Foundations for Functions

Classify and order real numbers. Objective

Vocabulary Set- – a collection of items Elements- – what make up the set Subset – – a set whose elements all belong to another set Empty Set- – set containing no elements

The diagram shows some important subsets of the real numbers.

Example 1B: Ordering and Classifying Real Numbers NumbersRealRationalIntegerWholeNaturalIrrational 2.3  Consider the numbers Classify each number by the subsets of the real numbers to which it belongs.

Vocabulary Roster Notation – You are able to list all numbers in a set. Interval Notation – Notation for a set of all numbers between two endpoints. – The symbols [ and ] are used to include the endpoints – The symbols ( and ) are used to exclude the endpoints Set Builder Notation – Notation for a set that uses a rule to describe the properties of the elements of the set.

There are many ways to represent sets. For instance, you can use words to describe a set. You can also use roster notation, in which the elements in a set are listed between braces, { }.

The set of real numbers between 3 and 5, which is also an infinite set, can be represented on a number line or by an inequality < x < 53 < x < 5

An interval is the set of all numbers between two endpoints, such as 3 and 5. In interval notation the symbols [ and ] are used to include an endpoint in an interval, and the symbols ( and ) are used to exclude an endpoint from an interval. (3, 5)(3, 5) The set of real numbers between but not including 3 and < x < 53 < x < 5

An interval that extends forever in the positive direction goes to infinity (∞), and an interval that extends forever in the negative direction goes to negative infinity (–∞) –∞ ∞

Use interval notation to represent the set of numbers. 7 < x ≤ 12 (7, 12] Example 2A: Interval Notation 7 is not included, but 12 is.

There are two intervals graphed on the number line. [–6, –4] (5, ∞) –6 and –4 are included. 5 is not included, and the interval continues forever in the positive direction. The word “or” is used to indicate that a set includes more than one interval. [–6, –4] or (5, ∞) Example 2B: Interval Notation –6 –4 – Use interval notation to represent the set of numbers.

Use interval notation to represent each set of numbers. Check It Out! Example 2 a. (–∞, –1] b. x ≤ 2 or 3 < x ≤ 11 (–∞, 2] or (3, 11] –1 is included, and the interval continues forever in the negative direction. (–∞, 2] 2 is included, and the interval continues forever in the negative direction. (3, 11] 3 is not included, but 11 is.

The set of all numbers x such 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.” 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.” Helpful Hint

Some representations of the same sets of real numbers are shown.

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 The order of elements is not important. Example 3: Translating Between Methods of Set Notation {10, 20, 30, …} {x | x ≤ –2} ; set-builder notation C.

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

Identify and use properties of real numbers. Objective

For all real numbers n, WORDS Multiplicative Identity Property The product of a number and 1, the multiplicative identity, is the original number. NUMBERS ALGEBRA n  1 = 1  n = n Properties Real Numbers Identities and Inverses

For all real numbers n, WORDS Additive Inverse Property The sum of a number and its opposite, or additive inverse, is 0. NUMBERS 5 + (–5) = 0 ALGEBRA n + (–n) = 0 Properties Real Numbers Identities and Inverses

For all real numbers n, WORDS Multiplicative Inverse Property The product of a nonzero number and its reciprocal, or multiplicative inverse, is 1. NUMBERS ALGEBRA Properties Real Numbers Identities and Inverses

Find the additive and multiplicative inverse of each number. Example 1A: Finding Inverses 12 The opposite of 12 is –12. additive inverse: –12 Check – = 0 The Additive Inverse Property holds. The reciprocal of 12 is. multiplicative inverse: The Multiplicative Inverse Property holds. Check

Find the additive and multiplicative inverse of each number. Example 1B: Finding Inverses additive inverse: multiplicative inverse: The opposite of is. The reciprocal of is

