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Classify These Functions: Has Inverse or No Inverse
publish date: 2026/05/23 21:16:59.334499 UTC
volume_muteClassify each function below as Has Inverse or No Inverse (as functions from \(\mathbb{R}\) to \(\mathbb{R}\) unless otherwise noted).
drag and drop the selected option to the right place
Correct Answer
(1) f(x) = 3x - 2,Has Inverse
(2) f(x) = x/(1-x),Has Inverse
(3) f(x) = |x|,No Inverse
(4) f(x) = cos x,No Inverse
Explanation
\(3x-2\): linear, strictly increasing → bijective → has inverse. \(x/(1-x)\): injective and surjective → has inverse \(x/(1+x)\). \(|x|\): \(|-2|=|2|=2\) → not injective → no inverse on \(\mathbb{R}\). \(\cos x\): periodic, not injective (\(\cos 0 = \cos 2\pi\)) and range \([-1,1] \ne \mathbb{R}\) → no inverse on \(\mathbb{R}\).
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012