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starved rock cabins - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Now, solve \( f(f(x)) = x \). This includes solutions to \( f(x) = x \) (fixed points) and solutions where \( f(f(x)) = x \) but \( f(x)

📌 e x \) â i.e., 2-cycles.

📌 Letâs check the range and behavior. As \( x o \pm\infty \), \( f(x) \sim x^3 / x^2 = x o \mp\infty \), so \( f(x) \) is odd asymptotically.

📌 Now, compute \( f(f(x)) - x = 0 \). This is a rational equation; clear denominators:

📌 Let \( y = f(x) \), so \( f(y) = x \). Then:

📌 f(f(x)) = rac{(f(x))^3 - 3f(x)}{(f(x))^2 + 1} = x

📌 Substitute \( f(x) = rac{x^3 - 3x}{x^2 + 1} \). Let \( N = x^3 - 3x \), \( D = x^2 + 1 \), so \( f(x) = N/D \). Then:

📌 f(f(x)) = rac{(N/D)^3 - 3(N/D)}{(N/D)^2 + 1} = rac{ rac{N^3 - 3ND^2}{D^3} }{ rac{N^2 + D^2}{D^2} } = rac{N^3 - 3ND^2}{D^3} \cdot rac{D^2}{N^2 + D^2} = rac{N^3 - 3ND^2}{D(N^2 + D^2)}

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