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If x*f(x)+x*f(1/x)=x+1, what is the value of f(2024)?

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Answer to a math question If x*f(x)+x*f(1/x)=x+1, what is the value of f(2024)?

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Gene
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56 Answers
Veja como resolver para f(2024): **1. Substitua x por 1/x:** Na equação x*f(x) + x*f(1/x) = x + 1, substitua x por 1/x: * (1/x) * f(1/x) + (1/x) * f(x) = 1/x + 1 **2. Simplificar:** * f(1/x) + f(x) = 1 + x **3. Seja x = 2024:** Substitua x por 2024 na equação original e na equação da etapa 2: * Original: 2024 *f(2024) + 2024 *f(1/2024) = 2025 * Etapa 2: f(1/2024) + f(2024) = 1 + 2024 = 2025 **4. Resolva o sistema de equações:** Agora você tem duas equações com duas incógnitas: * 2024 * f(2024) + 2024 * f(1/2024) = 2025 * f(1/2024) + f(2024) = 2025 Como ambas as equações são iguais a 2025, podemos igualá-las: * 2024 * f(2024) + 2024 * f(1/2024) = f(1/2024) + f(2024) Simplificar: * 2023 * f(2024) = -2023 * f(1/2024) Divida ambos os lados até 2023: *f(2024) = -f(1/2024) **Não podemos determinar o valor exato de f(2024) sem mais informações sobre a função. No entanto, sabemos que f(2024) é o negativo de f(1/2024).**

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