Question: Find the average rate of change for the function.
f(x) = \frac{1}{x - 4} between x = -3 and x = 3
The average rate of change is \boxed{\text{□}}.
(Type an integer or a simplified fraction.)
1. **State the problem:** We need to find the average rate of change of the function $$f(x) = \frac{1}{x - 4}$$ between $$x = -3$$ and $$x = 3$$.
2. **Recall the formula for average rate of change:**
$$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$
where $$a = -3$$ and $$b = 3$$.
3. **Calculate $$f(a)$$ and $$f(b)$$:**
$$f(-3) = \frac{1}{-3 - 4} = \frac{1}{-7} = -\frac{1}{7}$$
$$f(3) = \frac{1}{3 - 4} = \frac{1}{-1} = -1$$
4. **Substitute values into the formula:**
$$\frac{f(3) - f(-3)}{3 - (-3)} = \frac{-1 - \left(-\frac{1}{7}\right)}{3 + 3} = \frac{-1 + \frac{1}{7}}{6}$$
5. **Simplify the numerator:**
$$-1 + \frac{1}{7} = -\frac{7}{7} + \frac{1}{7} = -\frac{6}{7}$$
6. **Rewrite the expression:**
$$\frac{-\frac{6}{7}}{6}$$
7. **Divide by 6:**
$$= -\frac{6}{7} \times \frac{1}{6} = -\frac{6}{7 \times 6} = -\frac{6}{42}$$
8. **Simplify the fraction:**
$$-\frac{6}{42} = -\frac{\cancel{6}}{7 \times \cancel{6}} = -\frac{1}{7}$$
**Final answer:** The average rate of change is $$-\frac{1}{7}$$.