section 2.1 operations with numbers. number sets natural numbers -> 1, 2, 3, … whole numbers...

Section 2.1

Operations With Numbers

Number Sets

• Natural numbers -> 1, 2, 3, …• Whole numbers -> 0, 1, 2, 3, …• Integers -> …, -3, -2, -1, 0, 1, 2, 3, …• Rational numbers -> (p/q), where p and q are

integers and q ≠ 0.• Irrational numbers -> numbers whose decimal

part does not terminate or repeat.• Real numbers -> all rational and irrational

numbers.

Real Numbers

Rational

Integer

Whole

Natural Irrational

Properties of Addition and Multiplication

• Closure• Commutative• Associative• Identity

• Inverse

Addition Multiplicationa + b is a real #. ab is a real #.a + b = b + a ab = ba(a+b)+c = a+(b+c) (ab)c = a(bc)

0 1Example: 0+a=a Example: a(1) = aFor every real # a, For every nonzero there is a real # -a real number a, … a+(-a)=0 there is a real #

1/a

The Distributive Property

For all real #’s a, b, and c: a(b+c) = ab + ac and (b+c)a = ba+ca

Order of Operations

If an expression involves only #’s and operations1. Perform operations within the innermost

grouping symbols according to Steps 2-4.2. Perform operations indicated by exponents.3. Perform multiplication and division in order

from left to right.4. Perform addition and subtraction in order

from left to right.

Section 2.2

Properties of Exponents

Definition of Integer Exponents

• Let a be a real number.• If n is a natural number, then aⁿ = a x a x a x … x a, n times.• If a is nonzero, then a⁰ = 1• If n is a natural #, then a ⁿ = ⅟aⁿˉ

Properties of Exponents

Let a and b be nonzero real #s.

• Product of Powers• Quotient of Powers• Power of a Power• Power of a Product• Power of a Quotient

Let m and n be integers.

(a) � (a)ⁿ = a � ⁺ ⁿ a � / aⁿ = a � ⁻ ⁿ(a � )ⁿ = a � ⁿ(ab)ⁿ = aⁿbⁿ(a/b)ⁿ = aⁿ/bⁿ

Definition of Rational Exponents

For all positive real numbers a:

• If n is a nonzero integer, then a¹΅ⁿ = ⁿ√a• If m and n are integers and n ≠ 0, thena � ΅ⁿ = (a¹΅ⁿ) � = (ⁿ√a) � = (ⁿ√a � )

Section 2.3

Introduction to Functions

Definition of Function

• A function is a relationship between two variables such that each value of the first variable is paired with exactly one value of the second variable.

• The domain of a function is the set of all possible values of the first variable. The range of a function is the set of all possible values of the second variable.

Examples

Example of a Function

Domain, x Range, y

1 -2

2 -5

3 4

7 15

8 4

15 -7

Example that is not a function

Domain, x Range, y

4 10

5 53

5 27

6 20

12 1

13 -9

Vertical-Line Test

If every vertical line intersects a given graph at no more than one point, then the graph represents a function.

Definition of Relation

• A relationship between two variables such that each value of the first variable is paired with one or more values of the second variable is called a relation.

Function Notation

• If there is a correspondence between values of the domain, x, and values of the range, y, that is a function, then y = f(x), and (x,y) can be written as (x,f(x)). The notation f(x) is read “f of x.” The number represented by f(x) is the value of the function f at x.

• The variable x is called the independent variable.• The variable y, or f(x), is called the dependent

variable.

Functions and Function Notation

• An equation can represent a function. The equation y = 2x + 5 represents a function. To express this equation as a function, use function notation and write y = 2x + 5 as f(x) = 2x + 5.

x --------f(x) = 2x + 5-----------f(x)-2 f(-2) = 2(-2) + 5 10 f(0) = 2(0) + 5 56 f(6) = 2(6) + 5 17

Comparing Terms

• x-variable

• Domain

• Independent Variable

• Input

• y-variable

• Range

• Dependent Variable

• Output

Section 2.4

Operations With Functions

Operations With Functions

For all functions f and g:• Sum (f + g)(x) = f(x) + g(x)• Difference (f – g)(x) = f(x) – g(x)• Product (f · g)(x) = f(x) · g(x)• Quotient (f/g)(x) = f(x)/g(x),

where g(x) ≠ 0

Composition of Functions

• Let f and g be functions.• The composition of f with g, denotes f ∘ g, is

defined by f(g(x)).• The domain of y = f(g(x)) is the set of domain

values of g whose range values are in the domain of f. The function f ∘ g is called the composition function of f with g.

