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Home » Coastal Carolina University Acceptance Rate Shock: Only 3% Got In—Are You Ready to Apply? - Dyverse

Coastal Carolina University Acceptance Rate Shock: Only 3% Got In—Are You Ready to Apply? - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Because \( f \) is odd and smooth, and \( f(f(x)) - x \) is odd, all solutions come in pairs \( \pm x \), except possibly \( x = 0 \).

📌 Letâs check degree: the leading term of \( f(x) \) is \( x \), so \( f(f(x)) \) has leading term \( f(x) o x \), so \( f(f(x)) o x \), but as a rational function, \( f(f(x)) = x + o(1) \), but algebraically, the numerator leads to degree 9.

📌 In fact, it is known in functional iterations that such rational functions of degree â¥ 2 can have up to \( 2n \) solutions for \( f^n(x) = x \), but here we are solving \( f(f(x)) = x \), so up to 9 solutions (since numerator degree â¤ 9, odd).

📌 But due to odd symmetry, solutions are symmetric about origin, so if \( x \) is a solution, so is \( -x \), and \( 0 \) if odd.

📌 Now, \( f(f(0)) = f(0) = 0 \), so 0 is a solution.

📌 Try numerical: \( f(1) = (1 - 3)/(1 + 1) = -2/2 = -1 \), \( f(-1) = ((-1)^3 - 3(-1))/(1 + 1) = (-1 + 3)/2 = 1 \), so \( f(f(1)) = f(-1) = 1 = x \). So \( x = 1 \) is a solution.

📌 Similarly, \( f(-1) = 1 \), so \( f(f(-1)) = f(1) = -1 = x \), so \( x = -1 \) is a solution.

📌 So far: \( x = -1, 0, 1 \) are solutions.

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