3.2 Domain and Range

3.2 Domain and Range
Chapter 3 Functions

Concepts & Objectives
⚫ Objectives for this section are:
⚫ Find the domain of a function defined by an equation
⚫ Graph piecewise-defined functions

Finding the Domain
⚫ In this section, we will practice determining domains
and ranges for specific functions.
⚫ 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,
as we did with inequalities.
⚫ 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.

Interval Notation Summary
b
a
b
a
b
a b
a
b
a
b
a
Name of
Interval
Notation
Inequality
Description
Number Line Representation
finite, open
(a, b) a < x < b
finite, closed
[a, b] a ≤ x ≤ b
finite, half-
open
(a, b]
[a, b)
a < x ≤ b
a ≤ x < b
infinite, open
(a, )
(‒, b)
a < x < 
‒ < x < b
infinite,
closed
[a, )
(‒, b]
a ≤ x < 
‒< x ≤ b
b
a
b
a
b
a
b
a
b
a
b
a

Find the Domain From a List
⚫ Example: Find the domain of the following function:
( ) ( ) ( ) ( ) ( )
 
2,10 , 3,10 , 4,20 , 5,30 , 6,40

Find the Domain From a List
⚫ Example: Find the domain of the following function:
First identify the input values. The input value is the
first coordinate in an ordered pair (the x-coordinate).
Since the ordered pairs are listed, the domain is the set
of the first coordinates of the ordered pairs.
{2, 3, 4, 5, 6}
( ) ( ) ( ) ( ) ( )
 
2,10 , 3,10 , 4,20 , 5,30 , 6,40

Finding the Domain of a Function
⚫ To find the domain of a function written in equation
form
⚫ Identify the input values
⚫ Identify any restrictions on the input and exclude
those values from the domain
⚫ Write the domain in interval form, if possible

⚫ Example: Find the domain of the function
The input value, x, is squared and then the result is
decreased by 1. Any real number may be squared and
then lowered by 1, so there are no restrictions on the
domain of this function. Therefore, the domain is the
set of real numbers.
In interval form, the domain of f is (‒, ).
( ) 2
1
f x x
= −

⚫ How to identify domain restrictions:
⚫ If there is a denominator with a variable, set the
denominator equal to zero and solve for the variable.
⚫ If the there is a variable in the radicand (inside) of an
even root (square root, fourth root, etc.), set the
radicand greater than or equal to zero and solve.
⚫ When you write the domain in interval form, make sure
to exclude any restricted values from the domain.

The domain cannot include any values that would force
the denominator to be zero, so we will set the
denominator equal to zero and solve for x.
Now, we will exclude 2 from the domain:
( means “union” ‒ combine the two intervals together)
( )
1
2
x
f x
x
+
=
−
2 0
2
x
x
− =
=
( ) ( )
,2 2,
− 

In the real number system, the square root of a negative
number does not exist.
So our domain must be numbers less than or equal to 7,
or (‒, 7].
( ) 7
f x x
= −
7 0
7
7
x
x
x
− 
−  −


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: In set notation, the braces { } are read as “the
set of,” and the vertical bar is read as “such that,” so the
statement {x | 10 ≤ x < 30} 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.)
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

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”.

Domain and Range from Graphs
⚫ Another way to identify the domain and range of
functions is by using graphs.
⚫ Because the domain refers to the set of possible input
values, the domain of a graph consists of all the input
values shown on the x-axis.
⚫ The range is the set of possible output values, which are
shown on the y-axis.
⚫ Keep in mind that if the graph continues beyond the
portion of the graph we can see, the domain and range
may be greater than the visible values.

⚫ We can see that the
graph extends horizon-
tally from ‒5 to the right
without bound, so the
domain is [‒5, ).
⚫ The vertical extent of the
graph is all range values
5 and below, so the
range is (‒, 5]
Domain
Range

Example: Find the domain
and range of the function f
whose graph is shown.

Example: Find the domain
and range of the function f
whose graph is shown.
⚫ The horizontal extent of
the graph is ‒3 to 1, so
the domain of f is (‒3, 1].
⚫ The vertical extent of the
graph is 0 to ‒4, so the
range is [‒4, 0).
Domain
Range



Finding the Domain and Range
⚫ Example: Find the domain and range of
The square root of a negative number is not real, so the
value inside the radical must be nonnegative.
Domain of f(x) is [‒4, )
We know that f(‒4) = 0, and the function increases as x
increases without any upper limit; therefore, the range
of f is [0, ).
( ) 2 4
f x x
= +
4 0
4
x
x
+ 
 −

Piecewise-Defined Functions
⚫ The absolute value function is defined by different rules
over different intervals of its domain. Such functions are
called piecewise-defined functions.
⚫ If you are graphing a piecewise function by hand, graph
each piece over its defined interval. If necessary, use
open and closed circles to mark discontinuities.
⚫ If you are using Desmos to graph a piecewise function,
you can control the interval graphed by putting braces
after the function.
⚫ You can make open circles by plotting the point and
changing the type of point used.

Piecewise-Defined Functions
⚫ Example: Graph the function.
( )
2 5 if 2
1 if 2
x x
f x
x x
− + 

= 
+ 


Classwork
⚫ College Algebra 2e
⚫ 3.2: 6-24 (even); 3.1: 28-58 (even); 2.7: 34-52 (even)
⚫ 3.2 Classwork Check
⚫ Quiz 3.1

3.2 Domain and Range

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