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Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10.

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Presentation on theme: "Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10."— Presentation transcript:

1 Section 10.1 The Algebra of Functions

2 Section 10.1 Exercise #1 Chapter 10

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8 Section 10.1 Exercise #3 Chapter 10

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13 Section 10.1 Exercise #4 Chapter 10

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17 OBJECTIVES A Find the sum, difference, product, and quotient of two functions.

18 OBJECTIVES B Find the composite of two functions.

19 OBJECTIVES C Find the domain of (ƒ + g )( x ), (ƒ – g )( x ), (ƒ g )( x ), and

20 OBJECTIVES D Solve an application.

21 DEFINITION OPERATIONS WITH FUNCTIONS

22 DEFINITION COMPOSITE FUNCTION If ƒ and g are functions:

23 Section 10.2 Inverse Functions

24 OBJECTIVES A Find the inverse of a function when the function is given as a set of ordered pairs.

25 OBJECTIVES B Find the equation of the inverse of a function.

26 OBJECTIVES C Graph a function and its inverse and determine whether the inverse is a function.

27 OBJECTIVES D Solve applications involving functions.

28 DEFINITION The relation obtained by reversing the order of x and y. INVERSE OF A FUNCTION

29 FINDING THE EQUATION OF AN INVERSE FUNCTION PROCEDURE 1.Interchange the roles of x and y. 2.Solve for y.

30 DEFINITION If y = ƒ(x) is one-to-one, the inverse of ƒ is also a function, denoted by y = ƒ –1 (x).

31 Section 10.2 Exercise #6 Chapter 10

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36 Section 10.2 Exercise #8 Chapter 10

37 The inverse is not a function.

38 Section 10.3 Exponential Functions

39 OBJECTIVES A Graph exponential functions of the form a x or a – x ( a > 0).

40 OBJECTIVES B Determine whether an exponential function is increasing or decreasing.

41 OBJECTIVES C Solve applications involving exponential functions.

42 DEFINITION EXPONENTIAL FUNCTION A function defined for all real values of x by:

43 DEFINITION Increasing: rises left to right. Decreasing: falls left to right. INCREASING AND DECREASING FUNCTIONS

44 DEFINITION NATURAL EXPONENTIAL FUNCTION, BASE e

45 Section 10.3 Exercise #9 Chapter 10

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47 Yes x y

48 increasing

49 Section 10.3 Exercise #10 Chapter 10

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53 Section 10.4 Logarithmic Functions and their Properties

54 OBJECTIVES A Graph logarithmic functions.

55 OBJECTIVES B Write an exponential equation in logarithmic form and a logarithmic equation in exponential form.

56 OBJECTIVES C Solve logarithmic equations.

57 OBJECTIVES D Use the properties of logarithms to simplify logarithms of products, quotients, and powers.

58 OBJECTIVES E Solve applications involving logarithmic functions.

59 DEFINITION Means the exponent to which we raise 3 to get x. LOG 3 x

60 DEFINITION LOGARITHMIC FUNCTION ƒ ( x ) = y = log b x is equivalent to: b y = x ( b > 0, b ≠ 1, and x > 0)

61 DEFINITION EQUIVALENCE PROPERTY For any b > 0, b ≠ 1, b x = b y is equivalent to x = y.

62 DEFINITION PROPERTIES OF LOGARITHMS

63 DEFINITION OTHER PROPERTIES OF LOGARITHMS

64 Section 10.4 Exercise #11 Chapter 10

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66 x y

67 x y

68 Section 10.4 Exercise #12 Chapter 10

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71 Section 10.4 Exercise #13 Chapter 10

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73 Section 10.5 Common and Natural Logarithms

74 OBJECTIVES A Find logarithms and their inverses base 10.

75 OBJECTIVES B Find logarithms and their inverses base e.

76 OBJECTIVES C Change the base of a logarithm.

77 OBJECTIVES D Graph exponential and logarithmic functions base e.

78 OBJECTIVES E Solve applications involving common and natural logarithms.

79 DEFINITION NATURAL LOGARITHMIC FUNCTION ƒ ( x ) = ln x, where x means log e x and x > 0

80 FORMULA CHANGE-OF-BASE

81 Section 10.5 Exercise #18 Chapter 10

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84 Section 10.5 Exercise #19 Chapter 10

85 x y

86 Section 10.5 Exercise #20 Chapter 10

87 x y

88 x y

89 Section 10.5 Exercise #21 Chapter 10

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92 Section 10.5 Exercise #22 Chapter 10

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95 Section 10.5 Exercise #23 Chapter 10

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97 or

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100 Section 10.5 Exercise #24 Chapter 10

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103 It takes 8.66 years to double the money.

104 Section 10.5 Exercise #25 Chapter 10

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107 1.386 years is the half-life of this substance.

108 Section 10.6 Exponential and Logarithmic Equations and Applications

109 OBJECTIVES A Solve exponential equations.

110 OBJECTIVES B Solve logarithmic equations.

111 OBJECTIVES C Solve applications involving exponential or logarithmic equations.

112 DEFINITION An equation in which the variable occurs in an exponent. EXPONENTIAL EQUATION

113 DEFINITION EQUIVALENCE PROPERTY For any b > 0, b ≠ 1, b x = b y is equivalent to x = y.

114 DEFINITION EQUIVALENCE PROPERTY FOR LOGARITHMS log b M = log b N is equivalent to M = N

115 SOLVING LOGARITHMIC EQUATIONS PROCEDURE 1.Write equation: log b M = N 2.Write equivalent exponential equation. Solve. 3.Check answer and discard values for M ≤ 0.


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