3.6 Introduction to Functions
1
Understand the definition of a relation.
2
Understand the definition of a function.
3
Decide whether an equation defines a function.
4
Find domains and ranges.
5
Use function notation.
6
Apply the function concept in an application.
Understand the definition of a relation.
In an ordered pair (x, y), x and y are called the components of the
ordered pair.
Any set of ordered pairs is called a relation.
The set of all first components of the ordered pairs of a relation is the
domain of the relation, and the set of all second components of the
ordered pairs is the range of the relation.
EXAMPLE 1 Identifying Domains and Ranges of Relations
Identify the domain and range of the relation.
 4,0 ,  6,1 ,  7,1 ,  3, 2
Solution:
Domain:
3, 4,6,7
Range:
0,1, 2
Understand the definition of a function.
A very important type of relation called a function.
Function
A function is a set of ordered pairs in which each first component
corresponds to exactly one second component.
By definition, the relation in the following order pairs is not a function,
because the same first component, 3, corresponds to more then one
second component.
3, 5 , 3, 63,73,8
If the ordered pairs from this example were interchanged, giving the
relation
5, 3 ,  6, 3 7, 38,3,
the result would be a function.
In that case, each domain element (first component) corresponds
to exactly one range element (second component).
Mapping
Relations and functions can also be expressed as
a correspondence or mapping from one set to
another. In the example below the arrows from 1
to 2 indicates that the ordered pair (1, 2) belongs
to F. Each first component is paired with exactly
one second component.
x-axis values
y-axis values
1
–2
3
2
4
–1
Mapping
In the mapping for relations H, which is not a
function, the first component –2 is paired with two
different second components, 1 and 0.
x-axis values
y-axis values
–4
–2
1
0
EXAMPLE 2 Determining Whether Relations Are Functions
Determine whether each relation is a function.
 2,8 ,  1,1 , 0,0 , 1,1 ,  2,8 ,
Solution: function
5, 2 , 5,1 , 5,0
Solution: not a function
Decide whether an equation defines a function.
Given the graph of an equation, the definition of a function can be
used to decide whether or not the graph represents a function. By the
definition of a function, each x-value must lead to exactly one y-value.
Vertical Line Test
If a vertical line intersects a graph in more than one point, then the
graph is not the graph of a function.
Any nonvertical line is the graph of a function. For this reason, any
linear equation of the form y = mx + b defines a function. (Recall
that a vertical line has an undefined slope.)
USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each
relation graphed is a function.
y
a.
(1, 2)
(– 1, 1)
This graph
represents a
x function.
(0, – 1)
(4, – 3)
EXAMPLE 3 Determining Whether Relations Define Functions
Determine whether each relation is a function.
Solution: not a function
Solution: function
USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each
relation graphed is a function.
y
b.
6
x
–4
4
–6
This graph fails the
vertical line test, since
the same x-value
corresponds to two
different y-values;
therefore, it is not the
graph of a function.
USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each
relation graphed is a function.
y
c.
x
This graph
represents a
function.
USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each
relation graphed is a function.
y
d.
x
This graph
represents a
function.
Find domains and ranges.
By the definitions of domain and range given for relations, the set of
all numbers that can be used as replacements for x in a function is the
domain of the function. The set of all possible values of y is the range
of the function.
FINDING DOMAINS AND RANGES
OF RELATIONS
Give the domain and range of the relation.
Tell whether the relation defines a function.
a. (3, 1),(4,2),(4,5),(6,8)
The domain, the set of x-values, is {3, 4, 6};
the range, the set of y-values is {–1, 2, 5, 8}.
This relation is not a function because the
same x-value, 4, is paired with two different
y-values, 2 and 5.
FINDING DOMAINS AND RANGES
OF RELATIONS
Give the domain and range of the relation.
Tell whether the relation defines a function.
b.
4
6
7
–3
10
0
20
0
30
0
The domain is {4, 6, 7, –3}; the range is
{100, 200, 300}. This mapping defines a
function. Each x-value corresponds to
exactly one y-value.
FINDING DOMAINS AND RANGES
OF RELATIONS
Give the domain and range of the relation.
Tell whether the relation defines a function.
c.
x
–5
0
5
y
2
2
2
This relation is a set of
ordered pairs, so the
domain is the set of xvalues {–5, 0, 5} and the
range is the set of y-values
{2}. The table defines a
function because each
different x-value
corresponds to exactly one
y-value.
FINDING DOMAINS AND RANGES
FROM GRAPHS
Give the domain and range of each relation.
y
a.
(1, 2)
(– 1, 1)
x
(0, – 1)
The domain is the set
of x-values which are
{– 1, 0, 1, 4}.
The range is the set of
y-values which are
{– 3, – 1, 1, 2}.
