Transcript Interval Notation

Domain and Interval
Notation
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
Upper Bound
• 1st Number
• 2nd Number
• Lowest Possible x-value
• Highest Possible x-value
Interval Notation – Symbols

Has 2 types of symbols: brackets and
parentheses
[4, 12)
[ ] → brackets



Inclusive (the number is
included)
=, ≤, ≥
● (closed circle)
( ) → parentheses



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
6
.

On a Number Line:
to
Example – Domain: –16 < x < –8

Interval Notation:

How We Say It: The domain is –16
–8
.

On a Number Line:
to
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

On a Number Line:
to
Example – Domain: x is 

Interval Notation:

How We Say It: The domain is

On a Number Line:
all real
numbers
to
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
Combining Multiple Domain
Restrictions, cont.
1.
2.
3.
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.
8.
9.
10.
11.
12.
13.
14.
Golf !!!
1.
2.
3.
4.
5.
Answers
6. (–∞,4)
(–2, 7)
(–3, 1]
7. (–1, 2) U (2, ∞)
[–9, –4]
8. [–5, ∞)
[–7, –1]
9. (–2, ∞)
(–∞, 6) U (6, 10) U (10, ∞)
Experiment

What happens we type the following expressions
into our calculators?


16
 16

0

5
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).
2
f (x ) 
x  6x
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) = x2 + 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 ) 
 2x
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 
1
2
x  5x  6
Additional Example

Find the domain of f(x).
2
f (x ) 
x 1
4
Your Turn:


Complete problems 1 – 10 on the “Solving for
the Domain Algebraically” handout
#8 – Typo!
f (x ) 
1
2
x x6
Answers:
1.
2.
3.
4.
5.
Answers, cont:
6.
7.
8.
9.
10.