Functions and Relations
Definitions
Domain, Independent variable,
Input – the set of x-values
Example:
Given {(2, 3), (6, 4), (-1, 0)}
The domain is {2, 6, -1}
Range, Dependent variable,
Output – The set of y-values
Example:
Given {(2, 3), (6, 4), (-1, 0)}
The range is {3, 4, 0}
Function - relates each element of
a set with exactly one element of
another set.
Example:
The relation {(2, 6),(1, -4),(0, 6)} is
a function because all the x-values
are different
Relation – a set of ordered pair
(x/y coordinate pair)
Example:
(9, 6)
There are a few ways to determine if a relation is a function or not.
1.
Ordered pairs – look at the x-value, if any of the numbers
are the same it is NOT a function. If all x-values are a different
number, it is a function.
Example:
Function
Not a Function
2.
Table – look at only the x column, if any of the numbers
are the same it is NOT a function. If all x-values are a different
number, it is a function.
Example:
Function
Not a Function
Mapping – an illustration that pairs
each domain with its range
Example:
The relation {(2, 6),(1, -4),(0, 6)}
can be mapped as follows:
x
y
2
1
0
6
-4
x/y table – A 2 column table
where the domain is paired up with
its range in each row
Example:
The table for the relation
{(2, 6),(1, -4),(0, 6)} would be:
x
2
1
0
y
6
-4
6
3.
Mapping – look at the arrows coming off the numbers in
the x circle. If there is only 1 arrow for each x-value, it is a
function. If there is more than 1 arrow coming off any of the xvalues, it is NOT a function.
Example:
Function
Not a Function
4. Graph – when given a graph, we use the vertical line test to determine if it is a function or not.
When you move a line across your graph vertically (the same way as the y-axis):
 If the vertical line you are moving touches the graph in more than one place at any time,
it is NOT a function.
 If when you move the line across your graph and it only touches in one place at all times,
it is a function.
Example:
Function:
Notice that where each of the solid lines
are crossing the given graphed line
(where the arrows are pointing), there
is only ONE point of intersection.
Not a Function:
Notice where each of the vertical lines
cross the graph, they intersect with the
graph in at least 2 separate places. This
means it is NOT a function.
Is it a function or not? If not, circle or highlight the part that makes it not a function.
1. {(9, 3), (4, 6), (3, 9), (-3, 8)}
4.
x
1
3
2
4
2. {(2, 2), (1, 0), (-1, 0), (5, 2)}
5.
y
8
7
6
5
7.
y
3
-1
5
3
8.
x
-6
4
0
4
10.
x
-1
0
1
2
11.
6.
9.
x
-6
4
0
4
y
7
8
2
-1
3. {(2, 3), (4, 9), (2, 0), 1, 6)}
y
7
8
2
-1
12.
x
4
4
4
4
x
-6
4
0
4
y
8
2
3
5
y
7
8
2
Use the given illustration to fill in the rest of the chart. Determine if the relation is a function or not.
Ordered pair
x/y table
Mapping
1
{(2, 6), (-3, 5),
(0, 1), (-2, 2)}
2
x
3
2
-1
2
3
x
4
5
2
3
4.
y
3
-2
1
0
y
5
0
3
4
Graph
Use the given illustration to fill in the rest of the chart. Determine if the relation is a function or not.
Ordered pair
x/y table
Mapping
5.
{(1, -3), (3, 5),
(0, 1), (1, 2)}
6.
x
3
0
-1
3
7.
x
2
5
-1
3
8.
y
1
1
1
1
y
5
2
3
-4
Graph
Use the given illustration to fill in the rest of the chart. Determine if the relation is a function or not.
Ordered pair
x/y table
Mapping
9
{(2, 0), (3, 5),
(0, 0), (-4, 1)}
10.
x
4
-1
2
11.
x
-3
4
2
3
12.
y
3
3
1
0
y
5
0
3
4
Graph