Answer
$ f^{-1}(x)=sin^{-1}(\frac{x+5}{3})$
range of $f(x)$: $[-8,-2]$
domain and range of $f^{-1}(x)$: $[-8,-2]$ and $[-\frac{\pi}{2},\frac{\pi}{2}]$
Work Step by Step
1. $f(x)=3sin(x)-5\Longrightarrow y=3sin(x)-5\Longrightarrow x=3sin(y)-5 \Longrightarrow y=sin^{-1}(\frac{x+5}{3})\Longrightarrow f^{-1}(x)=sin^{-1}(\frac{x+5}{3})$
2. We can find the range of $f(x)$: $[-8,-2]$
3. We can find the domain and range of $f^{-1}(x)$: $[-8,-2]$ and $[-\frac{\pi}{2},\frac{\pi}{2}]$
