Lesson 1 – Characteristics & Properties of Functions
Warm-up: Prerequisite Skills
Relation:
A relationship between two variables, values of the independent variable
are with values of the dependent variable.
Relations may be described in various ways:
1. LISTING the elements.
2. TABLE OF VALUES
Ex. { (1,2), (3,4), (5,6), (7,8)}
x
y = 3x + 1
–1
–2
0
1
1
4
2
7
3. GRAPHICALLY
4. MAPPING DIAGRAM
𝑥
𝑦
0
–4
–2
0
2
4
2
4
5. EQUATIONS
ex. 𝑦 = 2𝑥 − 5
DOMAIN:
The set of all values for which the independent variable is defined
(usually x).
RANGE:
The set of all values for which the dependent variable is defined
(usually y).
Example 
State the domain and range for each of the following:
a)
b)
x
y
5
2
5
4
5
6
5
8
D=
R=
D=
R=
c)
{ (1,3), (2, -4), (3,5), (4,5)}
d)
x
y
D=
0
R=
2
4
D=
R=
–4
–2
0
2
4
Function:
A relation in which there is only one value of the dependent variable for
each value of the independent variable.
THE DOMAIN CANNOT REPEAT!!
If each vertical line drawn intersects the graph of a relation in at most one
point, then the relation is a function.
Vertical Line
Test (VLT)
Example 2 For each of the following relations, determine the domain and range using real
numbers. State whether or not the relation is a function.
a)
b)
c)
D=
D=
D=
R=
R=
R=
Function?
Function?
Function?
FUNCTION NOTATION:
Example 
a)
f(–3)
f(x) is read “the value of f at x” or “f of x”. It is used to
represent the value of the dependent variable
(y) for a given value of the independent variable (x).
(y and f(x) are the same!)
If f(x) = 3x2 – 4, determine the value of
b)
f(½)