Determining Whether a Relation Is a Function

Determining whether a relation is a function means checking that no input repeats with a different output — by scanning pairs, reading a table, or running the vertical line test on a graph.

Test {(1, 2), (2, 3), (1, 5)}: the input 1 appears twice, paired with 2 and then with 5 — one input, two answers, not a function. Test {(1, 2), (2, 2), (3, 2)}: three inputs share the output 2, and that is legal — outputs may pool; inputs may not split. On a graph, the same scan becomes geometry: a vertical line is one input's whole territory, and a double hit is a split input caught in the act. Three formats, one question: does any input lead two lives?