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When Does an Inverse Function Exist?
publish date: 2026/05/23 21:16:54.268964 UTC
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The diagram below shows a function \(f\) and its reverse correspondence \(f^{-1}\), reading arrows both ways.
Under which condition does a function \(f: A \to B\) have an inverse function \(f^{-1}\)?
Correct Answer
When f is bijective (both one-one and onto)
Explanation
An inverse function \(f^{-1}\) exists if and only if \(f\) is bijective. Why one-one? If two domain elements shared the same image, the reverse would not be a function (the image would have two pre-images to choose from). Why onto? Every element of \(B\) must be an image so that \(f^{-1}\) is defined on all of \(B\).
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012