Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 1 Introduction to Functions and Graphs

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1

Chapter 1

Introduction to Functions and

Graphs

2Copyright © 2014, 2010, 2006 Pearson Education, Inc.

Numbers, Data and Problem Solving

♦ Recognize common sets of numbers♦ Evaluate Expressions by applying the order of

operations♦ Learn scientific notation and use it in

applications♦ Apply problem solving strategies

1.1


Natural Numbers and Integers

Natural Numbers (or counting numbers)are numbers in the set N = {1, 2, 3, ...}.

Integers are numbers in the set I = {… 3, 2, 1, 0, 1, 2, 3, ...}.

These are the natural numbers, their additive inverses (negatives), and 0.


Rational Numbers

Rational Numbers are numbers which can be expressed as the ratio of two integers p/q where q 0

Examples:

Note that:• Every integer is a rational number. • Rational numbers can be expressed as decimals

that either terminate (end) or repeat a sequence of digits.

2

1,1

3,

1

4, 50

2,22

7,0, 25,1.2


Irrational Numbers Numbers

Irrational Numbers are numbers which are not rational numbers. Irrational numbers:• Cannot be expressed as the ratio of two

integers.• Have a decimal representation which does

not terminate and does not repeat a sequence of digits.

Examples:2, 15,


Real Numbers Numbers

Real Numbers are can be represented by decimal numbers. Real numbers include both the rational and irrational numbers.

Examples:

2, 10, 131.3337,1

30.3,

5 2.2361, 11 3.3166


Example: Classify Numbers

Classify each number as one or more of the following: natural number, integer, rational number, irrational number.

5, 1.2,13

7, 7, 12, 16 4

5: natural number, integer, rational number

–1.2: rational number

Solution

13

7: rational number


Example: Classify Numbers

5, 1.2,13

7, 7, 12, 16 4

7 : irrational number

16 4 : natural number, integer, rational number

–12: integer, rational number

Solution (continued)


Order of Operations

Using the following order of operations, first perform all calculations within parentheses, square roots, and absolute value bars and above and below fraction bars. Then use the same order of operations to perform any remaining calculations.

1. Evaluate all exponents. Then do any negation after evaluating exponents.

2. Do all multiplication and division from left to right.

3. Do all addition and subtraction from left to right.


Example: Evaluating Arithmetic Expressions

Evaluate each expression by hand.

b. 10 6

5 3 4 7 2a. 3 1 5 2 42

Solutiona. 3 1 5 2 42 b.

10 6

5 3 4 7 2

3 4 2 42

3 16 16

48 16

32

4

2 4 5

2 4 5

2 5

7


Scientific Notation

A real number r is in scientific notation when r is written as c 10n, where 1 ≤ |c| < 10and n is an integer.

Examples:• The distance to the sun is 93,000,000 mi. • In scientific notation this is 9.3 107 mi.

• The size of a typical virus is 0.000005 cm.• In scientific notation this is 5 106 cm.


Example: Evaluating Expressions by Hand

Evaluate each expression. Write your result in scientific notation and standard form.

3 4a. 3 10 2 10

3 5b. 5 10 6 10

1

2

4.6 10c.

2 10


b. 5 10 3 6 105 5 6 10 3 105

30 102

3103

3000

Solutiona. 3103 2 104

32 103 104

6 1034

6 107

60,000,000



c. 4.6 10 1

2 102

4.6

2

10 1

102

2.310 1 2

2.310 3

0.0023

Solution (continued)



Problem Solving

Possible Solution Strategies •Make a sketch.•Apply formulas.


The volume V of the cylindrical soda can is given by V = r2h, where r is its radius and h is its height.a. If r = 1.4 inches and h = 5 inches, find the volume of the can in cubic inches.b. Could this can hold 16 fluid ounces? (Hint: 1 cubic inch equals 0.55 fluid ounces.)

Solution

Example: Finding the Volume of a soda can

2V hr2( ) ( )

30.8 cubic inches

51.4V


b. Could this can hold 16 fluid ounces? (Hint: 1 cubic inch equals 0.55 fluid ounces.)

To find the number of fluid ounces, multiply the number of cubic inches by 0.55.

Yes, the can could hold 16 fluid ounces.

Example: Finding the Volume of a soda can

30.8 0.55 16.94

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Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 1 Introduction to Functions and Graphs