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Is f(x) = x³ Bijective on ℝ?
publish date: 2026/05/23 18:47:50.147439 UTC
volume_mute
Consider \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^3\). What type is it?
Correct Answer
Bijective (one-one and onto)
Explanation
\(f(x) = x^3\) is bijective on \(\mathbb{R}\). Injective: if \(a_1^3 = a_2^3\) then \(a_1 = a_2\). Surjective: for every \(y \in \mathbb{R}\) there exists \(x = \sqrt[3]{y} \in \mathbb{R}\) with \(f(x) = y\). Because it is bijective, \(f(x)=x^3\) has an inverse: \(f^{-1}(y) = \sqrt[3]{y}\).
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012