Single correct option
A function f satisfies f(x) + 2f(1/x) = 3x for every non-zero real x. What is the value of f(2)?
- A. 1
- B. 2
- C. −2
- D. −1
Show answer and solution
Answer: (D) −1
Put x = 2: f(2) + 2f(1/2) = 6 … (i)
Put x = 1/2: f(1/2) + 2f(2) = 3/2 … (ii)
Let a = f(2), b = f(1/2). From (ii), b = 3/2 − 2a. Substituting in (i): a + 3 − 4a = 6 ⇒ −3a = 3 ⇒ a = −1.
So f(2) = −1 (and f(1/2) = 7/2).
In general, solving the pair gives f(x) = (2 − x²)/x; f(2) = (2 − 4)/2 = −1 ✓.
Tip: Whenever a functional equation mixes f(x) and f(1/x), substitute x → 1/x to get a second equation and solve the linear system.
Put x = 1/2: f(1/2) + 2f(2) = 3/2 … (ii)
Let a = f(2), b = f(1/2). From (ii), b = 3/2 − 2a. Substituting in (i): a + 3 − 4a = 6 ⇒ −3a = 3 ⇒ a = −1.
So f(2) = −1 (and f(1/2) = 7/2).
In general, solving the pair gives f(x) = (2 − x²)/x; f(2) = (2 − 4)/2 = −1 ✓.
Tip: Whenever a functional equation mixes f(x) and f(1/x), substitute x → 1/x to get a second equation and solve the linear system.
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