For all real numbers a and b, WORDS Closure Property The sum or product of any two real numbers is a real number NUMBERS = 5 2(3) = 6 ALGEBRA a + b   ab   Properties Real Numbers Addition and Multiplication

For all real numbers a and b, WORDS Commutative Property You can add or multiply real numbers in any order without changing the result. NUMBERS = (11) = 11(7) ALGEBRA a + b = b + a ab = ba Properties Real Numbers Addition and Multiplication

For all real numbers a and b, WORDS Associative Property The sum or product of three or more real numbers is the same regardless of the way the numbers are grouped. NUMBERS (5 + 3) + 7 = 5 + (3 + 7) (5  3)7 = 5(3  7) ALGEBRA a + (b + c) = a + (b + c) (ab)c = a(bc) Properties Real Numbers Addition and Multiplication

For all real numbers a and b, WORDS Distributive Property When you multiply a sum by a number, the result is the same whether you add and then multiply or whether you multiply each term by the number and add the products. NUMBERS 5(2 + 8) = 5(2) + 5(8) (2 + 8)5 = (2)5 + (8)5 ALGEBRA a(b + c) = ab + ac (b + c)a = ba + ca Properties Real Numbers Addition and Multiplication

Identify the property demonstrated by each question. Example 2: Identifying Properties of Real Numbers A. 2  3.9 = 3.9  2 Numbers are multiplied in any order without changing the results. B. The numbers have been regrouped. Commutative Property of Multiplication Associative Property of Addition

Example 3: Consumer Economics Application Use mental math to find a 5% tax on a $42.40 purchase. A 5% tax on a $42.40 is $ %(42.40) Move the decimal point left 1 place. Think: 10% of $ % is half of 10%, so find half of Think: 5% = (10%) = = 4.24

Example 4A: Classifying Statements as Sometimes, Always, or Never True a  b = a, where b = 3 Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers. True and false examples exist. The statement is true when a = 0 and false when a ≠ 0. sometimes true true example: 0  3 = 0 false example: 1  3 ≠ 1

Example 4B: Classifying Statements as Sometimes, Always, or Never True 3(a + 1) = 3a + 3 Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers. Always true by the Distributive Property. always true

Daily Practice Practice B – Sections 1-1 and 1-2

Estimate square roots. Simplify, add, subtract, multiply, and divide square roots. Objectives

Find a perfect square factor of 32. Simplify each expression. Example 2: Simplifying Square–Root Expressions Product Property of Square Roots Quotient Property of Square Roots A. B.

Product Property of Square Roots Simplify each expression. Example 2: Simplifying Square–Root Expressions Quotient Property of Square Roots C.C. D.

Simplify by rationalizing the denominator. Example 3A: Rationalizing the Denominator Multiply by a form of 1. = 2

Simplify the expression. Example 3B: Rationalizing the Denominator Multiply by a form of 1.

Add. Example 4A: Adding and Subtracting Square Roots

Subtract. Example 4B: Adding and Subtracting Square Roots Simplify radical terms. Combine like radical terms.

Simplify and evaluate algebraic expressions. Objective

Write an algebraic expression to represent each situation. Example 1: Translating Words into Algebraic Expressions 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 Add n to 12. Divide 100 by M n

To evaluate an algebraic expression, substitute a number for each variable and simplify by using the order of operations. One way to remember the order of operations is by using the mnemonic PEMDAS. Order of Operations 1. Parentheses and grouping symbols. 2. Exponents. 3. Multiply and Divide from left to right. 4. Add and Subtract from left to right.

Evaluate the expression for the given values of the variables. Example 2A: Evaluating Algebraic Expressions Substitute 5 for x and 2 for y. Multiply from left to right. Add and subtract from left to right. 2(5) – (5)(2) + 4(2) 10 – x – xy + 4y for x = 5 and y = 2

Evaluate the expression for the given values of the variables. Example 2B: Evaluating Algebraic Expressions q 2 + 4qr – r 2 for r = 3 and q = 7 Substitute 3 for r and 7 for q. Multiply from left to right. Add and subtract from left to right (7)(3) – – Evaluate exponential expressions. (7) 2 + 4(7)(3) – (3) 2

Recall that the terms of an algebraic expression are separated by addition or subtraction symbols. Like terms have the same variables raised to the same exponents. Constant terms are like terms that always have the same value.

