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costco mount prospect il - SUpost

📅 July 5, 2026 👤 scraface

Jul 05, 2026

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📌 Observe that the right-hand side is a perfect cube:

📌 Question: A postdoctoral researcher defines a function $ f $ recursively by $ f(f(x)) = 2f(x) - x $ for all real $ x $, and $ f(1) = 3 $. Find $ f(2025) $.

📌 and $ f(1) = 3 $. We aim to find $ f(2025) $.

📌 Since $ f(1) = 3 $, this gives $ f(3) = 5 $.

📌 Now apply the equation at $ x = 3 $:

📌 f(f(x)) = f(x + 2) = (x + 2) + 2 = x + 4,

📌 2f(x) - x = 2(x + 2) - x = 2x + 4 - x = x + 4.

📌 So $ f(f(x)) = 2f(x) - x $ holds. Also, $ f(1) = 3 $, which matches.

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