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) =, ≤, ≥ ( ) → parentheses Exclusive (the number is excluded) ≠, <, >

Understanding Interval Notation 4 ≤ x < 12 Interval Notation: How We Say It: The domain is 4 to 12 . What It Means: In the function, y is defined (there is a value for y) when x is at least 4 and up to, but not including, 12.

Example – Domain: –2 < x ≤ 6 Interval Notation: How We Say It: The domain is –2 to 6 . What It Means: In the function, y is defined when x is close to, but not including –2, and up to 6.

Example – Domain: –16 < x < –8 Interval Notation: How We Say It: The domain is –16 to –8 . What It Means: In the function, y is defined when x is close to, but not including –16, and up to, but not including, –8.

Your Turn: Complete problems 1 – 3 on the “Domain and Interval Notation” 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 . . What It Means: In the function, y is defined when x is at least 4.

Example – Domain: x < –7 Interval Notation: How We Say It: The domain is to –7 . What It Means: In the function, y is defined when x is less than –7.

Example – Domain: x is Interval Notation: all real numbers Interval Notation: How We Say It: The domain is to . . What It Means: In the function, y is defined of all values of x.

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

Domain Restrictions: x ≥ 4, x ≠ 11 Write the interval notation for one of the domains. Alter the interval notation to include the other domain. Include a “U” in between each set of intervals (if you have more than one). Interval Notation: 1. 2. 3.

Domain Restrictions: –10 ≤ x < 14, x ≠ 0 Interval Notation: 1. 2. 3.

Domain Restrictions: x ≥ 0, x < 12 Interval Notation: 1. 2. 3.

Domain Restrictions: x ≥ 0, x ≠ 0 Interval Notation: 1. 2. 3.

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” handout

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 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 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 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 #1 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 #2 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 #3 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 #4 Find the domain of f(x).

Example #5 Find the domain of f(x).

Example #6 Find the domain of f(x).

Example #7 Find the domain of f(x).

Your Turn: Complete the last section of problems on the “Domain and Interval Notation” handout