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A function f defined for all nonzero real x satisfies f(x) + 2 f(1/x)…
Question
A function f defined for all nonzero real x satisfies f(x) + 2 f(1/x) = 3x for every x ≠ 0. Let S = f(2) + f(3) + f(4). Writing S = -p/q in lowest terms with positive integers p and q, find p + q.
✓ Verified answer: 47checked by our engine — not a guess
Step-by-step solution
Replace x by 1/x in f(x)+2f(1/x)=3x to get f(1/x)+2f(x)=3/x.
Treat these as two linear equations in f(x), f(1/x).
Multiply the second by 2 and subtract the first: 3 f(x) = 6/x - 3x, so f(x) = 2/x - x.
Then f(2) = 1 - 2 = -1, f(3) = 2/3 - 3 = -7/3, f(4) = 1/2 - 4 = -7/2.
S = -1 - 7/3 - 7/2 = (-6 - 14 - 21)/6 = -41/6. So p = 41, q = 6, p + q = 47.
Final answer47
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