Question: find the domain of the function:$\frac{x-4}{x^3-16x}$
1. **State the problem:** Find the domain of the function $$f(x) = \frac{x-4}{x^3 - 16x}$$.
2. **Recall the domain rule for rational functions:** The domain includes all real numbers except where the denominator is zero because division by zero is undefined.
3. **Set the denominator equal to zero and solve:**
$$x^3 - 16x = 0$$
4. **Factor the denominator:**
$$x(x^2 - 16) = 0$$
5. **Further factor the quadratic term:**
$$x(x - 4)(x + 4) = 0$$
6. **Find the roots:**
$$x = 0, x = 4, x = -4$$
7. **Exclude these values from the domain:**
The domain is all real numbers except $$x = 0, 4, -4$$.
8. **Write the domain in interval notation:**
$$(-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty)$$