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Home » Crush a Procedurally Perky Smile: The Ultimate Face Pulls Workout Revealed! - American Beagle Club

Crush a Procedurally Perky Smile: The Ultimate Face Pulls Workout Revealed! - American Beagle Club

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 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 $:

📌 f(x) + f(x) = 2f(x) + 2f(0) \Rightarrow 2f(x) = 2f(x) + 2f(0) \Rightarrow f(0) = 0.

📌 So $ f(-y) = f(y) $, hence $ f $ is even.

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Combine like terms in the numerator:
\frac{a + 4a + 3a + 4 + 1 + 7}{3} = 15
\frac{8a + 12}{3} = 15
Multiply both sides by 3 to eliminate the denominator:
8a + 12 = 45
8a = 33
Divide both sides by 8:
a = \frac{33}{8}
Thus, the value of \( a \) is \(\boxed{\frac{33}{8}}\).
Question:** In an evolutionary genomics study, a researcher is examini...

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