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Relations and Functions

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Relations

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:

**Mapping Diagram of Relation**
Lines connect the inputs with their outputs. The relation can also be represented as:

Graph of Relation

###
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*:

Line Test