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birthday themes for first birthday - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📝 Recent Posts

Question: A behavioral health researcher studies patient engagement ov...
Solution: We are given the functional equation:
f(x + y) + f(x - y) = 2f(x) + 2f(y) \quad \text{for all } x, y \in \ma...
This is a well-known equation. Set $ x = 0 $:
f(y) + f(-y) = 2f(0) + 2f(y) \Rightarrow f(-y) = 2f(0) + f(y).
Now set $ y = 0 $:
f(x) + f(x) = 2f(x) + 2f(0) \Rightarrow 2f(x) = 2f(x) + 2f(0) \Rightar...
So $ f(-y) = f(y) $, hence $ f $ is even.
Now, the equation becomes:
f(x + y) + f(x - y) = 2f(x) + 2f(y).

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