To solve this, we need to find the average rate of change of the function $f(x) = \frac{1}{x + 5}$ on the interval $[2, 2 + h]$.

Average Rate of Change Formula:

The average rate of change of a function $f(x)$ over the interval $[x_1, x_2]$ is given by:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Here, $x_1 = 2$ and $x_2 = 2 + h$. So, we need to compute:

$\frac{f(2 + h) - f(2)}{(2 + h) - 2}$

This simplifies to:

$\frac{f(2 + h) - f(2)}{h}$

Step-by-step Solution:

1. Evaluate $f(2)$:

$f(2) = \frac{1}{2 + 5} = \frac{1}{7}$

2. Evaluate $f(2 + h)$:

$f(2 + h) = \frac{1}{(2 + h) + 5} = \frac{1}{h + 7}$

3. Substitute into the average rate of change formula:

$\text{Average Rate of Change} = \frac{\frac{1}{h + 7} - \frac{1}{7}}{h}$

4. Simplify the numerator:

To subtract the fractions, find a common denominator:

$\frac{1}{h + 7} - \frac{1}{7} = \frac{7 - (h + 7)}{7(h + 7)} = \frac{-h}{7(h + 7)}$

5. Substitute this back into the formula:

$\text{Average Rate of Change} = \frac{\frac{-h}{7(h + 7)}}{h}$

6. Simplify the expression:

$\text{Average Rate of Change} = \frac{-h}{7(h + 7)} \times \frac{1}{h} = \frac{-1}{7(h + 7)}$

Final Answer:

The average rate of change of $f(x)$ on the interval $[2, 2 + h]$ is:

$\boxed{\frac{-1}{7(h + 7)}}$

Would you like a further explanation or details on any step?

Here are some related questions you can explore:

1. How would the rate of change differ if we used another interval?
2. What does the negative sign in the rate of change represent geometrically?
3. How would this problem change if $f(x)$ was a polynomial instead of a rational function?
4. What is the instantaneous rate of change of $f(x)$ at $x = 2$?
5. Can you find the general formula for the average rate of change of $f(x) = \frac{1}{x + 5}$ over any interval?

Tip: The average rate of change can give insight into how steep or shallow the function's graph is between two points.