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Graph Symmetry of Inverse Functions

publish date: 2026/05/23 21:16:57.243908 UTC

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The diagram below illustrates the geometric relationship between the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\).

Inverse Functions Function f(x) x → y Multiply by 2 Add 3 Inverse f⁻¹(x) y → x Subtract 3 Divide by 2 An inverse function reverses the operations of the original function.

With respect to what line are the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) symmetric?

Correct Answer

The line \(y = x\)

Explanation

Interchanging the roles of \(x\) and \(y\) in the formula for \(f\) gives \(f^{-1}\). Geometrically, swapping coordinates \((x,y) \leftrightarrow (y,x)\) is exactly a reflection across the line \(y = x\). So the graphs of \(f\) and \(f^{-1}\) are mirror images of each other in the line \(y = x\).

Reference

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

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