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Imagine running a secret code machine: typing "HELLO" outputs "KHOOR". To read the original text, you need a reverse machine that takes "KHOOR" and converts it right back to "HELLO".
In mathematics, an inverse function does exactly that — it reverses the assignment of inputs and outputs of the original function.
How do we describe this relationship with formal mathematical notation?

The Inverse Relationship

If a function ff takes an input xx to an output yy, written as f(x)=yf(x) = y, then its inverse f−1f^{-1} takes yy back to xx, meaning f−1(y)=xf^{-1}(y) = x.
This means the domain of ff becomes the range of f−1f^{-1}, and the range of ff becomes the domain of f−1f^{-1}.
Can every function be reversed this way, or do some cause a mathematical breakdown?

When Inverses Exist: One-to-One Functions

A function has an inverse function if and only if it is one-to-one (or injective), meaning every unique output comes from exactly one unique input.
Graphically, we test this using the Horizontal Line Test: if any horizontal line crosses the graph of f(x)f(x) more than once, ff is not one-to-one and cannot have an inverse function over that domain.
How can we algebraically prove that two functions completely undo each other?