What Makes a Rule a Function — Cheat sheet

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Key method

A vertical line $x=a$ may meet the graph at at most one point. A formula that fails at an input can define a function on a smaller domain, but cannot silently retain the forbidden input.

Example

The circle $x^2+y^2=1$ fails the vertical-line test: at $x=0$ it has $y=1$ and $y=-1$ . Its upper half $y=\sqrt{1-x^2}$ is a function on $[-1,1]$ . The rule $1/x$ is a function on $\mathbb R\setminus\{0\}$ , but not on all of $\mathbb R$ .

Avoid this mistake

The horizontal-line test checks one-to-one behavior, not whether the relation is a function in the first place.