3.1 Functions and Function Notation

3.1 Functions and Function
Notation
Chapter 3 Functions

Concepts and Objectives
⚫ Objectives for this section are:
⚫ Determine whether a relation represents a function.
⚫ Find the value of a function.
⚫ Determine whether a function is one-to-one.
⚫ Use the vertical line test to identify functions.
⚫ Graph the functions listed in the library of functions.

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 1
y 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 1
y x
= −
2 1 0
x − 
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.
⚫ We usually read this as “y = f of x”.
( ),
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
4
x y

One-to-One Functions
⚫ In a one-to-one function, each x-value corresponds to
only one y-value, and each y-value corresponds to only
one x-value. In a 1-1 function, neither the x nor the y can
repeat.
⚫ We can also say that f (a) = f (b) implies a = b.
A function is a one-to-one function if, for
elements a and b in the domain of f,
a ≠ b implies f (a) ≠ f (b).

One-to-One Functions (cont.)
⚫ Example: Decide whether is one-to-one.
( )= − +
3 7
f x x

We want to show that f (a) = f (b) implies that a = b:
Therefore, f is a one-to-one function.
( )= − +
3 7
f x x
( ) ( )
f a f b
=
3 7 3 7
a b
− + = − +
3 3
a b
− = −
a b
=

( ) 2
2
f x x
= +

This time, we will try plugging in different values:
Although 3 ≠ ‒3, f (3) does equal f (‒3). This means that
the function is not one-to-one by the definition.
( ) 2
2
f x x
= +
( ) 2
3 3 2 11
f = + =
( ) ( )
2
3 3 2 11
f − = − + =

⚫ Another way to identify whether a function is one-to-one
is to use the horizontal line test, which says that if any
horizontal line intersects the graph of a function in more
than one point, then the function is not one-to-one.
•
•
one-to-one not one-to-one

“Toolkit Functions”
⚫ When working with functions, it is helpful to have a base
set of building-block elements.
⚫ We call these our “toolkit functions,” which form a set of
basic named functions for which we know the graph,
formula, and special properties.
⚫ For these definitions we will use x as the input variable
and y = f(x) as the output variable.
⚫ We will see these functions and their combinations and
transformations throughout this course, so it will be
very helpful if you can recognize them quickly.

Constant Function f(x) = c
⚫ f(x) = c is constant on its entire domain, [c, c].
⚫ It is continuous on its entire domain, (–∞ ∞)
Domain: (–∞ ∞) Range: [c, c]
x y
–2 c
–1 c
0 c
1 c
2 c
c

Identity Function f(x) = x
⚫ f(x) = x is increasing on its entire domain, (–∞ ∞).
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
–2 –2
–1 –1
0 0
1 1
2 2

Absolute Value Function
⚫ decreases on the interval (–∞ 0] and
increases on the interval [0, ∞).
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
-2 2
-1 1
0 0
1 1
2 2
( )
f x x
=
( )
f x x
=

Quadratic Function f(x) = x2
⚫ f(x) = x2 decreases on the interval (–∞ 0] and increases
on the interval [0, ∞).
Domain: (–∞ ∞) Range: [0 ∞)
x y
–2 4
–1 1
0 0
1 1
2 4
vertex

Cubic Function f(x) = x3
⚫ f(x) = x3 increases on its entire domain, (–∞ ∞).
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
–2 –8
–1 –1
0 0
1 1
2 8

Reciprocal Function
⚫ f(x) = 1/x has a vertical asymptote at x = 0
⚫ It is not continuous on its entire domain, (–∞, 0)  (0 ∞)
Domain: (–∞, 0)  (0 ∞) Range: (–∞, 0)  (0 ∞)
x y
–2 ‒0.5
–1 –1
‒0.5 ‒2
0.5 2
1 1
2 0.5
( )
1
f x
x
=

Reciprocal Squared
⚫ f(x) = 1/x2 has asymptotes at x = 0 and y = 0
⚫ It is not continuous on its entire domain, (–∞, 0)  (0 ∞)
Domain: (–∞, 0)  (0 ∞) Range: (0 ∞)
x y
–2 0.25
–1 1
‒0.5 4
0.5 4
1 1
2 0.25
( ) 2
1
f x
x
=

Square Root Function
⚫ increases on its entire domain, [0 ∞).
⚫ It is continuous on its entire domain, [0 ∞)
Domain: [0 ∞) Range: [0 ∞)
x y
0 0
1 1
4 2
9 3
16 4
( )
f x x
=
( )
f x x
=

Cube Root Function
⚫ increases on its entire domain, (–∞ ∞).
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
-8 -2
-1 -1
0 0
1 1
8 2
( ) 3
f x x
=
( ) 3
f x x
=

Classwork
⚫ College Algebra 2e
⚫ 3.1: 6-26 (even); 2.7: 24-32 (even); 2.6: 30-40 (even)
⚫ 3.1 Classwork Check
⚫ Quiz 2.7

3.1 Functions and Function Notation

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