Inverse Functions

A function is a rule that always returns a UNIQUE output for a given input. For a function to be invertible — for the reverse mapping (output back to input) to itself be a function — the original function must satisfy the same uniqueness requirement in reverse: distinct inputs must give distinct outputs. A function with this property is called one-to-one (injective).

Why this matters — a concrete analogy. Consider a set of unrelated people (no identical twins). Each person has a blood type and a DNA sequence — both are functions of the person. Many people can share a blood type, so "blood type $\to$ person" cannot be inverted (told a blood type, you cannot say WHICH person it came from). But a DNA sequence is unique to each individual, so "DNA sequence $\to$ person" CAN be inverted — this reverse mapping is called the inverse function, and informally, the inverse function undoes what the original function does.

The trigonometric problem. In a right triangle, given one acute angle and one side, every other angle and side follows directly. But given only two SIDES (a ratio), recovering the ANGLE needs the inverse of a trigonometric function. Unfortunately, none of the six trigonometric functions is one-to-one over its full domain — e.g. $\sin\theta=0.5$ is satisfied by infinitely many angles $\theta=\ldots,-\tfrac{7\pi}6,\tfrac{\pi}6,\tfrac{5\pi}6,\tfrac{13\pi}6,\ldots$ (any of these plus a period), so knowing $\sin\theta$ alone can never pin down a unique $\theta$. …