Let e be the base of natural logarithm and let f: 1, 2, 3, 4 1, e, e^2, e^3 and g: 1, e, e^2, e^3 1, 1 2 , 1 3 , 1 4 be two bijective functions such that f is strictly decreasing and g is strictly increasing. If (x) = [f⁻¹ g⁻¹ ( 1 2 ) ]^x , then the area of the region R = (x, y): x^2 y (x), 0 x 1 is:

Step-by-step solution

Given f: 1, 2, 3, 4 1, e, e^2, e^3 is a strictly decreasing bijective function. Arranging the domain and codomain in increasing order, we get f(1) = e^3 , f(2) = e^2 , f(3) = e , and f(4) = 1 . Given g: 1, e, e^2, e^3 1, 1 2 , 1 3 , 1 4 is a strictly increasing bijective function. Arranging the domain and codomain in increasing order, we get g(1) = 1 4 , g(e) = 1 3 , g(e^2) = 1 2 , and g(e^3) = 1 . Evaluating (x) = [f⁻¹ g⁻¹ ( 1 2 ) ]^x : From the mapping of g , g⁻¹ ( 1 2 ) = e^2 . From the mapping of f , f⁻¹(e^2) =