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Can We Always Find a Formula for f⁻¹?
publish date: 2026/05/23 21:16:58.359257 UTC
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Consider \(f(x) = x^5 + 2x + 1\). Can we find an explicit formula for \(f^{-1}(x)\) by solving for \(x\)?
Correct Answer
No — general polynomial equations of degree ≥ 5 cannot be solved in terms of their coefficients
Explanation
A famous result in algebra (proved by Abel and Galois) states that polynomial equations of degree \(\ge 5\) have no general formula for their roots in terms of the coefficients. This means we cannot always write \(f^{-1}\) as an explicit formula, even when we know \(f^{-1}\) must exist. Fortunately, criteria (such as strict monotonicity) can confirm existence without requiring an explicit formula.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012