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Classify These Functions
publish date: 2026/05/23 18:47:51.523085 UTC
volume_muteClassify each function as Injective only, Surjective only, Bijective, or Neither when viewed as functions from \(\mathbb{R}\) to \(\mathbb{R}\).
drag and drop the selected option to the right place
Correct Answer
(1) f(x) = x³,Bijective
(2) f(x) = x²,Neither
(3) f(x) = 2x + 1,Bijective
(4) f(x) = sin x,Neither
Explanation
\(x^3\): strictly increasing, range = \(\mathbb{R}\) → bijective. \(x^2\): \(f(2)=f(-2)\) (not injective); range = \([0,\infty) \ne \mathbb{R}\) (not surjective) → neither. \(2x+1\): strictly increasing, range = \(\mathbb{R}\) → bijective. \(\sin x\): \(\sin(0) = \sin(\pi)\) (not injective); range = \([-1,1]\) (not surjective) → neither.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012