Answer
The vertical asymptote moves to $x=1$.
The horizontal asymptote remains $y=0.$
The domain changes to $(-\infty,1)\cup(1,\infty)$
The range remains $(-\infty,0)\cup(0,\infty)$
Work Step by Step
Let $f(x)=\displaystyle \frac{1}{x}$
The asymptotes are $x=0$ (vertical) and $y=0$ (horizontal).
The domain and range are both $\mathbb{R}/\{0\}=(-\infty,0)\cup(0,\infty)$
The graph and the asymptotes of f are graphed with red dashed lines.
$r(x)=\displaystyle \frac{1}{x-1}=f(x-1),$
so its graph (blue solid line) is obtained from the graph of f by shifting it to the right by 1 unit.
The vertical asymptote moves to $x=1$.
The horizontal asymptote remains $y=0.$
The domain changes to $(-\infty,1)\cup(1,\infty)$
The range remains $(-\infty,0)\cup(0,\infty)$
