Let f: R R be a function defined by f(x) = 2x^3 - 3x^2 + 2x , and let g(x) be the inverse function of f(x) . Then the value of 2 _ k=1 ³⁹ g ( k 40 ) is equal to

Step-by-step solution

Given f(x) = 2x^3 - 3x^2 + 2x . Let us evaluate f(x) + f(1-x) : f(1-x) = 2(1-x)^3 - 3(1-x)^2 + 2(1-x) = 2(1 - 3x + 3x^2 - x^3) - 3(1 - 2x + x^2) + 2 - 2x = 2 - 6x + 6x^2 - 2x^3 - 3 + 6x - 3x^2 + 2 - 2x = -2x^3 + 3x^2 - 2x + 1 Adding f(x) and f(1-x) : f(x) + f(1-x) = (2x^3 - 3x^2 + 2x) + (-2x^3 + 3x^2 - 2x + 1) = 1 Since g(x) is the inverse of f(x) , let f(x) = y x = g(y) . Substitute x = g(y) into the identity: y + f(1 - g(y)) = 1 f(1 - g(y)) = 1 - y Taking g on both sides: 1 - g(y) = g(1 - y) g(y) + g(1 - y) = 1 T