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Relations and Functions
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
The Vertical and Horizontal Line Tests for Graphs
To determine whether y is a function of x,
given a graph of a relation, use the following criterion: if every vertical line
you can draw goes through only 1 point, y is a function of x.
If you can draw a vertical line that goes through 2 points, y is not a function of x.
This is called the vertical line test. Example 1: In the following graph, y
is a function of x:
Passes Vertical Line Test
Example 2: In the following graph, y
is not a function of x:
 Fails Vertical Line Test
To determine whether x is a function of y,
given a graph of a relation, use the following criterion: If every horizontal line
you can draw passes through only 1 point, x is a function of y.
If you can draw a horizontal line that passes through 2 points, x is not a function of y. This is called the horizontal line test. Example 1: In the following graph, x
is a function of y:
 Passes Horizontal Line Test
Example 2: In the following graph, x
is not a function of y:
 Fails Horizontal Line Test
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