1. Statement of the problem: Identify which graph corresponds to the function $$f(x)=\frac{1}{x-2}$$. 2. Formula and rules: The parent reciprocal function is $$g(x)=\frac{1}{x}$$ and shifting it right by 2 units gives $$f(x)=g(x-2)=\frac{1}{x-2}$$. 3. Important rules: Vertical asymptotes occur where the denominator is zero. Set $x-2=0$ which gives $x=2$. Horizontal asymptote for a rational function with numerator degree less than denominator degree is $y=0$. 4. Intermediate work: The domain is $x\neq 2$. As $x\to 2^+$, $f(x)\to +\infty$. As $x\to 2^-$, $f(x)\to -\infty$. As $x\to \pm\infty$, $f(x)\to 0$. 5. Sign analysis and sample points: For $x=3$, $f(3)=\frac{1}{3-2}=1$ which is in the first quadrant. For $x=1$, $f(1)=\frac{1}{1-2}=-1$ which is below the x-axis and right of the y-axis (fourth quadrant). For $x=-1$, $f(-1)=\frac{1}{-1-2}=\frac{1}{-3}\approx -0.333$ which is in the third quadrant. 6. Conclusion: The graph has a vertical asymptote at $x=2$ and horizontal asymptote at $y=0$ and the hyperbola branches follow the standard reciprocal shape shifted right by 2, matching option (c). Final answer: (c)