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📅 March 10, 2026 👤 scraface

Mar 10, 2026

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Solution:** We seek the number of real solutions to \( f(f(x)) = x \),...
Note that \( f(x) \) is an odd function: \( f(-x) = -f(x) \). Also, ob...
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)...
rac{x^3 - 3x}{x^2 + 1} = x \Rightarrow x^3 - 3x = x(x^2 + 1) = x^3 + x...
So, only one fixed point at \( x = 0 \).
Now, solve \( f(f(x)) = x \). This includes solutions to \( f(x) = x \...
e x \) â i.e., 2-cycles.
Letâs check the range and behavior. As \( x o \pm\infty \), \( f(x)...

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