Answer
$f(g(x))=x$
$g(f(x))=x$
Work Step by Step
$f(x)=\frac{2x+3}{x+4},$
$g(x)=\frac{4x-3}{2-x},$
$f(g(x))=\frac{2(\frac{4x-3}{2-x})+3}{\frac{4x-3}{2-x}+4}=\frac{8x-6+6-3x}{4x-3+8-4x}=\frac{5x}{5}=x$
Domain of $f(g(x))$ is $x\in \mathbb{R} \ne \{ 2\}$
$g(f(x))=\frac{4(\frac{2x+3}{x+4})-3}{2-(\frac{2x+3}{x+4})}=\frac{5x}{5}=x,$
Domain of $f(g(x))$ is $x\in \mathbb{R} \ne \{-4\}$
