Answer
$f^{-1}(x)=cos^{-1}(x-1)-2$
range of $f(x)$: $[0,2]$,
domain and range of $f^{-1}(x)$: $[0,2]$ and $[-2, \pi-2]$.
Work Step by Step
1. $f(x)=cos(x+2)+1 \Longrightarrow y=cos(x+2)+1 \Longrightarrow x=cos(y+2)+1 \Longrightarrow y=cos^{-1}(x-1)-2 \Longrightarrow f^{-1}(x)=cos^{-1}(x-1)-2$
2. We can find the range of $f(x)$: $[0,2]$,
3. We can find the domain and range of $f^{-1}(x)$: $[0,2]$ and $[-2, \pi-2]$.