Numerical answer
A function f satisfies f(x) + 2f(1/x) = 3x for every non-zero real x. Find f(2).
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Answer: -1
Put x = 2: f(2) + 2f(1/2) = 6. …(i)
Put x = 1/2: f(1/2) + 2f(2) = 3/2. …(ii)
Multiply (ii) by 2: 2f(1/2) + 4f(2) = 3. Subtract (i): 3f(2) = 3 − 6 = −3, so f(2) = −1.
In general, solving the pair gives f(x) = 2/x − x, and f(2) = 1 − 2 = −1 ✓.
Tip: Whenever f(x) and f(1/x) appear together, substitute x → 1/x to get a second equation.
Put x = 1/2: f(1/2) + 2f(2) = 3/2. …(ii)
Multiply (ii) by 2: 2f(1/2) + 4f(2) = 3. Subtract (i): 3f(2) = 3 − 6 = −3, so f(2) = −1.
In general, solving the pair gives f(x) = 2/x − x, and f(2) = 1 − 2 = −1 ✓.
Tip: Whenever f(x) and f(1/x) appear together, substitute x → 1/x to get a second equation.
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