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Is f(x) = x² Surjective on ℝ?
publish date: 2026/05/23 18:47:49.888646 UTC
volume_mute
Consider \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^2\). Is this function onto (surjective)?
Correct Answer
No — negative numbers in ℝ cannot be outputs of x²
Explanation
\(f(x) = x^2\) is not surjective as \(f: \mathbb{R} \to \mathbb{R}\). For any \(x \in \mathbb{R}\), \(x^2 \ge 0\), so negative numbers like \(-1, -2\) are never in the range. The range is \([0, \infty)\) which is a proper subset of \(\mathbb{R}\).
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012