To find the average rate of change of the function $f(x) = \frac{x - 6}{x + 9}$ on the interval $[x, x+h]$, we use the following formula for the average rate of change of a function between two points:

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

Step 1: Find $f(x+h)$

Substitute $x+h$ into the function $f(x) = \frac{x - 6}{x + 9}$:

$f(x+h) = \frac{(x+h) - 6}{(x+h) + 9} = \frac{x + h - 6}{x + h + 9}$

Step 2: Compute the difference $f(x+h) - f(x)$

Now, compute the difference between $f(x+h)$ and $f(x)$:

$f(x+h) - f(x) = \frac{x + h - 6}{x + h + 9} - \frac{x - 6}{x + 9}$

To subtract these two fractions, we'll need a common denominator, which is $(x+h+9)(x+9)$:

$f(x+h) - f(x) = \frac{(x + h - 6)(x + 9) - (x - 6)(x + h + 9)}{(x+h+9)(x+9)}$

Step 3: Simplify the numerator

Now expand and simplify the numerator:

$(x + h - 6)(x + 9) = x^2 + hx - 6x + 9h - 54$ $(x - 6)(x + h + 9) = x^2 + hx - 6x + 9x - 6h - 54$

Now subtract the two expressions:

$(x^2 + hx - 6x + 9h - 54) - (x^2 + hx + 3x - 6h - 54)$

Simplify this:

$= (x^2 - x^2) + (hx - hx) + (-6x - 3x) + (9h - (-6h)) + (-54 + 54)$ $= -9x + 15h$

Step 4: Divide by $h$

Now, divide this by $h$, the difference $(x+h) - x = h$:

$\frac{f(x+h) - f(x)}{h} = \frac{-9x + 15h}{h(x+h+9)(x+9)}$

Simplify the fraction:

$= \frac{-9x + 15h}{h(x+h+9)(x+9)} = \frac{-9(x - \frac{5h}{3})}{(x+h+9)(x+9)}$

Thus, the average rate of change of $f(x) = \frac{x - 6}{x + 9}$ on the interval $[x, x+h]$ is:

$\boxed{\frac{-9x + 15h}{h(x+h+9)(x+9)}}$

Would you like more details on any step or a different approach?

Here are some related questions you might find interesting:

1. How does the average rate of change differ from the instantaneous rate of change?
2. What is the derivative of $f(x) = \frac{x - 6}{x + 9}$?
3. How would the answer change if the function was $f(x) = \frac{x + 6}{x - 9}$?
4. What is the significance of the common denominator in this computation?
5. How would this concept apply to real-world rate of change problems?

Tip: To solve problems involving rates of change, breaking down the equation step-by-step helps ensure accuracy and reduces complexity when simplifying expressions.