A relation is a function if every allowed input value goes to exactly one output value.
Core idea
In math, a function is a rule that takes each element in a set called the domain (the inputs) and assigns to it one and only one element in another set called the codomain (the outputs). Different inputs are allowed to share the same output, but a single input is not allowed to have two different outputs.
What makes it a function
A relation “is” a function precisely when:
- Every input that is allowed by the rule actually gets an output (no missing outputs in the domain).
- No input is paired with two or more different outputs (no splitting one x into several y’s).
If there is any input value that would produce two different outputs, then the relation is not a function.
Simple examples
- y=x2+1y=x^2+1y=x2+1 is a function because each xxx gives exactly one value of yyy.
- The relationship given by y2=x+1y^2=x+1y2=x+1 is not a function of xxx because some xxx-values would correspond to two different yyy-values (one positive and one negative).
Graphically, this is why the “vertical line test” works: if any vertical line hits a graph in more than one point, then some input xxx has more than one output yyy, so it fails to be a function.
