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 AdditionMultiplication a + b is a real #.ab is a real #. a + b = b + aab = ba (a+b)+c = a+(b+c)(ab)c = a(bc) 01 Example: 0+a=aExample: a(1) = a For every real # a,For every nonzero there is a real # -areal number a, … a+(-a)=0there 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 operations 1.Perform operations within the innermost grouping symbols according to Steps 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 aˉⁿ = ⅟aⁿ

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, then a ͫ ΅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, xRange, y Example that is not a function Domain, xRange, y

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 f(x) -2f(-2) = 2(-2) f(0) = 2(0) f(6) = 2(6)

Comparing Terms x-variable Domain Independent Variable Input y-variable Range Dependent Variable Output

Section 2.4 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  4 0  1  10  1 4  5  254  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

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 One A 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 – 420if 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 f(x) = [x] f(x) = ⌈ x ⌉ x f(x) = ⌈ x ⌉ - 303

Absolute-Value Functions The absolute-value function, denoted by f(x) = |x|, can be defined as a piecewise function as follows: |x| = xif x ≥ 0 f(x) = |x| = - x if x < 0

Section 2.7 A Preview of Transformations

Exploring Translations of Data Year Number of years after 1960 Number of years after 1970 Cases reported (in thousands) The table of data gives the number of reported cases of chickenpox in the U.S. in thousands from

Exploring Translations of Data 1Enter 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. 2Enter the data from columns 3 and 4 into your graphing calculator. Make a scatter plot. Then find an equation for the least-squares line. 3How 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)