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Home » crzzy shooter made gamers lose their minds—this is the game you NEED to Play! - RTA

crzzy shooter made gamers lose their minds—this is the game you NEED to Play! - RTA

📅 May 12, 2026 👤 scraface

May 12, 2026

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📌 Thus, the smallest such \( n \) is 11.

📌 oxed{r = 3^{1/5}, \; n = 11}

📌 Question:** A high-altitude atmospheric carbon isotope tracking researcher uses quantum spectroscopy data modeled by the function \( f(x) = rac{x^3 - 3x}{x^2 + 1} \). Find the number of real solutions to the equation \( f(f(x)) = x \).

📌 Solution:** We seek the number of real solutions to \( f(f(x)) = x \), where \( f(x) = rac{x^3 - 3x}{x^2 + 1} \).

📌 Note that \( f(x) \) is an odd function: \( f(-x) = -f(x) \). Also, observe the behavior:

📌 Letâs analyze \( f(x) \). We compute:

📌 f(x) = rac{x(x^2 - 3)}{x^2 + 1}

📌 This is smooth and differentiable. Letâs study fixed points: \( f(x) = x \) implies:

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