Always, Sometimes, Never

A claim goes up — “the inverse of a function is a function” — and everyone must file it under always, sometimes, or never. “Sometimes” is not a shrug: it obliges you to produce a case where the claim holds, one where it fails, and the exact boundary between them.

How we play

1. Classify silently first. Gut verdicts welcome; they get audited.
2. “Always” and “never” demand an argument covering every case.
3. “Sometimes” demands an example, a counter-example, and the boundary.

The inverse claim, argued

• ”$y=2x$ undoes cleanly — halve whatever you got. Not never.”
• ”$y=x^2$ fails: an output of 9 came from 3 or from $-3$, and the undoing cannot decide. Sometimes.”
• “The boundary: a function whose outputs never repeat can be undone without ambiguity — no horizontal line crosses it twice.”
• “Then the sharpened claim — the inverse of an increasing function is always a function — is one we could actually defend.”

One variation

Claims from the rest of the course: "$\sin \theta \le 1$" — always, and the reason lives inside the triangle. “A geometric sequence eventually outgrows an arithmetic one” — sometimes, and saying when is a line Sequences and Their Rules teaches you to draw.

"Sometimes" is where the mathematics is

The boundary of a claim is its content. For the inverse claim, the boundary can even be moved: restrict the inputs of $y=x^2$ and the undoing survives — a repair job that belongs to Domain and Range and pays off in The Inverse of a Function.