Question

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

Explanation:

Step1: Find vertical asymptote

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

Step2: Find horizontal asymptote

As $x\to\pm\infty$, $\frac{1}{x-1}\to0$, so $f(x)\to0-2=-2$

Step3: Test a left-side point

Use $x=0$: $f(0)=\frac{1}{0-1}-2=-1-2=-3$

Step4: Test a right-side point

Use $x=2$: $f(2)=\frac{1}{2-1}-2=1-2=-1$

Step5: Match to graph

The top-right graph has vertical asymptote $x=1$, horizontal asymptote $y=-2$, passes through $(0,-3)$ and $(2,-1)$.

Answer:

The top-right graph (the second option)