Algebra II: Functions
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:
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
:
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
:
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
