[tex]\bf \begin{cases}
f(x)=\cfrac{2}{x-3}\\\\
g(x)=\cfrac{2}{x}+3
\end{cases}
\\\\\\
f(~~g(x)~~)=\cfrac{2}{g(x)-3}\implies f(~~g(x)~~)=\cfrac{2}{\frac{2}{x}+3 -3}
\\\\\\
f(~~g(x)~~)=\cfrac{2}{\frac{2}{x}}\implies f(~~g(x)~~)=\cfrac{\quad \frac{2}{1}\quad }{\frac{2}{x}}
\\\\\\
f(~~g(x)~~)=\cfrac{2}{1}\cdot \cfrac{x}{2}
\implies \boxed{f(~~g(x)~~)=x}[/tex]
[tex]\bf g(~~f(x)~~)=\cfrac{2}{f(x)}+3\implies g(~~f(x)~~)=\cfrac{2}{\frac{2}{x-3}}+3
\\\\\\
g(~~f(x)~~)=\cfrac{\frac{2}{1}}{\frac{2}{x-3}}+3\implies g(~~f(x)~~)=\cfrac{2}{1}\cdot \cfrac{x-3}{2}+3
\\\\\\
g(~~f(x)~~)=x-3+3\implies \boxed{g(~~f(x)~~)=x}[/tex]
