Question

What is f(g(x)) if f(x)is


`(2)/(x - 3)`
And g(x) is

`(2)/(x) + 3`
and also what is g(f(x))?

Answer 1

`\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}{(2)/(x)+3 -3} \\\\\\ f(~~g(x)~~)=\cfrac{2}{(2)/(x)}\implies f(~~g(x)~~)=\cfrac{\quad (2)/(1)\quad }{(2)/(x)} \\\\\\ f(~~g(x)~~)=\cfrac{2}{1}\cdot \cfrac{x}{2} \implies \boxed{f(~~g(x)~~)=x}`


`\bf g(~~f(x)~~)=\cfrac{2}{f(x)}+3\implies g(~~f(x)~~)=\cfrac{2}{(2)/(x-3)}+3 \\\\\\ g(~~f(x)~~)=\cfrac{(2)/(1)}{(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}`