Let _{$f\left(x\right)=\frac{e^x-e^{-x}}{2}$}  and if _{$f\left(g\left(x\right)\right)=x$}  then _{$g\left(\frac{e^{1002}-1}{2e^{501}}\right)=\_\_\_$}

_{$y=\frac{e^x-e^{-x}}{2}\Rightarrow e^{2x}-2y{e}^x-1=0$} put _$e^x=t$

_{$\begin{array}{c}\Rightarrow t^2-2yt-1=0\\ t=y\pm \sqrt{y^2+1}\\ x=\log \left(y+\sqrt{y^2+1}\right)\\ f^{-1}\left(x\right)=\log \left(x+\sqrt{x^2+1}\right)\end{array}$}

_{$\therefore g\left(\frac{e^{1002}-1}{2e^{501}}\right)=\log \left(\frac{e^{1002}-1}{2e^{501}}+\frac{e^{1002}+1}{2e^{501}}\right)$}

_{$=\log e^{501}=501$}