Single correct option
A function f, defined for all non-zero real x, satisfies f(x) + 2f(1/x) = 3x. What is the value of f(2)?
- A. −2
- B. −1
- C. 1
- D. 2
Show answer and solution
Answer: (B) −1
Put x = 2: f(2) + 2f(1/2) = 6 … (1)
Put x = 1/2: f(1/2) + 2f(2) = 3/2 … (2)
Let a = f(2), b = f(1/2). From (2), b = 3/2 − 2a. Substitute in (1): a + 3 − 4a = 6 → −3a = 3 → a = −1.
In general, solving the pair for any x gives f(x) = 2/x − x, and f(2) = 1 − 2 = −1 ✓.
Tip: For equations linking f(x) and f(1/x) (or f(x) and f(−x)), substitute the 'partner' value to get a second equation and solve the 2 × 2 system.
Put x = 1/2: f(1/2) + 2f(2) = 3/2 … (2)
Let a = f(2), b = f(1/2). From (2), b = 3/2 − 2a. Substitute in (1): a + 3 − 4a = 6 → −3a = 3 → a = −1.
In general, solving the pair for any x gives f(x) = 2/x − x, and f(2) = 1 − 2 = −1 ✓.
Tip: For equations linking f(x) and f(1/x) (or f(x) and f(−x)), substitute the 'partner' value to get a second equation and solve the 2 × 2 system.
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