Short answer
A relation is any set of ordered pairs that connects input values to output values. A function is a special type of relation in which every input has exactly one output.
For example, ({(1,2),(2,4),(3,6)}) is a function because each input appears with only one output. However, ({(1,2),(1,3),(2,4)}) is not a function because the input (1) is paired with two different outputs.
On a graph, you can test this using the vertical line test. If any vertical line crosses the graph more than once, one input has multiple outputs, so the relation is not a function. A circle fails this test, while a straight line such as (y=2x+1) passes it.
In IB Mathematics, remember that every function is a relation, but not every relation is a function. Also check the stated domain: an equation may define a function for one restricted domain even if it would fail under a wider domain.
Exam tip
When asked whether a relation is a function, identify the input variable, check for repeated inputs with different outputs, and state your conclusion clearly.
