One Output Each

A rule is a function only if every input gives exactly one output. Ask it about 3 and it must give one answer, the same one every time. That is what lets us write f(3) and mean a single number.

A table can break this rule. If the input 2 appears twice, once with output 5 and once with output 7, the table is not a function: asking about 2 has no single answer. Two different inputs sharing one output is fine, though. In a table of squares, both -3 and 3 give 9.

On a graph, the same idea becomes the vertical line test. A vertical line at x = 2 collects every point with input 2. If some vertical line meets the graph twice, one input has two outputs, so the graph is not a function. A parabola opening up passes; a circle fails.