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Is the Inverse Function Unique?

publish date: 2026/05/23 21:16:55.405396 UTC

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If a function \(f\) has an inverse, how many inverse functions does it have?

None — inverse functions do not exist for real functions

Exactly one — the inverse is uniquely determined

Two — one for each direction

Infinitely many — one for each element of the codomain

Correct Answer

Exactly one — the inverse is uniquely determined

Explanation

It can be proved that if \(f\) has an inverse, that inverse is unique. There is only one function \(f^{-1}: B \to A\) that satisfies both \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(y)) = y\). This uniqueness makes the inverse well-defined and useful.

Reference

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

Quizzes you can take where this question appears

Inverse Functions