The inverse of the function $f\left(x\right)$ $=\frac{e^x-e^{-x}}{e^x+e^{-x}}+2$ is given by :

Let $y=f\left(x\right)$ $=\frac{e^x-e^{-x}}{e^x+e^{-x}}+2$ 

$\Rightarrow y-2=\frac{e^{2x}-1}{e^{2x}+1}$ $\Rightarrow (y-2)e^{2x}+\left(y-2\right)=e^{2x}-1$

$\Rightarrow (y-3)e^{2x}=-y+1$

$\Rightarrow e^{2x}=\frac{1-y}{y-3}$

$\Rightarrow e^{2x}=\frac{y-1}{3-y}$

$\Rightarrow 2x={\log }_e\left(\frac{y-1}{3-y}\right)$

$=f^{-1}\left(y\right)={\log }_e{\left(\frac{y-1}{3-y}\right)}^{\frac{1}{2}}\Rightarrow f^{-1}\left(x\right)={\log }_e{\left(\frac{x-1}{3-x}\right)}^{\frac{1}{2}}$