A relation is a set of ordered pairs where each pair consists of an input (x-value) and an output (y-value). What makes a relation a function is the rule that each input (x-value) must be related to exactly one output (y-value). In other words, no x-value can be paired with more than one y-value in a function
. Key points that define a function from a relation:
- Each x-value appears only once or, if repeated, it always corresponds to the same y-value.
- There is a unique output for every input.
- A relation where an x-value maps to multiple y-values is not a function.
- This uniqueness can be visually tested using the vertical line test on a graph: if any vertical line intersects the graph more than once, the relation is not a function
For example, the relation {(1, 0), (1, 5), (2, 10)} is not a function because the input 1 corresponds to two different outputs (0 and 5). However, {(0, 1), (1, 5), (2, 4)} is a function since each input has exactly one output
. In summary, a function is a special type of relation in which every input has a single, well-defined output. This ensures a precise and unambiguous mapping from inputs to outputs
