Question

PROBLEMA TOP- POLINOMIOS ​

Answer 1

$g(x)=\dfrac{x+1}{x-1}\\ \\ \\gg(x)=\dfrac{g(x)+1}{g(x)-1}=x\\ \\ \\\{g^3\}(x)=g(x)=\dfrac{x+1}{x-1}\\ \\ \hspace*{2cm}\vdots\\ \\\{g^{2n+1}\}(x)=\dfrac{x+1}{x-1}$

$\dfrac{x+1}{x-1}=\dfrac{ax+b+1}{ax-b+1}\\ \\\\\dfrac{x+1}{x-1}-\dfrac{ax+b+1}{ax-b+1}=0\\ \\ \\\dfrac{(x+1)(ax-b+1)-(x-1)(ax+b+1)}{(x-1)(ax-b+1)}=0\\ \\ \\\dfrac{(a-b)x+1}{(x-1)(ax-b+1)}=0\\ \\ \\x=\dfrac{1}{b-a}~, x\notin\left\{1,\dfrac{b-1}{a}\right\}, \text{ etc}$