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Home » Substitute $ r = 12 $, $ \theta = 60^\circ $: - Londonproperty

Substitute $ r = 12 $, $ \theta = 60^\circ $: - Londonproperty

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 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)}

📌 So, \( f(f(x)) = x \) becomes:

📌 rac{N^3 - 3ND^2}{D(N^2 + D^2)} = x

📌 Multiply both sides by \( D(N^2 + D^2) \):

📌 N^3 - 3ND^2 = x D (N^2 + D^2)

📌 Now substitute back \( N = x^3 - 3x \), \( D = x^2 + 1 \):

📌 \( N^2 = (x^3 - 3x)^2 = x^6 - 6x^4 + 9x^2 \)

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