30.2 Function Criterion

Function Criterion is the specific requirement that determines whether a given relation qualifies as a function, distinguishing this restricted and especially useful category of relation from the broader collection of ordered pairs that any relation, as established in Relation Structure, may represent.

Function as a Restricted Relation frames every function as fundamentally a relation, an Ordered Pair Collection satisfying Input-Output Association, but one that additionally satisfies an extra condition beyond what a general relation requires, meaning that while every function is a relation, not every relation qualifies as a function.

One Output for Each Input is the precise statement of the extra condition itself: for a relation to qualify as a function, every distinct input value appearing anywhere within the Ordered Pair Collection must be associated with exactly one output value, never more than one, throughout the entire relation.

Repeated Input Inspection is the diagnostic action required to test One Output for Each Input against an actual relation: examining the full collection of ordered pairs to identify whether any input value, occupying First Coordinate as Input, appears more than once across different pairs within the collection, since only a repeated input creates the possibility of a violation.

Same Input with the Same Output describes the acceptable outcome when Repeated Input Inspection finds a repeated input value: if every ordered pair sharing that repeated input also shares the identical output value, the repetition does not violate One Output for Each Input, since the relation is still associating that input with only a single output overall, merely recording that association more than once within the collection.

Same Input with Different Outputs describes the unacceptable outcome when Repeated Input Inspection finds a repeated input value: if the ordered pairs sharing that repeated input record two or more different output values, this directly violates One Output for Each Input, since the same starting value would then be associated with more than one distinct result, an outcome the function criterion explicitly forbids.

Function Classification is the conclusion reached when Repeated Input Inspection, applied across the entire Ordered Pair Collection, finds no instance of Same Input with Different Outputs, confirming that One Output for Each Input holds throughout the relation and that the relation may therefore be properly classified as a function.

Nonfunction Classification is the opposite conclusion, reached as soon as even a single instance of Same Input with Different Outputs is found anywhere within the relation, confirming that One Output for Each Input has been violated and that the relation, regardless of how many of its other input values behave acceptably, must be classified as a relation that is not a function.