2 Domain The set of all possible input values (generally x values) We write the domain in interval notationInterval notation has 2 important components:PositionSymbols
3 Interval 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
4 Interval 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)
5 Understanding Interval Notation 4 ≤ x < 12Interval Notation:How We Say It: The domain is toOn a Number Line:
6 Example – Domain: –2 < x ≤ 6 Interval Notation:How We Say It: The domain is – toOn 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 InfinityInfinity is always exclusive!!!– The symbol for infinity
11 Example – Domain: x ≥ 4 Interval Notation: How We Say It: The domain is toOn a Number Line:
12 Example – Domain: x is all real numbers Interval Notation: How We Say It: The domain is toOn 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 < x5 ≤ 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 restrictionsExamples: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:
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 ExperimentWhat 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 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