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• 1st Number• Lowest Possible x-value

Upper Bound• 2nd Number

• Highest Possible x-value

Interval Notation – Symbols

[ ] → brackets

Inclusive (the number is included)

=, ≤, ≥ ● (closed circle)

( ) → parentheses

Exclusive (the number is excluded)

≠, <, > ○ (open circle)

[4, 12) Has 2 types of symbols: brackets and parentheses

Understanding Interval Notation4 ≤ x < 12

Interval Notation:

How We Say It: The domain is 4 to

12 .

On a Number Line:

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 Interval Notation:

How We Say It: The domain is to

On a Number Line:

all real numbers

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. Sketch one of the domains on a number line.2. Add a sketch of the other domain.3. 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

Answers7. 8.

9. 10.

11. 12.

13. 14.

Golf !!!

Answers1. (–2, 7) 6. (–∞,4)2. (–3, 1] 7. (–1, 2) U (2, ∞)3. [–9, –4] 8. [–5, ∞)4. [–7, –1] 9. (–2, ∞)5. (–∞, 6) U (6, 10) U (10, ∞)

Experiment What happens we type the following expressions

into our calculators?

50

05

16

16

*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).

2x)x(f

*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).

1xx6x)x(f

2

*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).

30xxx2)x(f 2

Additional Example Find the domain of f(x).

172x214)x(f

***Additional Example Find the domain of f(x).

6x5x1x510)x(f 2

Additional Example Find the domain of f(x).

41x)x(f

2

Your Turn:

Complete problems 1 – 10 on the “Solving for the Domain Algebraically” handout

#8 – Typo!6xx

1)x(f 2

Answers:1. 2.

3. 4.

5.

Answers, cont:6. 7.

8. 9.

10.