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Home » Master the Mother-in-Law Suite: Secrets to Surviving (and Thriving) with Her! - Abbey Badges

Master the Mother-in-Law Suite: Secrets to Surviving (and Thriving) with Her! - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Alternatively, suppose we assume $ p(x) $ is cubic: $ p(x) = ax^3 + 2x^2 + x $. Then $ p(1) = a + 2 + 1 = 3 \Rightarrow a = 0 $. So $ a = 0 $, same result.

📌 Thus, $ p(0) = 0 $.

📌 Question: A behavioral health researcher studies patient engagement over time and defines a function $ f : \mathbb{R} \to \mathbb{R} $ satisfying $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real $ x, y $. If $ f(1) = 4 $, find $ f(3) $.

📌 Solution: We are given the functional equation:

📌 f(x + y) + f(x - y) = 2f(x) + 2f(y) \quad \text{for all } x, y \in \mathbb{R}.

📌 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 $:

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