Sure, let's determine the average rate of change of a function [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 8]\)[/tex].
The average rate of change of a function [tex]\( f(x) \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[ \frac{f(b) - f(a)}{b - a} \][/tex]
In this case, the interval is [tex]\([2, 8]\)[/tex], so [tex]\(a = 2\)[/tex] and [tex]\(b = 8\)[/tex].
The expression for the average rate of change over the interval [tex]\([2, 8]\)[/tex] becomes:
[tex]\[ \frac{f(8) - f(2)}{8 - 2} \][/tex]
### Step-by-Step Solution:
1. Identify the given points in the interval: [tex]\(a = 2\)[/tex] and [tex]\(b = 8\)[/tex].
2. Write down the general formula for the average rate of change: [tex]\(\frac{f(b) - f(a)}{b - a}\)[/tex].
3. Substitute the specific values into the formula: [tex]\(\frac{f(8) - f(2)}{8 - 2}\)[/tex].
4. Calculate [tex]\(8 - 2 = 6\)[/tex].
5. The values of the function [tex]\(f\)[/tex] at the specific points are [tex]\(f(8) = 9\)[/tex] and [tex]\(f(2) = 2\)[/tex].
6. Compute [tex]\(f(8) - f(2) = 9 - 2 = 7\)[/tex].
7. Finally, the average rate of change is given by:
[tex]\[ \frac{7}{6} \][/tex]
Thus, the expression that determines the average rate of change over [tex]\([2, 8]\)[/tex] is [tex]\(\frac{f(8) - f(2)}{8 - 2}\)[/tex].
Given the options provided in the original question:
- [tex]\(f(9-2)\)[/tex]
- [tex]\(f(9)-f(2)\)[/tex]
Neither option directly gives us the average rate of change. The correct formula involves using [tex]\(\frac{f(8) - f(2)}{8 - 2}\)[/tex], so ensure that the expressions are appropriately understood in context.
However, the specific calculation shows that [tex]\(f(8) - f(2) = 7\)[/tex], evaluating the change in function values.
