Answer
$f^{-1}(x)=-tan^{-1}(x+3)-1$
range of $f(x)$: $(-\infty,\infty)$,
domain and range of $f^{-1}(x)$: $(-\infty,\infty)$ and $(-1-\frac{\pi}{2}, \frac{\pi}{2}-1)$.
Work Step by Step
1. $f(x)=-tan(x+1)-3 \Longrightarrow y=-tan(x+1)-3 \Longrightarrow x=-tan(y+1)-3 \Longrightarrow y=tan^{-1}(-x-3)-1=-tan^{-1}(x+3)-1 \Longrightarrow f^{-1}(x)=-tan^{-1}(x+3)-1$
2. We can find the range of $f(x)$: $(-\infty,\infty)$,
3. We can find the domain and range of $f^{-1}(x)$: $(-\infty,\infty)$ and $(-1-\frac{\pi}{2}, \frac{\pi}{2}-1)$.