Simplify the expression. Example 3A: Simplifying Expressions Identify like terms. Combine like terms. 3x 2 + 4x 2 = 7x 2 3x 2 + 2x – 3y + 4x 2 7x 2 + 2x – 3y

Simplify the expression. Example 3B: Simplifying Expressions Distribute, and identify like terms. Combine like terms. 7jk – 7jk = 0 j(6k 2 + 7k) + 9jk 2 – 7jk 6jk 2 + 7jk + 9jk 2 – 7jk 15jk 2

Example 4A: Application Apples cost $2 per pound, and grapes cost $3 per pound. Write and simplify an expression for the total cost if you buy 10 lb of apples and grapes combined. 2A + 3(10 – A) Combine like terms. Let A be the number of pounds of apples. Distribute 3. = 30 – A Then 10 – A is the number of pounds of grapes. = 2A + 30 – 3A

Apples cost $2 per pound, and grapes cost $3 per pound. What is the total cost if 2 lb of the 10 lb are apples? Evaluate 30 – A for A = – (2)= 28 The total cost is $28 if 2 lb are apples. Example 4B: Application

Daily Practice Practice B – Sections 1-3 and 1-4 Quiz next time on all sections 1-1 through 1-4

Simplify expressions involving exponents. Use scientific notation. Objectives

Write the expression in expanded form. Example 1A: Writing Exponential Expressions in Expanded Form The base is 5z, and the exponent is 2. 5z is a factor 2 times. (5z) 2 (5z)(5z)

Write the expression in expanded form. Example 1B: Writing Exponential Expressions in Expanded Form The base is s, and the exponent is 4. s is a factor 4 times. –s4–s4 –s4–s4 –(s  s  s  s) = –s  s  s  s

Write the expression in expanded form. There are two bases: h and k + 3. h is a factor 3 times, and k + 3 is a factor 2 times. Example 1C: Writing Exponential Expressions in Expanded Form 3h 3 (k + 3) 2 3(h)(h)(h) (k + 3)(k + 3)

Simplify the expression. Example 2A: Simplifying Expressions with Negative Exponents The reciprocal of. 3 –2

Simplify the expression. Example 2B : Simplifying Expressions with Negative Exponents The reciprocal of.

Check It Out! Example 3a Simplify the expression. Assume all variables are nonzero. Power of a Power Power of a Product (5x 6 ) 3 53(x6)353(x6)3 125x (6)(3) 125x 18

Negative Exponent Property Power of a Power Check It Out! Example 3b Simplify the expression. Assume all variables are nonzero. (–2a 3 b) –3

Scientific notation is a method of writing numbers by using powers of 10. In scientific notation, a number takes a form m  10 n, where 1 ≤ m <10 and n is an integer. You can use the properties of exponents to calculate with numbers expressed in scientific notation.

Simplify the expression. Write the answer in scientific notation. Example 4A: Simplifying Expressions Involving Scientific Notation Divide 4.5 by 1.5 and subtract exponents: –5 – 6 = –  10 –11

Because 22.1 > 10, move the decimal point left 1 place and add 1 to the exponent. Multiply 2.6 and 8.5 and add exponents: = 11. Simplify the expression. Write the answer in scientific notation. Example 4B: Simplifying Expressions Involving Scientific Notation 22.1  (2.6)(8.5)  (10 4 )(10 7 ) (2.6  10 4 )(8.5  10 7 ) 2.21  10 12

Daily Practice Practice B – Sections 1-5 Review day next time – Have all your questions ready to be asked. – Test the following meeting day.