Problem 9 - Entrance Test

We are given the functional equation: f(x) + 2f(1/x) = 3x (Equation 1) We want to find f(2). Let's substitute x = 2 into Equation 1: f(2) + 2f(1/2) = 3(2) f(2) + 2f(1/2) = 6 (Equation 2) Now, to get another equation involving f(2) and f(1/2), let's substitute x = 1/2 into Equation 1: f(1/2) + 2f(1/(1/2)) = 3(1/2) f(1/2) + 2f(2) = 3/2 (Equation 3) We now have a system of two linear equations with two unknowns, f(2) and f(1/2). Let A = f(2) and B = f(1/2). Equation 2 becomes: A + 2B = 6 Equation 3 becomes: B + 2A = 3/2 From Equation 3, we can express B in terms of A: B = 3/2 - 2A. Substitute this expression for B into the first equation: A + 2(3/2 - 2A) = 6 A + 3 - 4A = 6 -3A + 3 = 6 -3A = 6 - 3 -3A = 3 A = -1. Since A = f(2), we have f(2) = -1.