Single correct option
A function g satisfies g(x) + 3g(1/x) = 4x − 4 for every non-zero real x. What is g(3)?
- A. −2
- B. 2
- C. −4
- D. 4
Show answer and solution
Answer: (A) −2
Put x = 3: g(3) + 3g(1/3) = 4(3) − 4 = 8. … (i)
Put x = 1/3: g(1/3) + 3g(3) = 4/3 − 4 = −8/3. … (ii)
Let a = g(3), b = g(1/3). From (i), b = (8 − a)/3. Substitute in (ii): (8 − a)/3 + 3a = −8/3 → 8 − a + 9a = −8 → 8a = −16 → a = −2.
Check: b = (8 − (−2))/3 = 10/3, and b + 3a = 10/3 − 6 = −8/3. ✓
Put x = 1/3: g(1/3) + 3g(3) = 4/3 − 4 = −8/3. … (ii)
Let a = g(3), b = g(1/3). From (i), b = (8 − a)/3. Substitute in (ii): (8 − a)/3 + 3a = −8/3 → 8 − a + 9a = −8 → 8a = −16 → a = −2.
Check: b = (8 − (−2))/3 = 10/3, and b + 3a = 10/3 − 6 = −8/3. ✓
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