• Example: f ∘ g, or f(g(x)) reads “f of g of x.”

Composition of functions

• f = {(- 3, - 2), (0, 1), (4, 5)}• g = {(- 2, 4), (1, 1), (5, 25)}• The range of f is the domain of g.

f g g ∘ f- 3 - 2 4 -3 40 1 1 0 14 5 25 4 25

Section 2.5

Inverses of Functions

Inverse of a Relation

• The inverse of a relation consisting of the ordered pairs (x, y) is the set of all ordered pairs (y, x).

• The domain of the inverse is the range of the original relation.

• The range of the inverse is the domain of the original relation.

The inverse of the relation

• Relation: {(1, 2), (2, 4), (3, 6), (4, 8)}• The given relation is a function because each

domain value is paired with exactly one range value.

• Inverse: {(2, 1), (4, 2), (6, 3), (8, 4)}• The inverse is also a function because each

domain value is paired with exactly one range value.

Horizontal-Line Test

• The inverse of a function is a function if and only if every horizontal line intersects the graph of the given function at no more than one point.

Horizontal-Line Test

The inverse is not a function.• More than one point.

The inverse is a function.• No more than one point.

If a function has an inverse that is also a function, then the function is a one-to-one function. Every one-to-one function passes the horizontal-line test and has an inverse that is a function.

Composition and Inverses

• If f and g are functions and (f g)(x) = (g f)(x) = ∘ ∘ I(x) = x, then f and g are

inverses of one another.

Just as the graphs of f and f¯¹ are reflections of one another across the line y = x, the composition of a function and its inverse are related to the identity function.

Section 2.6

Special Functions

Special Functions

• Piecewise Functions

• Step Functions

• Absolute-Value Functions

• A function that consists of different function rules for different parts of the domain.

• A function whose graph looks like a series of steps.

• A function described by f(x) = |x|

Piecewise Functions

Example OneA truck driver ears $21.00 per hour for the first 40 hours worked in one week. The driver ears time-and-a-half, or $31.50, for each hour worked in excess of 40. The pair of function rules represent the driver’s wage, w(h), as a function of the hours worked in one week, h.

Piecewise Function

21h if 0 < h ≤ 40

w(h) = 31.5h – 420 if h >

40

Constant Function

The graph of a linear function with a slope of 0 is a horizontal line. This type of function is called a constant function because every function value is the same number.

Step Functions Greatest-integer function, or rounding down function

• f(x) = [x], or f(x) = x⌊ ⌋

Round-up function

x - 3 -1.5 0 2.8

f(x) = [x] - 3 - 2 0 2

f(x) = x⌈ ⌉

x - 3 -1.5 0 2.8

f(x) = x⌈ ⌉ - 3 -1 0 3

Absolute-Value Functions

• The absolute-value function, denoted by f(x) = |x|, can be defined as a piecewise function as follows:

|x| = x if x ≥ 0f(x) =

|x| = - x if x < 0

Section 2.7

A Preview of Transformations

Exploring Translations of Data

YearNumber of years after

1960

Number of years after

1970

Cases reported (in

thousands)

1989 29 19 185.4

1990 30 20 173.1

1991 31 21 147.1

1992 32 22 158.4

1993 33 23 134.7

1994 34 24 151.2

The table of data gives the number of reported cases of chickenpox in the U.S. in thousands from 1989-1994.

Exploring Translations of Data

1 Enter the data from columns 2 and 4 into your graphing calculator. Make a scatter plot. Then use linear regression to find an equation for the least-squares line.

2 Enter the data from columns 3 and 4 into your graphing calculator. Make a scatter plot. Then find an equation for the least-squares line.

3 How are the equations for the least-squares lines different? How are the graphs of the least-squares lines similar?

Summary of Transformations

Transformations of y = f(x)• Vertical translation of k

units up• Vertical translation of |k|

units down• Horizontal translation of h

units to the right• Horizontal translation of |h|

units to the left

Transformed function• y = f(x) + k, where k > 0

• y = f(x) + k, where k < 0

• y = f(x – h), where h > 0

• y = f(x – h), where h < 0

Summary of Transformations

Transformations of y = f(x)• Vertical stretch by a factor

of a• Vertical compression by a

factor of a• Horizontal stretch by a

factor of ⅟ b• Horizontal compression by a

factor of ⅟ b

Transformed function• y = af(x), where a > 1

• y = af(x), where 0 < a < 1

• y = f(bx), where 0 < b < 1

• y = f(bx), where b > 1

Summary of Transformations

Transformations of y = f(x)• Reflection across the x-axis

• Reflection across the y-axis

Transformed function• y = - f(x)

• y = f(- x)