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inline water filter for home - Rice University

📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 Additivity: $ f(x + y) = f(x) + f(y) $

📌 Multiplicativity: $ f(xy) = f(x)f(y) $

📌 Thus, the only possible functions are $ f(x) = 0 $ and $ f(x) = x $.

📌 $ f(x) = 0 $: clearly additive and multiplicative.

📌 $ f(x) = x $: $ f(x + y) = x + y = f(x) + f(y) $, and $ f(xy) = xy = f(x)f(y) $.

📌 Hence, there are exactly two such functions.

📌 Question: Find the area of the region enclosed by the graph of

📌 This is a triangle with vertices at $ (0, 0), (4, 0), (0, 4) $.

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