Single correct option
A function f satisfies f(x) + 2f(1/x) = 3x for every non-zero real x. What is f(2)?
- A. −1
- B. 1
- C. 2
- D. −2
Show answer and solution
Answer: (A) −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: 4f(2) + 2f(1/2) = 3. Subtract (i): 3f(2) = −3 ⇒ f(2) = −1.
(Then f(1/2) = 7/2; check (ii): 7/2 − 2 = 3/2 ✓.)
Tip: When a functional equation links f(x) and f(1/x), substitute both x and 1/x and solve the pair simultaneously.
Put x = 1/2: f(1/2) + 2f(2) = 3/2 ... (ii)
Multiply (ii) by 2: 4f(2) + 2f(1/2) = 3. Subtract (i): 3f(2) = −3 ⇒ f(2) = −1.
(Then f(1/2) = 7/2; check (ii): 7/2 − 2 = 3/2 ✓.)
Tip: When a functional equation links f(x) and f(1/x), substitute both x and 1/x and solve the pair simultaneously.
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