Chapter 6 - Section 6.2 - One-to-One Functions; Inverse Functions - 6.2 Assess Your Understanding - Page 421: 44

$f(g(x))=x,$ $g(f(x))=x$

$f(x)=\frac{x-5}{2x+3},$ $g(x)=\frac{3x+5}{1-2x},$ $f(g(x))=\frac{(\frac{3x+5}{1-2x})-5}{2(\frac{3x+5}{1-2x})+3}=\frac{13x}{13}=x,$ Domain of $f(g(x))$ is, $x \in \mathbb{R} \ne \{ \frac{1}{2}\}$ $g(f(x))=\frac{2(\frac{x-5}{2x+3})+5}{1-2(\frac{x-5}{2x+3})}=\frac{13x}{13}=x$ Domain of $f(g(x))$ is, $x \in \mathbb{R} \ne \{ -\frac{3}{2}\}$