Inverse Function Definition

Pre-Calculus · Axiom Academy

LESSON Inverse Function Definition What an inverse function does, the round-trip that defines it, and the notation trap that catches almost everyone. 1. The Round Trip That Defines an Inverse Let f be a one-to-one function with domain A and range B . Its inverse f^ -1 is the function with domain B and range A that undoes f : feed x into f to get an output, then feed that output into f^ -1 , and you arrive back at the very same x . f takes the input forward to the output f^ -1 carries the output back to the input 2. The Two Identities — and How to Verify a Pair "Undoes" is made precise by two composition identities . They are also exactly how you check whether two functions are inverses: compose them both ways, and if each one collapses to plain x , they're inverses. for every x in the domain of f . for every x in the domain of f^ -1 . Both compositions collapse to x , so this pair really are inverses. 3. The Notation Trap: f^ -1 Is Not a Reciprocal Here is the single biggest source of confusion. The " -1 " in f^ -1 is not an exponent — it is a symbol meaning "the inverse function." So f^ -1 (x) does not mean . Read it aloud as " f inverse," never " f to the negative one." 4. Domain and Range Switch Places Because f^ -1 runs the machine backward, what used to be the output set becomes the input set . The inputs of f^ -1 are exactly the outputs of f , so the domain and range simply trade roles. A case where the swap is dramatic: g(x)=e^ x

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