Question

I need help combining functions!

Answer 1

`\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}`


`\bf (g\circ f)(x)\iff g[\quad f(x)\quad ]=\cfrac{\boxed{(2)/(x)}}{\boxed{(2)/(x)}+2} \\\\\\ g[\quad f(x)\quad ]=\cfrac{(2)/(x)}{(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}`