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)
=, ≤, ≥
● (closed circle)
( ) → parentheses
Exclusive (the number is excluded)
≠, <, >
○ (open circle)

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

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

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

8. Your Turn:
Complete problems 1 – 3 on the “Domain and Interval Notation – Guided Notes” 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
On a Number Line:

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

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

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

15. 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

16. 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).

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

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

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

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

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 – Guided Notes” handout

25. Answers

26. Golf !!!

27. 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, ∞)

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 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.

32. 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

33. *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

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

35. *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.

36. Example
Solve for the domain of f(x).

37. *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 (–∞, ∞)!!!

38. Example
Find the domain of f(x).
f(x) = x2 + 4x – 5

39. *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

40. Example
Find the domain of f(x).

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

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

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

44. Your Turn:
Complete problems 1 – 10 on the “Solving for the Domain Algebraically” handout
#8 – Typo!

45. Answers:

46. Answers, cont: