Question

which graph represents the function $f(x) = \frac{2}{x - 1} + 3$?

Explanation:

Step1: Find vertical asymptote

Set denominator to 0: $x-1=0 \implies x=1$

Step2: Find horizontal asymptote

For rational functions, as $x\to\pm\infty$, $\frac{2}{x-1}\to0$, so $y=0+3=3$

Step3: Verify a point

Substitute $x=0$: $f(0)=\frac{2}{0-1}+3=-2+3=1$. The point $(0,1)$ lies on the top-left graph, which matches the asymptotes $x=1$ and $y=3$.

Answer:

The top-left graph (the first option)