Let f:(1, ) R be a function defined as f(x) = x-1 x+1 . Let f^ i+1 (x) = f(f^i(x)) , i=1, 2, , 25 , where f^1(x)=f(x) . If g(x) + f²⁶(x) = 0 , x (1, ) , then the area of the region bounded by the curves y=g(x) , 2y=2x-3 , y=0 and x=4 is:

Step-by-step solution

Given f(x) = x-1 x+1 . Let us find the first few compositions of f(x) : f^2(x) = f(f(x)) = x-1 x+1 -1 x-1 x+1 +1 = x-1-x-1 x-1+x+1 = -2 2x = - 1 x f^3(x) = f(f^2(x)) = f (- 1 x ) = - 1 x -1 - 1 x +1 = -1-x -1+x = x+1 1-x f^4(x) = f(f^3(x)) = f ( x+1 1-x ) = x+1 1-x -1 x+1 1-x +1 = x+1-1+x x+1+1-x = 2x 2 = x Since f^4(x) = x , the sequence of functions is periodic with a period of 4 . Therefore, f²⁶(x) = f^ 4 6 + 2 (x) = f^2(x) = - 1 x . We are given g(x) + f²⁶(x) = 0 , which implies: g(x) - 1 x = 0 g(x) = 1 x We ne