Relations and functions

Definition
- A set of ordered
pairs.
- The domain of a
relation is the set of
first coordinates.
- The range is the set
of second
coordinates.
Relation

Definition
x-values
are inputs,
domain,
independent
variable
y-values
are outputs,
range,
dependent
variable

Example of relation
{(0, -5) , (1, -4) , (2, -3), (3, -2) , (4, -1), (5, 0)}
{(1, 5),(4, -4),(6, -3)

Identifying domain and range
What are the domain?
{( , ),( , ),( , ),( , ),( , ),( , )}    0 5 1 4 2 3 3 2 4 1 5 0
What are the range?
(0, 1, 2, 3, 4, 5)
(-5, -4, -3, -2, -1, 0)

Definition
- a SPECIAL
RELATION in
which each element
of the domain
corresponds to
exactly one
element of the
range.
Function

Characteristics
of Functions
1. Each element
in domain X must
be matched with
exactly one
element in range
Y.

Characteristics
of Functions 2. Some elements
in Y may not be
matched with
any element in X.

Characteristics
of Functions
3. Two or more
elements in X
may be matched
with the same
element in Y.

A function can
be represented
in different
ways.
1. Table of
Values
2. Ordered Pairs
3. Mapping
4. Graph
5. An equation

Example of table of values
Is this
relation
a
function?
YES!
What
are the
domain?
(0,1,2,3)
What
are the
range?
(4,-1,7,6)
x y
0 4
1 -1
2 7
3 6

Example of ordered pairs
{(5,1), (7,2), (4,-9), (0,2)
Is this
relation
a
function?
YES!
What
are the
domain?
(5,7,4,0)
What
are the
range?
(1,2,-9,2)

Example of mapping
Is this
relation
a
function?
YES!
What
are the
domain?
(3,1,0)
What
are the
range?
(-1,2,3)
3
1
0
–1
2
3

Set A is the domain
1
2
3
4
5
Set B is the range
2
10
8
6
4
Must use all the x’s
Let’s look at another relation and decide if it is a function.
The x value can only be assigned to one y
This is a function ---it
meets our conditions

1
2
3
4
5
2
10
8
6
4
Is the relation shown above a function?
NO Why not???
2 was assigned both 4
and 10

Set A is the domain
1
2
3
4
5
Set B is the range
2
10
8
6
4
This is not a function---
it doesn’t assign each
x with a y
Check this relation out to determine if it is a function.

Example of graph
Is this
relation
a
function?
YES!

-
Vertical
Line Test
in function

Definition
- A graph represents
a function if and
only if no vertical
line intersects the
graph in more than
one point. Vertical
Line Test

y
x
5
5
-5
-5
y
x
5
5
-5
-5
Example in Vertical Line Test
Determine if the graph is a function,
a) b)
Not a
function function

Relations and functions

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Relations and functions