1. Domain and Interval Notation

2. 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

3. 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

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

5. Understanding Interval Notation
4 ≤ x < 12
Interval Notation:
How We Say It: The domain is to
What It Means: In the function, y is defined (there is a value for y) when x is at least 4 and up to, but not including, 12.

6. Example – Domain: –2 < x ≤ 6
Interval Notation:
How We Say It: The domain is – to
What It Means: In the function, y is defined when x is close to, but not including –2, and up to 6.

7. Example – Domain: –16 < x < –8
Interval Notation:
How We Say It: The domain is – to –
What It Means: In the function, y is defined when x is close to, but not including –16, and up to, but not including, –8.

8. Your Turn:
Complete problems 1 – 3 on the “Domain and Interval Notation” handout

9. Infinity
Infinity is always exclusive!!!
– The symbol for infinity

10. Infinity, cont.
Negative Infinity
Positive Infinity

11. Example – Domain: x ≥ 4 Interval Notation:
How We Say It: The domain is to
What It Means: In the function, y is defined when x is at least 4.

12. Example – Domain: x < –7
Interval Notation:
How We Say It: The domain is to –
What It Means: In the function, y is defined when x is less than –7.

13. Example – Domain: x is Interval Notation:
all real numbers
Interval Notation:
How We Say It: The domain is to
What It Means: In the function, y is defined of all values of x.

14. Your Turn:
Complete problems 4 – 6 on the “Domain and Interval Notation” handout

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

16. 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

17. Domain Restrictions: x ≥ 4, x ≠ 11
Write the interval notation for one of the domains.
Alter the interval notation to include the other domain.
Include a “U” in between each set of intervals (if you have more than one).
Interval Notation: 1. 2. 3.

18. Domain Restrictions: –10 ≤ x < 14, x ≠ 0
Interval Notation: 1. 2. 3.

19. Domain Restrictions: x ≥ 0, x < 12
Interval Notation: 1. 2. 3.

20. Domain Restrictions: x ≥ 0, x ≠ 0
Interval Notation: 1. 2. 3.

21. Challenge – Domain Restriction: x ≠ 2
Interval Notation:

22. Domain Restriction: –6 ≠ x
Interval Notation:

23. Domain Restrictions: x ≠ 1, 7
Interval Notation:

24. Your Turn:
Complete problems 7 – 14 on the “Domain and Interval Notation” handout

25. Activity!
Golf!!!

26. Answers
1. (–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, ∞)

27. 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.

28. Solving for the Domain of Functions Algebraically, cont.
Domain Convention – unless otherwise stated, the domain (input or x-value) of a function is every number that produces a real output (y-value)
No imaginary numbers or division by zero!

29. What are some situations give me an error or undefined in the calculator?

30. Experiment
What happens we type the following expressions into our calculators?

31. Solving for the Restricted Domain Algebraically
Determine if you have square roots and/or fractions in the function (If you have neither, then the domain is (–∞, ∞)!!!)
For square roots, set the radicand (the expression under the radical symbol) ≥ 0, then solve for x
For fractions, set the denominator ≠ 0, then solve for x
Rewrite the answer in interval notation
This is called restricting the domain

32. Example #1
Find the domain of f(x).

33. Example #2
Solve for the domain of f(x).

34. Example #3
Find the domain of f(x).

35. Example #4
Find the domain of f(x).

36. Your Turn:
Complete problems 4 – 12 on the “Domain and Interval Notation” handout