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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Question: A brain-computer interface algorithm uses a function $ f $ satisfying $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real $ x, y $. If $ f(1) = 3 $, compute $ f(5) $.

📌 Solution: The functional equation is $ f(x+y) + f(x-y) = 2f(x) + 2f(y) $. This is a known variant of the quadratic functional equation.

📌 Assume $ f(x) = ax^2 $. Test:

📌 Left: $ a(x+y)^2 + a(x-y)^2 = a(x^2 + 2xy + y^2 + x^2 - 2xy + y^2) = a(2x^2 + 2y^2) = 2a x^2 + 2a y^2 $

📌 Right: $ 2a x^2 + 2a y^2 $. So satisfies.

📌 Now $ f(1) = a(1)^2 = a = 3 $ â $ a = 3 $

📌 Thus $ f(x) = 3x^2 $, so $ f(5) = 3 \cdot 25 = 75 $

📌 To confirm no other solutions: standard result â all solutions are quadratic $ f(x) = ax^2 $, under mild regularity (which we assume for neural response modeling).

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