Foundations for Functions

5. Foundations for Functions

6. Objective Classify and order real numbers.

7. 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

8. The diagram shows some important subsets of the real numbers.

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

10. 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.

11. 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, { }.

12. -2 -1 0 1 2 3 4 5 6 7 8 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. 3<x<5

13. -2 -1 0 1 2 3 4 5 6 7 8 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) The set of real numbers between but not including 3 and 5. 3<x<5

14. 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 (–∞). ∞ –∞ -5 0 5

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

16. Example 2B: Interval Notation 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] –6 and –4 are included. 5 is not included, and the interval continues forever in the positive direction. (5, ∞) The word “or” is used to indicate that a set includes more than one interval. [–6, –4] or (5, ∞)

17. Check It Out! Example 2 Use interval notation to represent each set of numbers. a. -4 -3 -2 -1 0 1 2 3 4 (–∞, –1] –1 is included, and the interval continues forever in the negative direction. b. x ≤ 2 or 3 < x ≤ 11 (–∞, 2] 2 is included, and the interval continues forever in the negative direction. (3, 11] 3 is not included, but 11 is. (–∞, 2] or (3, 11]

18. 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.”

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

20. Example 3: Translating Between Methods of Set Notation -4 -3 -2 -1 0 1 2 3 4 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, …} The order of elements is not important. ; set-builder notation C. {x | x ≤ –2}

21. You try it! Example 3 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 The order of the elements is not important. {3, 4, 5, 6, 7} c. [99, ∞};set-builder notation {x | x ≥ 99}

22. Objective Identify and use properties of real numbers.

23. Properties Real Numbers Identities and Inverses

24. Properties Real Numbers Identities and Inverses

25. Properties Real Numbers Identities and Inverses

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

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

28. Properties Real Numbers Addition and Multiplication

29. Properties Real Numbers Addition and Multiplication

30. Properties Real Numbers Addition and Multiplication

31. Properties Real Numbers Addition and Multiplication

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

33. Think:5% = (10%) Example 3: Consumer Economics Application Use mental math to find a 5% tax on a $42.40 purchase. Think:10% of $42.40 Move the decimal point left 1 place. 10%(42.40) = 4.240 = 4.24 5% is half of 10%, so find half of 4.24. A 5% tax on a $42.40 is $2.12.

34. Example 4A: Classifying Statements as Sometimes, Always, or Never True Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers. a b = a, where b = 3 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

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

36. Daily Practice • Practice B • Sections 1-1 and 1-2

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

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

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

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

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

42. Example 4A: Adding and Subtracting Square Roots Add.

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

44. Objective Simplify and evaluate algebraic expressions.

45. Example 1: Translating Words into Algebraic Expressions 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.

46. 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.

47. Example 2A: Evaluating Algebraic Expressions Evaluate the expression for the given values of the variables. 2x – xy + 4y for x = 5 and y = 2 2(5) – (5)(2) + 4(2) Substitute 5 for x and 2 for y. 10 – 10 + 8 Multiply from left to right. 0 + 8 Add and subtract from left to right. 8

48. Example 2B: Evaluating Algebraic Expressions Evaluate the expression for the given values of the variables. q2 + 4qr – r2 for r = 3 and q = 7 (7)2 + 4(7)(3) – (3)2 Substitute 3 for r and 7 for q. 49 + 4(7)(3) – 9 Evaluate exponential expressions. 49 + 84 – 9 Multiply from left to right. 124 Add and subtract from left to right.

49. 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.

50. Example 3A: Simplifying Expressions Simplify the expression. 3x2 + 2x – 3y + 4x2 3x2 + 2x – 3y + 4x2 Identify like terms. Combine like terms. 3x2 + 4x2 = 7x2 7x2 + 2x – 3y