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Sullivan Algebra and
Trigonometry: Section 3.1
Objectives
• Determine Whether a Relation Represents a
Function
• Find the Value of a Function
• Find the Domain of a Function
• Identify the Graph of a Function
• Obtain Information from or about the Graph of a
Function
Let X and Y be two nonempty sets of real
numbers. A function from X into Y is a rule
or a correspondence that associates with
each element of X a unique element of Y.
The set X is called the domain of the
function.
For each element x in X, the
corresponding element y in Y is called the
image of x. The set of all images of the
elements of the domain is called the
range of the function.
f
x
y
x
y
x
X
DOMAIN
Y
RANGE
Example: Which of the following relations
are function?
{(1, 1), (2, 4), (3, 9), (-3, 9)}
A Function
{(1, 1), (1, -1), (2, 4), (4, 9)}
Not A Function
Functions are often denoted by letters such as f,
F, g, G, and others. The symbol f(x), read “f of
x” or “f at x”, is the number that results when x is
given and the function f is applied.
Elements of the domain, x, can be though of as
input and the result obtained when the function
is applied can be though of as output.
Restrictions on this input/output machine:
1. It only accepts numbers from the
domain of the function.
2. For each input, there is exactly one
output (which may be repeated for
different inputs).
2
f
(
x
)

2
x
5
Example: Given the function
Find: f (3)
f (3)  2(3)  5  23
2
f (x) is the number that results when the
number x is applied to the rule for f.
Find: f ( x  h)
f ( x  h)  2( x  h)  5
2
 2( x  2 xh  h )  5
2
2
 2 x  4 xh  2h  5
2
2
The domain of a function f is the set of real
numbers such that the rule of the function
makes sense.
Domain can also be thought of as the set of
all possible input for the function machine.
Example: Find the domain of the following function:
g ( x )  3x  5x  1
3
Domain: All real numbers
Example: Find the domain of the following function:
4
s( t ) 
t 1
Domain of s is t |t  1
.
Example: Find the domain of the following function:
h( z )  z  2
z20
z  2
Domain of h is z| z  2
.
When a function is defined by an equation in x
and y, the graph of the function is the graph of
the equation, that is, the set of all points (x,y)
in the xy-plane that satisfies the equation.
Vertical Line Test for Functions:
A set of points in the xy-plane is the graph of a
function if and only if a vertical line intersects
the graph in at most one point.
Example: Does the following graph
represent a function?
y
x
The graph does not represent a function, since
it does not pass the vertical line test.
Example: Does the following graph
represent a function?
y
x
The graph does represent a function, since it
does passes the vertical line test.
Determine the domain, range, and intercepts of
the following graph.
y
4
(2, 3)
(4, 0)
0
(0, -3)
-4
(1, 0)
(10, 0)
x