2-1: Relations and Functions
Algebra 2
What is a Relation
A set of inputs and outputs
 Can be represented in 4 different ways:

Ordered Pairs
Table of Values
1,9 
Mapping Diagram
 4, 2 
1, 11
 9,10 
1
4
9
2
11
10
9
Input
x
Output
y
1
9
1
-11
4
-2
9
10
Graph
10
B
A
5
10
C
-5
-10
D
In 2000, the 4 most populous states(in millions), were CA {32}, TX {21}, NY {19}
and FL {16}.The numbers of U.S. Representatives were CA {53}, TX {32}, NY {29}
and FL {25}. How can you represent a relation for these data in 4 different ways?
Ordered Pairs
Table of Values
 32,53
 21,32 
19, 29 
16, 25
Mapping Diagram
Input
x
Output
y
32
53
21
32
19
29
16
25
Graph
32
21
19
16
53
32
29
25
80
60
40
20
20
40
Domain and Range of a Relation
Domain


Range
Set of inputs
x coordinates of ordered
pair
x
y


Set of outputs
y coordinates of ordered
pair
1
2
3
4
5
6
$5.00
$10.00
$15.00
$20.00
$25.00
$30.00
1, 2,3, 4,5,6
Domain: _______________________________
5.00,10.00,15.00, 20.00, 25.00,30.00
Range: _________________________________
Identifying Functions

What is a Function?
◦ A relation in which each element of the
domain corresponds with one and only one
element in the range.
32
21
19
16
53
32
29
25
Complete Got It? p. 62 #3
1). No
2). Yes
1,9 ,  4, 2 , 1, 11 , 9,10
Look for duplicates in the Domain
Independent Variable
Solution depends on it.
Dependent Variable
Depends on the first
variable for a value.
Input
Output
first variable
Usually the ______
in a table
Usually the _______
second
variable in a table
Components of Functions
The Vertical Line Test

Used with graphs
◦ If the vertical line touches the graph in more
than one point, the graph is not a function.
Why the Vertical Line Test Works
 x, y1 
(3, 4 )
 x, y2 
(3, -4 )
Homework: Day 1 p. 65 #11, 13-16, 20, 22, 23, 26, 28, 45-51
odd
Types of Function Notation

Euhler’s Function Notation:
y  f (x )

Mapping Notation:
f : n  3n
T:xx
Euhler’s Notation: y  f (x )
• Notations are used as a short hand
• Used mainly with sentences (equations)
• Letter denotes the name of the function
y  3x  10
f ( x )  3 x  10
Equation
Function
Rule
•Value of the function
•Argument of the function
•Dependent variable
•Independent variable
•Output
•Input
Ex.1: Here, the three distances (thinking, braking, stopping)
are each a function of speed. Rewrite as Function Rules
using Euhler’s f(x).
Thinking distance at speed x:
Braking distance at speed x:
Stopping distance at speed x:
T ( x)  x
y= x.
2
x
B( x) 
20
2
x
y
20
2
x
x
y  x
S ( x)  x 
20
20
2
Using Function Notation
For f  x   3x  2 , what is the output for the
input -4, 0 and 0.2?
x
Input
Function Rule
f  x   3x  2
f  x
Output
-4
f  4  3  4  2
14
0
f  0  3  0   2
2
0.2
f  0.2  3  0.2  2
1.4
Complete Got It? p. 63 #5 a). 9
b). 1
c). -19
A pizza costs $14; the flat delivery fee is $1.50. What
function rule models the total cost of the number of pizzas
delivered? Evaluate for 5 pizzas.
Relate
Total
Cost

Cost per
Pizza
Define
p  # of pizzas
Write
C  p


# of
Pizzas
Delivery
Fee
C  p   Total cost
14

p
C  p   14 p  1.5
Evaluate

C  5  14  5  1.5  71.5
The total cost for 5 pizzas is $71.50

1.5
Complete Got It? p. 64 #6
Let x = # of bottles purchased
C  x   1.19 x
Let C = Total cost
C 15  1.19 15  17.85
Complete Lesson Check p. 64 # 1-4
1).
D  0,3, 4 R  2,1, 2, 4
2).
D  4, 3,0, 4 R  4, 3,0, 4
3). No, 3 has two values in the range.
4). Yes, each value in the domain has only one value in the range.
Homework: Day 2 p. 65 #8-10, 12, 17-19, 25, 29, 30, 32, 39-43,
44-52 even