Basics of Functions and Graphs

2.1 Basics of Functions and
Their Graphs
Chapter 2 Functions and Graphs

Concepts and Objectives
⚫ Objectives for this section are:
⚫ Determine whether a relation or an equation
represents a function.
⚫ Evaluate a function.
⚫ Use the vertical line test to identify functions.
⚫ Identify the domain and range of a function from its
graph
⚫ Identify intercepts from a function’s graph

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 , ,2 , ,3 , 2,3
4,1 , 3,1 , 2,0
1 1 1
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

Finding the Domain
⚫ Keep in mind that, in determining domains and ranges,
we need to consider what is physically possible or
meaningful in real-world examples.
⚫ For example, in a problem involving people, we
would not want to consider partial people, but only
whole ones.
⚫ Another example would be making sure we do not
end up dividing by zero.

Finding the Domain (cont.)
⚫ We can write the domain and range in interval notation,
or set-builder notation.
⚫ Recall that in interval notation, we use a square
bracket [ when the set includes the endpoint and a
parenthesis ( to indicate that the endpoint is either
not included or the interval is unbounded.
⚫ Example: Interpret (–2, 3]
⚫ The parenthesis indicates that the interval does not
include –2 and the bracket indicates that 3 is
included, therefore this could be written as –2 < x ≤ 3

Set-Builder Notation
⚫ In the previous examples, we used inequalities and lists
to describe the domain of functions. Another type of
statement we can use defines sets of values or data to
describe the behavior of the variable in set-builder
notation.
⚫ Example: Interpret {x|10 ≤ x < 30}
⚫ In set notation, the braces { } are read as “the set of,”
and the vertical bar is read as “such that,” so the
statement would be read as “the set of x-values such
that 10 is less than or equal to x, and x is less than
30.”

Set-Builder Notation (cont.)
⚫ If a domain or range is all real numbers, represented in
interval notation as (‒, ), is written in set-builder
notation as either the symbol for real numbers, , or
{x|x  }, where  means “is a member of” or “is an
element of”.
⚫ Since interval notation, by definition, describes an
interval of real numbers, it is not appropriate for
describing a set of distinct numbers such as {1, 2, 3}.
Set-builder notation, however, is capable of handling
such distinctions: {x|x = 1, 2, 3}

Interval vs. Set-Builder Notations
b
a
b
a
b
a b
a
b
a
b
a
Inequality
Interval
Notation
Set-builder Notation Number Line Representation
a < x < b (a, b) {x|a < x < b}
a ≤ x ≤ b [a, b] {x|a ≤ x ≤ b}
a < x ≤ b
a ≤ x < b
(a, b]
[a, b)
{x|a < x ≤ b}
{x|a ≤ x < b}
a < x < 
‒ < x < b
(a, )
(‒, b)
{x|x > a}
{x|x < b}
a ≤ x < 
‒< x ≤ b
[a, )
(‒, b]
{x|x ≥ a}
{x|x ≤ b}
b
a
b
a
b
a
b
a
b
a
b
a

Domain and Range From Graphs
a)

a)
Domain: (−, )
Range: (−, )
or
Domain: {x|x ∈ ℝ}
Range: {y|y ∈ ℝ}

b)

b)
Domain: [−4, 4]
Domain

b)
Domain: [−4, 4]
Range: [−6, 6]
or
Domain: {x|–4 ≤ x ≤ 4}
Range: {y|–6 ≤ y ≤ 6}
Domain
Range

c)

c)
Range: [−3, )
or
Domain: {x|x ∈ ℝ}
Range: {y|y ≥ –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

Intercepts
⚫ Consider the graph below. What do you notice?

Intercepts
Wherever the graph
crosses the x-axis, these
points are the x-intercepts.
They are also called the
zeros of the function.
The y-intercept occurs
where the graph crosses
the y-axis.

Intercepts (cont.)
The zeros of a function f are
the x-values for which
f(x) = 0. The y-intercept is
the y-value found by f(0).
A function can have
multiple x-intercepts, but at
most one y-intercept.

For Next Class
⚫ Section 2.1 in MyMathLab
⚫ Quiz 2.1 in Canvas
⚫ Optional: Read section 2.2 More on Functions

Basics of Functions and Graphs

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