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Is f(x) = x² Injective on ℝ?

publish date: 2026/05/23 18:47:49.628660 UTC

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Consider \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^2\). Is this function one-one (injective)?

Yes — every output value is the square of exactly one positive number

No — both x = 2 and x = −2 give f(x) = 4, so two distinct inputs share an output

Yes — because x² is always non-negative

No — because it is not defined for negative x

Correct Answer

No — both x = 2 and x = −2 give f(x) = 4, so two distinct inputs share an output

Explanation

\(f(x) = x^2\) on \(\mathbb{R}\) is not injective. Take \(a_1 = 2\) and \(a_2 = -2\): \(f(2) = 4 = f(-2)\) but \(2 \ne -2\). This violates the definition \(f(a_1)=f(a_2) \Rightarrow a_1=a_2\). It is therefore a many-one function.

Reference

Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012

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Types of Functions