1. **Problem:** Identify the vertical asymptote, horizontal asymptote, domain, and range of the function $$f(x) = -\frac{1}{x - 2} - 1$$ and sketch its graph. 2. **Vertical asymptote:** The vertical asymptote occurs where the denominator is zero, so solve $$x - 2 = 0$$ which gives $$x = 2$$. 3. **Horizontal asymptote:** For rational functions where the degree of numerator is less than denominator, the horizontal asymptote is $$y = $$ the constant outside the fraction, here $$y = -1$$. 4. **Domain:** All real numbers except where denominator is zero, so $$\{x \in \mathbb{R} : x \neq 2\}$$. 5. **Range:** The function approaches but never reaches the horizontal asymptote $$y = -1$$. Since the function is a transformed reciprocal function, the range is $$\{y \in \mathbb{R} : y \neq -1\}$$. 6. **Summary:** - Vertical asymptote: $$x = 2$$ - Horizontal asymptote: $$y = -1$$ - Domain: $$(-\infty, 2) \cup (2, \infty)$$ - Range: $$(-\infty, -1) \cup (-1, \infty)$$ 7. **Graph description:** The graph has two branches, one approaching the vertical asymptote at $$x=2$$ from left and right, and the horizontal asymptote at $$y=-1$$ from above and below.