2Domain The set of all possible input values (generally x values) We write the domain in interval notationInterval notation has 2 important components:PositionSymbols
3Interval Notation – Position Has 2 positions: the lower bound and the upper bound[4, 12)Lower Bound1st NumberLowest Possible x-valueUpper Bound2nd NumberHighest Possible x-value
4Interval Notation – Symbols Has 2 types of symbols: brackets andparentheses[4, 12)[ ] → bracketsInclusive (the number is included)=, ≤, ≥● (closed circle)( ) → parenthesesExclusive (the number is excluded)≠, <, >○ (open circle)
5Understanding Interval Notation 4 ≤ x < 12Interval Notation:How We Say It: The domain is toOn a Number Line:
6Example – Domain: –2 < x ≤ 6 Interval Notation:How We Say It: The domain is – toOn a Number Line:
7Example – Domain: –16 < x < –8 Interval Notation:How We Say It: The domain is – to–On a Number Line:
8Your Turn:Complete problems 1 – 3 on the “Domain and Interval Notation – Guided Notes” handout
9InfinityInfinity is always exclusive!!!– The symbol for infinity
11Example – Domain: x ≥ 4 Interval Notation: How We Say It: The domain is toOn a Number Line:
12Example – Domain: x is all real numbers Interval Notation: How We Say It: The domain is toOn a Number Line:
13Your Turn:Complete problems 4 – 6 on the “Domain and Interval Notation – Guided Notes” handout
14Restricted Domain When the domain is anything besides (–∞, ∞) Examples:3 < x5 ≤ x < 20–7 ≠ x
15Combining Restricted Domains When we have more than one domain restriction, then we need to figure out the interval notation that satisfies all the restrictionsExamples:x ≥ 4, x ≠ 11–10 ≤ x < 14, x ≠ 0
16Combining 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).
17Domain Restrictions: x ≥ 4, x ≠ 11 Interval Notation:
18Domain Restrictions: –10 ≤ x < 14, x ≠ 0 Interval Notation:
19Domain Restrictions: x ≥ 0, x < 12 Interval Notation:
20Domain Restrictions: x ≥ 0, x ≠ 0 Interval Notation:
21Challenge – Domain Restriction: x ≠ 2 Interval Notation:
28Solving 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!
29What are some situations give me an error or undefined in the calculator?
30ExperimentWhat 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.
32Solving 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 xFor fractions, set the denominator ≠ 0, then solve for xRewrite the answer in interval notationThis 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