The Undo Button

Algebra 2 · Axiom Academy

Every function has an undo. Encrypt a secret message, then get it back — and meet the inverse function. You want to send a message nobody else can read, so you scramble it with a rule: push every letter forward 3 places in the alphabet. Easy enough. But a code is worthless unless the person on the other end can unscramble it. Every function that does something needs a partner that puts it back — and that partner has a name. Watch HELLO travel. The encryption function f(x) = x + 3 pushes each letter's position 3 steps forward, and the message comes out as KHOOR. Then the trip runs in reverse: f^ -1 (x) = x - 3 pulls each letter back 3 steps — and every single one lands exactly on the dashed ring it started in. An inverse function reverses the effect of the original function — that is its whole job. Same code, shorter message: CAT encrypts to FDW. Now you are the one on the receiving end — you hold FDW, and you need CAT back. Turn the alphabet ring and hunt for the shift that lands every letter home. Exactly one value works, and that is not luck: the function that undoes f is called its inverse , written f^ -1 . Apply a function and then its inverse, and you land on the very input you started with: f^ -1 (f(x)) = x . It runs both ways — if f^ -1 undoes f , then f undoes f^ -1 . Two operations, undone backwards

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