[tex]\bf \begin{cases}
f(x)=\cfrac{2}{x}\\\\
g(x)=\cfrac{x}{x+2}
\end{cases}\qquad g[\quad f(x)\quad ]=\cfrac{[f(x)]}{[f(x)]+2}[/tex]
[tex]\bf (g\circ f)(x)\iff g[\quad f(x)\quad ]=\cfrac{\boxed{\frac{2}{x}}}{\boxed{\frac{2}{x}}+2}
\\\\\\
g[\quad f(x)\quad ]=\cfrac{\frac{2}{x}}{\frac{2x+2}{x}}\implies g[\quad f(x)\quad ]=\cfrac{2}{x}\cdot \cfrac{x}{2x+2}
\\\\\\
g[\quad f(x)\quad ]=\cfrac{2x}{2x^2+2x}\implies g[\quad f(x)\quad ]=\cfrac{2x}{2x(x+1)}
\\\\\\
g[\quad f(x)\quad ]=\cfrac{1}{x+1}[/tex]
