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Domain: a set of first elements in a relation (all
of the x values). This is the input.
Range: The second elements in a relation
(all of the y values). This is the output.
How would you use your calculator
to solve 52?
Input
5
Output
x2
25
The number you entered is the input
number (or x-value on a graph).
The result is the output number (or yvalue on a graph).
A function is a relation that gives a single
output number for every valid input number
(x values cannot be repeated).
A relation is a rule that produces one or more
output numbers for every valid input number
(x and y values may be repeated).
There are many ways to represent relations:
Graph
Table of values
A set of ordered pairs
Mapping
Equations
These are all ways of
showing a relationship
between two variables.
Function
X values are always located
on the left and y values are
on the right.
They can be represented by
words, symbols or numbers.
This represents a function as every input
value (x) has only been used once.
A Relation is a rule that produces one or more
output numbers for every valid input number
(x and y values may be repeated).
This represents only a relation
because the input value or
x-value of 2 was used twice.
Therefore this relation is not a
Function.
All functions are relations but
not all relations are functions!
Vertical Line Test:
1.Draw a vertical line through the
graph.
2. See how many times the vertical
line intersects the graph.
3. Only Once – Pass (function)
More than Once – Fail (not
function)
To determine if a graph is a function,
we perform the vertical line test.
-- Yes, it is a function.
-- No, it is not a
function.
Is this graph a function?
Only
crosses at
one point.
Yes, this is a function because it passes the vertical
line test.
Is this graph a function?
Crosses at more
than one point.
No, this is not a function because it does not pass
the vertical line test.
Is this graph a function?
Only crosses at
one point.
Yes, this is a function because it passes the vertical
line test.
To determine if a table is a function,
we look at the x column (domain).
If each number in the x column
appears only once in that column,
it is a function.
Also, each number in the x column
corresponds with one and only one
value in the y column.
FUNCTIONS
Many to One
Relationship
One to One
Relationship
X
1
2
3
Y
0
0
0
X
1
2
3
Y
5
7
9
Is this relation a function?
X
1
2
3
4
Y
5
4
3
2
Every number just
appears once.
Yes, this is a function because each number in the
column only appears once.
x
Is this relation a function?
X
7
9
10
10
Y
7
9
8
10
The number 10
appears more than
once.
No, this is not a function because 10 appears in the x column
more than once.
Is this relation a function?
X
1
2
3
4
Y
0
0
0
0
Every number just
appears once.
Yes, this is a function because each number in the
column only appears once.
x
Is this relation a function?
X
Y
0
1
2
2
4
3
2
4
The number 2
appears more than
once.
No, this is not a function because 2 appears in the x column
more than once.
How to Identify Functions and
Relations from Ordered Pairs
List of Ordered Pairs
{(2,3), (-1,5), (4,-2), (9,9),
(0,-6)}
Look at the X’s (numbers
on the left)
List them out.
2, -1, 4, 9, 0
If no X values are
repeated it is a
function.
If X values are
repeated it is a
relation.
Identifying Functions with
Equations