Domain
 The set of all possible input values (generally x values)
 We write the domain in interval notation

Interval notation has 2 important components:

Position

Symbols

Interval Notation – Position
 Has 2 positions: the lower bound and the upper bound
[ 4 , 12 )
Lower Bound
• 1 st Number
• Lowest Possible x-value
Upper Bound
• 2 nd Number
• Highest Possible x-value

Interval Notation – Symbols
 Has 2 types of symbols: brackets and parentheses
[ ] → brackets
[ 4, 12 )
( ) → parentheses



Inclusive (the number is
included)
=, ≤, ≥
● (closed circle)



Exclusive (the number is
excluded)
≠, <, >
○ (open circle)

Understanding Interval Notation
4 ≤ x < 12
 Interval Notation:
 How We Say It: The domain is 4
12 .
 On a Number Line: to


Example – Domain: –2 < x ≤ 6
Interval Notation:
 How We Say It: The domain is –2 to
6 .
 On a Number Line:


Example – Domain: –16 < x < –8
Interval Notation:
 How We Say It: The domain is –16 to
–8 .
 On a Number Line:

Your Turn:
 Complete problems 1 – 3 on the “Domain and
Interval Notation – Guided Notes” handout

Infinity


Infinity is always exclusive!!!

– The symbol for infinity

Infinity, cont.
Negative Infinity Positive Infinity
  


Example – Domain: x ≥ 4
Interval Notation:
 How We Say It: The domain is 4 to
 On a Number Line:


Example – Domain: x is
 all real
Interval Notation: numbers
 How We Say It: The domain is to
 On a Number Line:

Your Turn:
 Complete problems 4 – 6 on the “Domain and
Interval Notation – Guided Notes” handout


Restricted Domain
When the domain is anything besides (– ∞ , ∞ )
 Examples:

3 < x

5 ≤ x < 20

–7 ≠ x

Combining Restricted Domains
 When we have more than one domain restriction, then we need to figure out the interval notation that satisfies all the restrictions
 Examples:
 x ≥ 4, x ≠ 11

–10 ≤ x < 14, x ≠ 0

2.
3.
1.
Combining Multiple Domain
Restrictions, cont.
Sketch one of the domains on a number line.
Add a sketch of the other domain.
Write the combined domain in interval notation.
Include a “U” in between each set of intervals (if you have more than one).

Domain Restrictions: x ≥ 4, x ≠ 11
Interval Notation:

Domain Restrictions: –10 ≤ x < 14, x ≠ 0
Interval Notation:

Domain Restrictions: x ≥ 0, x < 12
Interval Notation:

Domain Restrictions: x ≥ 0, x ≠ 0
Interval Notation:

Challenge – Domain Restriction: x ≠ 2
Interval Notation:

Domain Restriction: –6 ≠ x
Interval Notation:

Domain Restrictions: x ≠ 1, 7
Interval Notation:

Your Turn:
 Complete problems 7 – 14 on the “Domain and Interval Notation – Guided Notes” handout

Answers
7.
9.
11.
13.
8.
10.
12.
14.

1. (–2, 7)
2. (–3, 1]
3. [–9, –4]
4. [–7, –1]
Answers
6. (–∞,4)
7. (–1, 2) U (2, ∞)
8. [–5, ∞)
9. (–2, ∞)
5. (–∞, 6) U (6, 10) U (10, ∞)


Experiment
What happens we type the following expressions into our calculators?
 16 
0
5


16 
5
0


*Solving for Restricted Domains
Algebraically
In order to determine where the domain is defined algebraically , we actually solve for where the domain is undefined!!!
 Every value of x that isn’t undefined must be part of the domain.

*Solving for the Domain Algebraically
 In my function, do I have a square root ?

Then I solve for the domain by: setting the radicand (the expression under the radical symbol) ≥ 0 and then solve for x

Example
 Find the domain of f(x).
f ( x )
 x

2

*Solving for the Domain Algebraically
 In my function, do I have a fraction ?

Then I solve for the domain by: setting the
denominator ≠ 0 and then solve for what x
is not equal to.

Example
 Solve for the domain of f(x).
f ( x )
 x
2 x

6 x

1

*Solving for the Domain Algebraically
 In my function, do I have neither ?

Then I solve for the domain by: I don’t have to solve anything!!!

The domain is (–∞, ∞)!!!

Example
 Find the domain of f(x).
f(x) = x 2 + 4x – 5

*Solving for the Domain Algebraically
 In my function, do I have both ?

Then I solve for the domain by: solving for each
of the domain restrictions independently

Example
 Find the domain of f(x).
f ( x )
 x
2

2 x
 x

30

Additional Example
 Find the domain of f(x).
f ( x )

14 2 x

2

17

***Additional Example
 Find the domain of f(x).
f ( x )

10

5 x
 x
2
1

5 x

6

Additional Example
 Find the domain of f(x).
x
2 
1 f ( x )

4

Your Turn:
 Complete problems 1 – 10 on the “Solving for the Domain Algebraically” handout
 #8 – Typo!
f ( x )
 x
2
1
 x

6

Answers:
1.
3.
5.
2.
4.

Answers, cont:
6.
8.
10.
7.
9.