Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

Answer 1

$f(x)= \frac{x-9}{x+5}$
$g(x)= \frac{-5x-9}{x-1}$
$f(g(x))= \frac{ \frac{-5x-9}{x-1} -9}{ \frac{-5x-9}{x-1}+5 }$
$f(g(x))= \frac{-5x-9-9(x-1)}{-5x-9+5(x-1)}$
$f(g(x))= \frac{-5x-9-9x+9}{-5x-9+5x-5}$
$f(g(x))= \frac{-14x}{-14}=x$
$g(f(x))= \frac{-5 \frac{x-9}{x+5}-9 }{ \frac{x-9}{x+5}-1 }$
$g(f(x))= \frac{-5x+45-9(x+5)}{x-9-1(x+5)}$
$g(f(x))= \frac{-5x+45-9x-45}{x-9-x-5}$
$g(f(x))= \frac{-14x}{-14}=x$