Define \(f(x)=\frac{1+x}{1-x}\) and \(g(x)=\frac{-2}{x+1}\). Find the value of \(g(f(g(f(\dotsb g(f(12)) \dotsb ))))\) where the function f is applied 8 times, and the function g is applied 8 times, alternating between the two.
+128407 1st iteration
f(12) = 13/-11 = -13/11
g (f(12)) = -2 / [ -13/11 + 1 ] = - 2 / [ -2/11] = 11
2nd iteration
f ( g ( f (12) )) = f (11) = 12 / - 10 = - 6/ 5
g ( f ( g ( f (12) )) ) = g (-6/5) = -2 / [ -6/5 + 1] = -2 / [ -1/5] = 10
Continuing like this
3rd iteration
f(10) = -11/9
g (-11/9) = -2 / [ -11/9 + 1 ] = -2 /[ -2/9] = 9
So...it appears that each iteration lessens the result by 1
So
11,10, 9, 8 ,7, 6, 5, 4
So....after 8 iterations....we get 4
