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Home » This Rare Bloom Is Oklahoma’s State Flower—Shocking Facts Inside! - Professionaloutdoormedia

This Rare Bloom Is Oklahoma’s State Flower—Shocking Facts Inside! - Professionaloutdoormedia

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 f(x) = rac{x(x^2 - 3)}{x^2 + 1}

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

📌 rac{x^3 - 3x}{x^2 + 1} = x \Rightarrow x^3 - 3x = x(x^2 + 1) = x^3 + x \Rightarrow -4x = 0 \Rightarrow x = 0

📌 So, only one fixed point at \( x = 0 \).

📌 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:

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