1. The problem involves evaluating and analyzing the rational function $$f(x) = \frac{2x - 1}{x - 3}$$ and understanding its behavior. 2. The formula for a rational function is $$f(x) = \frac{P(x)}{Q(x)}$$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. 3. Important rules: - The domain excludes values where the denominator is zero. - Vertical asymptotes occur where the denominator is zero. - Horizontal asymptotes depend on the degrees of numerator and denominator. 4. Find the domain: $$x - 3 \neq 0 \implies x \neq 3$$ 5. Evaluate $f(1)$: $$f(1) = \frac{2(1) - 1}{1 - 3} = \frac{2 - 1}{-2} = \frac{1}{-2} = -\frac{1}{2}$$ 6. Check if the point $(3,9)$ lies on the graph: Since $x=3$ is not in the domain, $f(3)$ is undefined, so $(3,9)$ is not on the graph. 7. Analyze asymptotes: - Vertical asymptote at $x=3$. - Degree numerator = 1, degree denominator = 1, so horizontal asymptote is ratio of leading coefficients: $$y = \frac{2}{1} = 2$$ 8. Summary: - Domain: $x \in \mathbb{R}, x \neq 3$ - Vertical asymptote: $x=3$ - Horizontal asymptote: $y=2$ - $f(1) = -\frac{1}{2}$ Final answer: $$f(1) = -\frac{1}{2}$$