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.
$r(x)=\displaystyle \frac{3}{x+1}=3\cdot f(x+1),$
so its graph (blue solid line) is obtained from the graph of f by
-shifting it to the left by $1$ unit, to obtain $f(x+1)$
and then,
- vertically stretching by a factor of 3.
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 $(-\infty,0)\cup(0,\infty)$
