Answer
$\text{Domain: }
(-\infty, \infty)
\\\text{Range: }
(-\infty,-1]
$
Work Step by Step
The black graph is the parent graph, $
f(x)=|x|
.$
The red graph is the graph of $
f(x)=-|x+2|-1
.$ It is the result of shifting the points of the parent graph $2$ units to the left, reflecting about the $x-$axis, and then shifting $1$ unit down.
The $x-$values of the given function, $
g(x)=-|x+2|-1
,$ can be any real number. The $y-$values are negative real numbers up to $-1$. Hence, the given function has the following characteristics:
\begin{array}{l}\require{cancel}
\text{Domain: }
(-\infty, \infty)
\\\text{Range: }
(-\infty,-1]
.\end{array}