4-3 Functions
•A relation is a function provided there is exactly
one output for each input.
•It is NOT a function if one input has more than
one output
In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE
SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE
OUTPUT
INPUT
Functions
(DOMAIN)
FUNCTION
MACHINE
OUTPUT (RANGE)
Example 6
Which of the following relations are
functions?
R= {(9,10, (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}
No two ordered pairs can have the
same first coordinate
(and different second coordinates).
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
1
3
-2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped
onto exactly one output
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
-2
4
1
4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1
1. {(2,5) , (3,8) , (4,6) , (7, 20)}
2. {(1,4) , (1,5) , (2,3) , (9, 28)}
3. {(1,0) , (4,0) , (9,0) , (21, 0)}
The Vertical Line Test
If it is possible for a vertical line
to intersect a graph at more
than one point, then the graph
is NOT the graph of a function.
Page 117
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(4,4)
(1,1)
(1,-2)
Function?
No, Two points are on
The same vertical line.
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line
Examples
 I’m
going to show you a series of
graphs. **don’t write 
 Determine whether or not these
graphs are functions.
 You do not need to draw the graphs in
your notes. **or write this note
#1
Function?
#2 Function?
#3 Function?
#4 Function?
#5
Function?
#6
Function?
#7 Function?
#8 Function?
#9 Function?
Function Notation
f (x )
“f of x”
Input = x
Output = f(x) = y
Before…
Now…
y = 6 – 3x
f(x) = 6 – 3x
x
y
x
f(x)
-2
12
-2
12
-1
9
-1
9
0
6
0
6
1
3
1
3
2
0
2
0
(x, y)
(input, output)
(x, f(x))
Example.
f(x) =
2
2x
–3
Find f(0), f(-3), f(5).