Answer
The vertical asymptote moves to $x=-4$.
The horizontal asymptote remains $y=0.$
The domain changes to $(-\infty,-4)\cup(-4,\infty)$
The range remains $(-\infty,0)\cup(0,\infty)$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/1132523e-9081-4568-af81-96dc599bd26a/result_image/1573944445.png?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240617%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240617T160113Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=fd68114f52f2766e23bd8aaca75e67fde0f8d05d3cc32e624c1b79e448224058)
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.
$r(x)=\displaystyle \frac{1}{x+4}=f(x+4),$
so its graph (blue solid line) is obtained from the graph of f by shifting it to the left by 4 units.
The vertical asymptote moves to $x=-4$.
The horizontal asymptote remains $y=0.$
The domain changes to $(-\infty,-4)\cup(-4,\infty)$
The range remains $(-\infty,0)\cup(0,\infty)$