A function satisfies f(x) + 2 f(1/x) = 3x for every nonzero real x. Find f in closed form and evaluate f(2).

A function satisfies f(x) + 2 f(1/x) = 3x for every nonzero real x. Find f in closed form and evaluate f(2).

Approach: Substitute the reciprocal to get a second equation in the same two unknown values, then treat f(x) and f(1/x) as two variables and eliminate one.

-1. Substituting 1/x for x gives the second relation f(1/x) + 2 f(x) = 3/x. Treat f(x) and f(1/x) as two unknowns in a linear system: the original equation is f(x) + 2 f(1/x) = 3x and twice the second is 4 f(x) + 2 f(1/x) = 6/x. Subtracting eliminates f(1/x) and leaves 3 f(x) = 6/x - 3x, so f(x) = 2/x - x. Verifying, f(x) + 2 f(1/x) = 2/x - x + 2(2x - 1/x) = 2/x - x + 4x - 2/x = 3x, so the closed form satisfies the functional equation for every nonzero x. At x = 2 the value is 1 - 2 = -1. The reciprocal substitution works because x to 1/x is an involution, so it closes the system after one application rather than generating new unknowns.

Follow-up: Solve f(x) + f((x-1)/x) = 1 + x, where the substitution has order 3 rather than 2?

Key concepts: functional equation, reciprocal substitution, linear system, elimination.