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