If g(x) = x^2 + x - 2 and 1 2 (g f)(x) = 2x^2 - 5x + 2 then f(x) is equal to

Step-by-step solution

Given 1 2 (g f)(x) = 2x^2 - 5x + 2 (g f)(x) = 4x^2 - 10x + 4 Since g(x) = x^2 + x - 2 , substituting f(x) in place of x gives: (f(x))^2 + f(x) - 2 = 4x^2 - 10x + 4 (f(x))^2 + f(x) - (4x^2 - 10x + 6) = 0 Solving for f(x) using the quadratic formula: f(x) = -1 1 - 4(1)(-(4x^2 - 10x + 6)) 2 f(x) = -1 16x^2 - 40x + 25 2 f(x) = -1 (4x - 5)^2 2 f(x) = -1 (4x - 5) 2 Taking the positive sign: f(x) = -1 + 4x - 5 2 = 2x - 3 Taking the negative sign: f(x) = -1 - (4x - 5) 2 = -2x + 2 From the given options, f(x) = 2x - 3 is th