Chapter 1:Foundations for Functions

Chapter 1:Foundations for Functions Algebra II

Table of Contents • 1.1 – Sets of Numbers • 1.2 – Properties of Real Numbers • 1.3 – Square Roots • 1.4- Simplifying Algebraic Expressions • 1.5- Properties of Exponents

1.1 Bell work (Algebra II) • Write down the following definitions • Asetis a collection of items called elements. • A subset is a set whose elements belong to another set. • The empty set, denoted , is a set containing no elements.

1.1 Sets of Numbers Algebra II

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1-1 Consider the numbers The numbers in order from least to great are Example 1 Order the numbers from least to greatest. Write each number as a decimal to make it easier to compare them.  ≈ 3.14

1-1 Consider the numbers Classify each number by the subsets of the real numbers to which it belongs.          

Math Humor • Q: Why do the other numbers refuse to take √2, √3, √5 and seriously? • A: They are completely irrational

1-1 Consider the numbers –2, , –0.321, and . Classify each number by the subsets of the real numbers to which it belongs.           

1-1 You can also use roster notation, in which the elements in a set are listed between braces, { }. Afinite sethas a definite, or finite, number of elements. An infinite sethas an unlimited, or infinite number of elements.

1-1 Pg. 8 Do Not Copy In interval notation, use [ ] to include an endpoint. Use ( ) to exclude an endpoint

1-1 Example 2 Use interval notation to represent the set of numbers. 7 < x ≤ 12 (7, 12] Use interval notation to represent the set of numbers. –6 –4 –2 0 2 4 6 There are two intervals graphed on the number line. [–6, –4] or (5, ∞)

1-1 Use interval notation to represent each set of numbers. a. -4 -3 -2 -1 0 1 2 3 4 (–∞, –1] b. x ≤ 2 or 3 < x ≤ 11 (–∞, 2] or (3, 11]

1-1 Helpful Hint The symbol  means “is an element of.” So xN is read “x is an element of the set of natural numbers,” or “x is a natural number.” The set ofall numbers xsuch that x has the given properties {x|8 < x ≤ 15 and x  N} Read the above as “the set of all numbers x such that x is greater than 8 and less than or equal to 15 and x is a natural number.”

1-1 -4 -3 -2 -1 0 1 2 3 4 Example 3 Rewrite each set in the indicated notation. A. {x | x > –5.5, x  Z }; words integers greater than 5.5 B. positive multiples of 10; roster notation {10, 20, 30, …} ; set-builder notation C. {x | x ≤ –2}

1-1 Rewrite each set in the indicated notation. a.{2, 4, 6, 8}; words even numbers between 1 and 9 b. {x | 2 <x < 8 and xN}; roster notation {3, 4, 5, 6, 7} The order of the elements is not important. c. [99, ∞};set-builder notation {x | x ≥ 99}

HW pg. 10 • 1.1 • 2, 5-11, 15-27, 30, 31, 52, 56 • Challenge: 44 • Write down original problems • Show all work • Use Pencil • On HW sheet put the following • Your Name • Hour • Section 1.1 • Problems on the assignment • Good bell work:43

1.2 Bell work (Algebra II) • Be prepared to share your answer to #56 to the class • Write down the following properties and leave two lines below each for notes • Additive Identity Proper • Multiplicative Identity Property • Additive Inverse Property • Multiplicative Inverse Property • Closure Property • Commutative Property • Associative Property • Distributive Property

1.2: Properties of Real Numbers Algebra II

1.2 Properties Real Numbers Identities and Inverses

1.2 multiplicative inverse: Check Example 1 Find the additive and multiplicative inverse of each number. 12 additive inverse: –12 additive inverse: Check –12 + 12 = 0  multiplicative inverse:

1.2 multiplicative inverse: Check 500 –0.01 additive inverse: –500 additive inverse: 0.01 Check 500 + (–500) = 0  multiplicative inverse: –100

1.2 Properties Real Numbers Addition and Multiplication

1.2 Example 2 Identify the property demonstrated by each question. A. 2  3.9 = 3.9  2 Commutative Property of Multiplication Associative Property ofAddition Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers. a b = a, where b = 3 sometimes true true example: 0 3 = 0 false example: 1 3 ≠ 1

1.2 3(a+ 1) = 3a + 3 a+ (–a) = b + (–b) always true Always true by the Distributive Property. Always true by the Additive Inverse Property.

HW pg. 17 • 1.2 • 1-9, 12-14, 26-33, 35-41, 62-65 • Challenge: 34, 50 • Write down original problems • Show all work • Use Pencil • On HW sheet put the following • Your Name • Hour • Section 1.2 • Problems on the assignment • Bell work 51, 52

1.3: Square Roots Algebra II

1.3 Bell work (Algebra II) • Look at #52 on pg. 19 and be prepared to explain the answer to the class • Put the following definitions in your notes • = radicalsymbol. • The number or expression under the radical symbol is called the radicand. • The radical symbol indicates only the positive square root of a number, called the principal root.

1.3 The side length of a square is the square root of its area. To indicate both the positive and negative square roots of a number, use the plus or minus sign (±). or –5

1.3 Pg. 22

1.3 Math Joke • Teacher: Lets find the square root of 1 million • Student: Don’t you think that’s a bit too radical?

1.3 Simplify each expression. Example 2 A. C. D. B.

1.3 Simplify each expression. A. C. B. D.

1.3 Example 3 Simplify by rationalizing the denominator.

1.3 Simplify by rationalizing the denominator.

1.3 Square roots that have the same radicand are called like radical terms.

1.3 Example 4 : Adding and Subtracting Square Roots

1.3

HW pg.24 • 1.3 • 7-17 (Odd), 22-32 (Even), 34-40, 42, 46, 67, 78-81 • Challenge 67 • Write down original problems • Show all work • Use Pencil • On HW sheet put the following • Your Name • Hour • Section 1.3 • Problems on the assignment • Bell work 43, 57, 66,

1.4 Bell work (Algebra II) Just Read There are three different ways in which a basketball player can score points during a game. There are 1-point free throws, 2-point field goals, and 3-point field goals. An algebraic expression can represent the total points scored during a game.

1.4: Simplifying Algebraic Expressions Algebra II

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1.4 Example 1 Write an algebraic expression to represent each situation. • A. the number of apples in a basket of 12 after n more are added • B. the number of days it will take to walk 100 miles if you walk M miles per day 12 + n Add n to 12. Divide 100 by M.

1.4 Write an algebraic expression to represent each situation. a. Lucy’s age y years after her 18th birthday 18 + y Add y to 18. b. the number of seconds in h hours 3600h Multiply h by 3600.

Chapter 1:Foundations for Functions