Question

Para cada n° real x ≠ 1, define-se f(x) = x/(x - 1). Então f(f(x)) é sempre igual a:

Answer 1

Bom dia Eric!

Solução!

Função f(x) composta com f(x).

$f(x)= \dfrac{x}{x-1}\\\\\\\ f(f(x))\\\\\\ f(x)= \dfrac{x}{x-1}\\\\\\f(f(x))= \dfrac{ \dfrac{x}{x-1} }{ \dfrac{x}{x-1} -1}\\\\\\\ f(f(x))=\dfrac{ \dfrac{x}{x-1} }{ \dfrac{x-1(x-1)}{x-1} }\\\\\\\ f(f(x))=\dfrac{ \dfrac{x}{x-1} }{ \dfrac{x+(-x+1}{x-1} }\\\\\\\ f(f(x))=\dfrac{ \dfrac{x}{x-1} }{ \dfrac{x-x+1)}{x-1} }\\\\\\\ f(f(x))=\dfrac{ \dfrac{x}{x-1} }{ \dfrac{1}{x-1} }\\\\\\\ f(f(x))=\dfrac{ x }{x-1}\times \frac{x-1}{1} \\\\\\\ f(f(x))=\dfrac{ x }{1} \\\\\\\ f(f(x))=x$

$\boxed{Resposta: Sempre~~sera~~f(f(x))=x}$

Bom dia!
Bons estudos!