A relation is a set of inputs and outputs, often written as ordered pairs
(input, output). We can also represent a relation as a mapping diagram or a
graph. For example, the relation can
be represented as:
Lines connect the inputs with their outputs. The relation can also be represented as:
Functions
A function is a relation in which each input has only one output.
In the relation , y is a function of x,
because for each input x (1, 2, 3, or 0), there is only one output y. x
is not a function of y, because the input y = 3 has multiple outputs: x = 1 and x = 2.
Examples:
\: y is a function of x, x is a function
of y.
: y is not a function of x (x = 3
has multiple outputs), x is a function of y.
: y is a function of x, x is not a
function of y (y = 9 has multiple outputs).
: y is not a function of x (x = 1
has multiple outputs), x is not a function of y (y = 2
has multiple outputs).
The Line Test for Mapping Diagrams
To check if a relation is a function, given a mapping diagram of the relation,
use the following criterion: If each input has only one line connected to it,
then the outputs are a function of the inputs.
Example: In the following mapping diagram, y is a function of x, but
x is not a function of y: