Domain and Interval Notation Domain n The set

Domain and Interval Notation

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

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

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

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

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

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

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

Infinity n n Infinity is always exclusive!!! – The symbol for infinity

Infinity, cont. Negative Infinity Positive Infinity

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

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

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

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

Combining Restricted Domains n n When we have more than one domain restriction, then we need to figure out the interval notation that satisfies all the restrictions Examples: q q 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: n Complete problems 7 – 14 on the “Domain and Interval Notation – Guided Notes” handout

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

Golf !!!

Experiment n What happens we type the following expressions into our calculators? q q

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

Example n Find the domain of f(x).

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

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

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

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

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

Example n Find the domain of f(x).

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

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

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

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

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

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