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