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Home » Why Mythical Animals Are More Real Than You Think—Revealed! - Veritas Home Health

Why Mythical Animals Are More Real Than You Think—Revealed! - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: An entrepreneur designing a sustainable irrigation system models soil moisture retention with a function $ f $ such that $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real $ x, y $, and $ f(1) = 3 $. Find $ f(5) $.

📌 Solution: The functional equation is:

📌 f(x + y) + f(x - y) = 2f(x) + 2f(y), \quad \forall x, y \in \mathbb{R}

📌 This is known as the **quadratic functional equation**. The general solution over $ \mathbb{R} $, assuming regularity (e.g., continuity or boundedness), is $ f(x) = cx^2 $. We verify this form.

📌 Let $ f(x) = cx^2 $. Then:

📌 f(x+y) + f(x-y) = c(x+y)^2 + c(x-y)^2 = c(x^2 + 2xy + y^2 + x^2 - 2xy + y^2) = c(2x^2 + 2y^2) = 2cx^2 + 2cy^2 = 2f(x) + 2f(y)

📌 So it satisfies the equation.

📌 Given $ f(1) = c(1)^2 = 3 \Rightarrow c = 3 $, so $ f(x) = 3x^2 $.

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