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 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 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 all real numbers Interval Notation: How We Say It: The domain is to On a Number Line:

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. 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 !!!

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, ∞)

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!

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

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

*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 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

*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).

*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).

*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).

Additional Example Find the domain of f(x).

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

Additional Example Find the domain of f(x).

Your Turn: 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.