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**Domain and Interval Notation**

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**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

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**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

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**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)

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**Understanding Interval Notation**

4 ≤ x < 12 Interval Notation: How We Say It: The domain is to On a Number Line:

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**Example – Domain: –2 < x ≤ 6**

Interval Notation: How We Say It: The domain is – to On a Number Line:

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**Example – Domain: –16 < x < –8**

Interval Notation: How We Say It: The domain is – to – On a Number Line:

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Your Turn: Complete problems 1 – 3 on the “Domain and Interval Notation – Guided Notes” handout

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Infinity Infinity is always exclusive!!! – The symbol for infinity

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Infinity, cont. Negative Infinity Positive Infinity

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**Example – Domain: x ≥ 4 Interval Notation:**

How We Say It: The domain is to On a Number Line:

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**Example – Domain: x is all real numbers Interval Notation:**

How We Say It: The domain is to On a Number Line:

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

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**Restricted Domain When the domain is anything besides (–∞, ∞)**

Examples: 3 < x 5 ≤ x < 20 –7 ≠ x

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**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

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**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).

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**Domain Restrictions: x ≥ 4, x ≠ 11**

Interval Notation:

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**Domain Restrictions: –10 ≤ x < 14, x ≠ 0**

Interval Notation:

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**Domain Restrictions: x ≥ 0, x < 12**

Interval Notation:

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**Domain Restrictions: x ≥ 0, x ≠ 0**

Interval Notation:

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**Challenge – Domain Restriction: x ≠ 2**

Interval Notation:

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**Domain Restriction: –6 ≠ x**

Interval Notation:

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**Domain Restrictions: x ≠ 1, 7**

Interval Notation:

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Your Turn: Complete problems 7 – 14 on the “Domain and Interval Notation – Guided Notes” handout

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Answers

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Golf !!!

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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, ∞)

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**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!

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

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

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***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.

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**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

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***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

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Example Find the domain of f(x).

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***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.

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Example Solve for the domain of f(x).

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***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 (–∞, ∞)!!!

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Example Find the domain of f(x). f(x) = x2 + 4x – 5

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***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

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Example Find the domain of f(x).

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Additional Example Find the domain of f(x).

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*****Additional Example**

Find the domain of f(x).

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Additional Example Find the domain of f(x).

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Your Turn: Complete problems 1 – 10 on the “Solving for the Domain Algebraically” handout #8 – Typo!

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Answers:

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Answers, cont:

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