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Mutual Inverse Functions Share One Curve

publish date: 2026/05/23 21:16:58.639475 UTC

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The functions \(y = f(x)\) and \(x = f^{-1}(y)\) — viewed in the same xy-plane — describe how many distinct curves?

Two distinct curves — one for each function

One and the same curve in the xy-plane

Three curves — the original, the inverse, and their intersection

None — they cancel each other out

Correct Answer

One and the same curve in the xy-plane

Explanation

Both \(y = f(x)\) and \(x = f^{-1}(y)\) describe the same set of ordered pairs and therefore the same curve in the xy-plane. The difference is only in which variable is treated as independent. Only when we rewrite the inverse with \(x\) as the independent variable — as \(y = f^{-1}(x)\) — do we get a different curve (the reflection across \(y = x\)).

Reference

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

Quizzes you can take where this question appears

Inverse Functions