2.3 Functions

2.3 Functions
Chapter 2 Graphs and Functions

Concepts and Objectives
 Relations and functions
 Determining functions from graphs or equations
 Function notation

Functions
 A relation is a set of ordered pairs.
 A function is a relation in which, for each distinct value of
the first component of the ordered pairs, there is exactly
one value of the second component.
 More formally:
If A and B are sets, then a function f from A to B
(written f: A  B)
is a rule that assigns to each element of A
a unique element of set B.

Functions (cont.)
 The set A is called the domain of the function f
 Every element of A must be included in the function.
 The set B is called the codomain of f
 The subset of B consisting of those elements that are
images under the function f is called the range.
 The range and the codomain may or may not be the
same.

Functions (cont.)
 In terms of ordered pairs, a function is the set of ordered
pairs A, f A.
 Historical note: The notation f x for a function of a
variable quantity x was introduced in 1748 by Leonhard
Euler in his text Algebra, which was the forerunner of
today’s algebra texts. Many other mathematical symbols
in use today (such as e and ) were introduced by Euler
in his writings.

Functions (cont.)
 Example: Decide whether each relation defines a
function.
      
        
      
1,2 , 2,4 , 3, 1
1,1 , 1,2 , 1,3 , 2,3
4,1 , 3,1 , 2,0
F
G
H
  

   

Functions (cont.)
 Example: Decide whether each relation defines a
function.
 F and H are functions, because for each different
x-value, there is exactly one y-value.
 G is not a function, because one x-value corresponds
to more than one y-value.
      
        
      
1,2 , 2,4 , 3, 1
1,1 , 1,2 , 1,3 , 2,3
4,1 , 3,1 , 2,0
F
G
H
  

   

Functions (cont.)
 Relations and functions can also be expressed as a
correspondence or mapping from one set to another.
 Note that H is a function, since the x-values don’t repeat,
even if the y-values do.
2
1
3
1
2
4
F
x-values y-values
F is a function.
1
2
1
2
3
G
x-values y-values
G is not a function.

Domain and Range
Example: Give the domain and range of each relation.
Determine whether the relation defines a function.
a)
b)
        3, 1 , 4,2 , 4,5 , 6,8
3
4
6
7
100
200
300

Domain and Range
Determine whether the relation defines a function.
a)
b)
        3, 1 , 4,2 , 4,5 , 6,8
3
4
6
7
100
200
300
D: {3, 4, 6}; R: {1, 2, 5, 8}
not a function (the 4s
repeat)
D: {3, 4, 6, 7};
R: {100, 200, 300}
function

Domain and Range From Graphs
a)

a)
Domain: , 
Range: , 

b)

b)
Domain: [4, 4]
Domain

b)
Domain: [4, 4]
Range: [6, 6]
Domain
Range

c)

c)
Range: [3, 

The Vertical Line Test
 Graphs (a) and (c) are relations that are functions – that
is, each x-value corresponds to exactly one y-value.
Since each value of x leads to only one value of y in a
function, any vertical line drawn through the graph of a
function must intersect the graph in at most one point.
This is the vertical line test for a function.
 Graph (b) is not a function because a vertical line
intersects the graph at more than one point.
 A practical way to test this is to move your pencil or pen
across the graph. If it touches the graph in more than
one place, it’s not a function.

Identifying Functions
Example: Decide whether each relation defines a function
and give the domain and range.
a) y = x + 4

a) y = x + 4
Since each value of x corresponds to one value of y, this
is a function. There are no restrictions on x, so the
domain is , . The value of y is always 4 greater
than x, so the range is also , .

b) 2 1y x 

b)
For any choice of x in the domain, there is exactly one
corresponding value for y since the radical is a
nonnegative number; so this equation defines a
function. The quantity under the radical cannot be
negative, thus,
Domain:
Range: [0, 
2 1y x 
2 1 0x  
1
2
x 
1
,
2
 
 

c)
2
y x

c)
If we look at the graph of this relation, we can see that it
fails the vertical line test, so it is not a function.
2
y x

c)
If we look at the graph of this relation, we can see that it
fails the vertical line test, so it is not a function.
Even without the graph, we can see
that the ordered pairs 4, 2 and
4, 2 both satisfy the equation.
Domain: [0, 
Range: , 
2
y x

Function Notation
 When a function f is defined with a rule or an equation
using x and y for the independent and dependent
variables, we say “y is a function of x” to emphasize that
y depends on x. We use the notation
called function notation, to express this.
 ,y f x

Function Notation (cont.)
 For the most part, we use f x and y interchangeably to
denote a function of x, but there are some subtle
differences.
 y is the output variable, while f x is the rule that
produces the output variable.
 An equation with two variables, x and y, may not be a
function at all.
Example: is a circle, but not a function 2 2
4x y

Classwork
 2.3 Assignment (College Algebra)
 2.3 – pg. 213: 4-36 (4); 2.2 – pg. 199: 20-30 (even);
2.1 – pg. 193: 44-54 (even)
 2.3 Classwork Check
 Quiz 2.2

2.3 Functions

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