(4, – 3)
FINDING DOMAINS AND RANGES
FROM GRAPHS
Give the domain and range of each relation.
y
b.
6
x
–4
4
The x-values of the
points on the graph
include all numbers
between –4 and 4,
inclusive. The yvalues include all
numbers between –6
and 6, inclusive.
The domain is [–4, 4].
The range is [–6, 6].
–6
FINDING DOMAINS AND RANGES
FROM GRAPHS
Give the domain and range of each relation.
y
c.
x
The arrowheads
indicate that the line
extends indefinitely
left and right, as well
as up and down.
Therefore, both the
domain and the range
include all real
numbers, written
(– , ).
FINDING DOMAINS AND RANGES
FROM GRAPHS
Give the domain and range of each relation.
y
d.
The arrowheads indicate
that the line extends
indefinitely left and right,
as well as upward. The
domain is (– , ).
Because there is at least
x y-value, –3, the range
includes all numbers
greater than, or equal to
–3 or [–3, ).
EXAMPLE 4 Finding the Domain and Range of Functions
Find the domain and range of the function y = x2 + 4.
Solution:
Domain:
 ,  
Range:
4,
Use function notation.
The letters f, g, and h are commonly used to name functions. For
example, the function y = 3x + 5 may be written
f  x   3x  5,
where f (x), which represents the value of f at x, is read “f of x.” The
notation f (x) is another way of writing y in a function. For the function
defined by f (x) = 3x + 5, if x = 7, then
f  7  3 7  5
 21  5  26.
Read this result, f (7) = 26, as “f of 7 equals 26.” The notation f (7)
means the values of y when x is 7. The statement f (7) = 26 says
that the value of y = 26 when x is 7. It also indicates that the point
(7,26) lies on the graph of f.
The notation f (x) does not mean f times x; f (x) means the value of x for
the function f. It represents the y –value that corresponds to x in the
function f.
Use function notation. (cont’d)
Function Notation
In the notation f (x),
f
is the name of the function,
x
is the domain value,
and
f (x)
is the range value y for the domain value x.
EXAMPLE 5 Using Function Notation
Find f (−1), for the function.
f (x) = 6x − 2
Solution:
f  1  6  1  2
f  1  6  2
f  1  8
USING FUNCTION NOTATION
For each function, find (3).
b.   ( 3,5),(0,3),(3,1),(6, 1)
Solution For  = {( – 3, 5), (0, 3), (3, 1), (6, – 9)},
we want (3), the y-value of the ordered pair
where x = 3. As indicated by the ordered pair
(3, 1), when x = 3, y = 1,so(3) = 1.
Increasing, Decreasing, and
Constant Functions
Suppose that a function  is defined over an
interval I. If x1 and x2 are in I,
(a)  increases on I if, whenever x1 < x2,
(x1) < (x2)
(b)  decreases on I if, whenever x1 < x2,
(x1) > (x2)
(c)  is constant on I if, for every x1 and x2,
(x1) = (x2)
DETERMINING INTERVALS OVER WHICH A
FUNCTION IS INCREASING, DECREASING, OR
CONSTANT
Determine the intervals over which the function
is increasing, decreasing, or constant.
y
6
2
x
–2
1
3
DETERMINING INTERVALS OVER WHICH A
FUNCTION IS INCREASING, DECREASING, OR
CONSTANT
Determine the intervals over which the function
is increasing,
decreasing, or constant.
y
Solution
6
2
–2
1
3
On the interval (–, 1), the
y-values are decreasing;
on the interval [1,3], the yvalues are increasing; on
x
the interval [3, ), the yvalues are constant (and
equal to 6).
EXAMPLE 6 Applying the Function Concept to Population
The median age at first marriage for women in the United States for
selected years is given in the table.
Write a set of ordered pairs that
defines a function f for these data.
Solution:
f  7  3 7  5
f   2000, 25.1 2002, 25.3 2004, 25.3 2006, 25.9
Give the domain and range of f.
domain: 2000, 2002, 2004, 2006
range: 25.1, 25.3, 25,9
Find f (2006).
25.9
Khan Academy Videos
http://www.khanacademy.org/math/algebra/algebrafunctions/v/introduction-to-functions
http://www.khanacademy.org/math/algebra/algebrafunctions/v/functional-relationships-1
http://www.khanacademy.org/math/algebra/algebrafunctions/v/testing-if-a-relationship-is-a-function
http://www.khanacademy.org/math/algebra/algebrafunctions/v/graphing-a--basic-function
http://www.khanacademy.org/math/algebra/algebrafunctions/v/domain-and-range-of-a-function
http://www.khanacademy.org/math/algebra/algebrafunctions/v/domain-and-range-1
http://www.khanacademy.org/math/algebra/algebrafunctions/v/domain-and-range-2