Inverse Functions and What They Undo

Think of f as a machine that takes an input and produces an output. The inverse, written f⁻¹, is the same machine wired to run backwards: feed it the output, and it hands you back the exact input that made it. That only works cleanly if each output came from exactly one input. If f(3) = 11, then f⁻¹(11) = 3, no exceptions and no other candidates.

This backwards relationship is why f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for every x where things are defined. Send x through f, then undo it with f⁻¹, and you land right back where you started, because f⁻¹ was built to reverse exactly the step f took. You can use this to evaluate an inverse without ever solving for its formula: to find f⁻¹(7) for f(x) = 3x - 5, just ask what input makes f equal 7. Solve 3x - 5 = 7 to get x = 4, so f⁻¹(7) = 4.

The one-input-one-output requirement connects inverses to the transformation work right next to this topic. A graph like f(x) = x² fails that test, since both 3 and -3 give 9, so there is no single value an inverse could assign back to 9. The function has to be restricted to half its domain, often just the piece where x ≥ 0, before an inverse exists. Graphically, an inverse is the reflection of f over the line y = x, which is the same flip-and-swap idea that shows up when you study how graphs transform.