Math Functions Functions as relations

Functions as relations

A relation is a set of ordered pairs. It assigns each element to at least one other element.

Functions are also relations, but only one assigned value may exist for each domain value.



A function is an explicit relation $x\mapsto y$.
Each $x$-value is associated to exactly one $y$-value.


Functions are labeled with a letter. In most cases you use the letter $f$.

The $y$-value is assigned to the $x$-value. To show this dependency, one often writes:


Read: y equals f of x

  • $y$ is the function value
  • $f(x)$ is the function term
  • $y=f(x)$ is the function equation


An example for a linear function would be:

$f(x)=5x+3$ or



There is another notation for functional equations, which is less common in school:

$f:x\mapsto y$

e.g. $f:x\mapsto 5x+3$