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The Inverse Undoes the Original Function
publish date: 2026/05/23 21:16:54.836106 UTC
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If \(f(x) = y\), which two identities hold for any \(x\) in the domain of \(f\) and any \(y\) in the range of \(f\)?
Correct Answer
\(f^{-1}(f(x)) = x\) and \(f(f^{-1}(y)) = y\)
Explanation
Each pair of inverse functions undoes what the other does: \(f^{-1}(f(x)) = x\) (applying \(f\) then \(f^{-1}\) returns to the original input), and \(f(f^{-1}(y)) = y\) (applying \(f^{-1}\) then \(f\) returns to the original output). Think of \(f\) as encoding and \(f^{-1}\) as decoding.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012