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Section 2.1

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.

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Section 2.1

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  1. Section 2.1 Operations With Numbers

  2. 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.

  3. Real Numbers

  4. Properties of Addition and Multiplication • Closure • Commutative • Associative • Identity • Inverse Addition Multiplication a + 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 1 Example: 0+a=a Example: a(1) = a For every real # a, For every nonzero there is a real # -a real number a, … a+(-a)=0 there is a real # 1/a

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

  6. Order of Operations If an expression involves only #’s and operations • Perform operations within the innermost grouping symbols according to Steps 2-4. • Perform operations indicated by exponents. • Perform multiplication and division in order from left to right. • Perform addition and subtraction in order from left to right.

  7. Section 2.2 Properties of Exponents

  8. 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

  9. Properties of Exponents Let a and b be nonzero real #s. Let m and n be integers. (a)ͫ (a)ⁿ = aͫ ⁺ ⁿ a ͫ / aⁿ= a ͫ⁻ⁿ (a ͫ )ⁿ = aͫⁿ (ab)ⁿ = aⁿbⁿ (a/b)ⁿ = aⁿ/bⁿ • Product of Powers • Quotient of Powers • Power of a Power • Power of a Product • Power of a Quotient

  10. 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 aͫ ΅ⁿ = (a¹΅ⁿ) ͫ = (ⁿ√a) ͫ = (ⁿ√a ͫ )

  11. Section 2.3 Introduction to Functions

  12. 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.

  13. Examples Example of a Function Example that is not a function

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

  15. 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.

  16. 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.

  17. 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 1 0 f(0) = 2(0) + 5 5 6 f(6) = 2(6) + 5 17

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

  19. Section 2.4 Operations With Functions

  20. 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

  21. 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.”

  22. 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  1 0  1 4  5  25 4  25

  23. Section 2.5 Inverses of Functions

  24. 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.

  25. 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.

  26. 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.

  27. Horizontal-Line Test The inverse is not a function. The inverse is a function. No more than one point. • More than one point.

  28. 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.

  29. 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.

  30. Section 2.6 Special Functions

  31. 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|

  32. Piecewise Functions Example One Piecewise Function 21h if 0 < h ≤ 40 w(h) = 31.5h – 420 if h > 40 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.

  33. 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.

  34. Step Functions Greatest-integer function, or rounding down function Round-up function • f(x) = [x], or f(x) = ⌊x⌋ f(x) = ⌈x⌉

  35. 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 ≥ 0 f(x) = |x| = - x if x < 0

  36. Section 2.7 A Preview of Transformations

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

  38. Exploring Translations of Data • 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. • 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. • How are the equations for the least-squares lines different? How are the graphs of the least-squares lines similar?

  39. Summary of Transformations Transformations of y = f(x) 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 • 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

  40. Summary of Transformations Transformations of y = f(x) 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 • 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

  41. Summary of Transformations Transformations of y = f(x) Transformed function y = - f(x) y = f(- x) • Reflection across the x-axis • Reflection across the y-axis

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