1. **State the problem:** Find the domain of the function $$f(x) = \frac{1}{\frac{12}{x-5} - 2}$$. 2. **Recall domain rules:** The domain of a function includes all real numbers except where the function is undefined. 3. **Identify restrictions:** The function is undefined where the denominator is zero or where any denominator inside the expression is zero. 4. **Check inner denominator:** The term $$\frac{12}{x-5}$$ requires $$x-5 \neq 0$$, so $$x \neq 5$$. 5. **Set the outer denominator not equal to zero:** $$\frac{12}{x-5} - 2 \neq 0$$ 6. **Solve for $$x$$:** $$\frac{12}{x-5} \neq 2$$ Multiply both sides by $$x-5$$ (not zero): $$\cancel{(x-5)} \cdot \frac{12}{\cancel{(x-5)}} \neq 2 \cdot (x-5)$$ $$12 \neq 2x - 10$$ 7. **Rearrange:** $$2x - 10 \neq 12$$ $$2x \neq 22$$ $$x \neq 11$$ 8. **Combine restrictions:** $$x \neq 5$$ and $$x \neq 11$$. 9. **Write domain in interval notation:** $$(-\infty, 5) \cup (5, 11) \cup (11, \infty)$$. **Final answer:** The domain of $$f(x)$$ is $$(-\infty, 5) \cup (5, 11) \cup (11, \infty)$$.