\text{start}})}{x_{\text{end}} - x_{\text{start}}}
\][/tex]
In this problem, the interval given is from [tex]\( x = 4 \)[/tex] to [tex]\( x = 8 \)[/tex]. So, [tex]\( x_{\text{start}} = 4 \)[/tex] and [tex]\( x_{\text{end}} = 8 \)[/tex].
From the table, we find the corresponding [tex]\( f(x) \)[/tex] values:
- [tex]\( f(4) = 7 \)[/tex]
- [tex]\( f(8) = 43 \)[/tex]
Now, substituting these values into our formula:
[tex]\[
\text{Average rate of change} = \frac{f(8) - f(4)}{8 - 4}
\][/tex]
Substitute [tex]\( f(8) = 43 \)[/tex] and [tex]\( f(4) = 7 \)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{43 - 7}{8 - 4} = \frac{36}{4} = 9
\][/tex]
Therefore, the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 4 \leq x \leq 8 \)[/tex] is [tex]\( 9 \)[/tex].
