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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 Upper Bound • 1st Number • 2nd Number • Lowest Possible x-value • 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 12 . On a Number Line: to Example – Domain: –2 < x ≤ 6 Interval Notation: How We Say It: The domain is –2 6 . On a Number Line: to Example – Domain: –16 < x < –8 Interval Notation: How We Say It: The domain is –16 –8 . On a Number Line: to 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 On a Number Line: to Example – Domain: x is Interval Notation: How We Say It: The domain is On a Number Line: all real numbers to 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. 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: Complete problems 7 – 14 on the “Domain and Interval Notation – Guided Notes” handout Answers 7. 8. 9. 10. 11. 12. 13. 14. Golf !!! 1. 2. 3. 4. 5. Answers 6. (–∞,4) (–2, 7) (–3, 1] 7. (–1, 2) U (2, ∞) [–9, –4] 8. [–5, ∞) [–7, –1] 9. (–2, ∞) (–∞, 6) U (6, 10) U (10, ∞) Experiment What happens we type the following expressions into our calculators? 16 16 0 5 5 0 *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). f(x) x 2 *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). x 6x f(x) x 1 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). 2x f(x) 2 x x 30 Additional Example Find the domain of f(x). f(x) 14 2x 2 17 ***Additional Example Find the domain of f(x). 1 f(x) 10 5x 2 x 5x 6 Additional Example Find the domain of f(x). x 1 f(x) 4 2 Your Turn: Complete problems 1 – 10 on the “Solving for the Domain Algebraically” handout #8 – Typo! 1 f(x) 2 x x 6 Answers: 1. 2. 3. 4. 5. Answers, cont: 6. 7. 8. 9. 10.