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Section 10.1 The Algebra of Functions
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Section 10.1 Exercise #1 Chapter 10
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Section 10.1 Exercise #3 Chapter 10
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Section 10.1 Exercise #4 Chapter 10
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OBJECTIVES A Find the sum, difference, product, and quotient of two functions.
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OBJECTIVES B Find the composite of two functions.
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OBJECTIVES C Find the domain of (ƒ + g )( x ), (ƒ – g )( x ), (ƒ g )( x ), and
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OBJECTIVES D Solve an application.
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DEFINITION OPERATIONS WITH FUNCTIONS
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DEFINITION COMPOSITE FUNCTION If ƒ and g are functions:
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Section 10.2 Inverse Functions
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OBJECTIVES A Find the inverse of a function when the function is given as a set of ordered pairs.
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OBJECTIVES B Find the equation of the inverse of a function.
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OBJECTIVES C Graph a function and its inverse and determine whether the inverse is a function.
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OBJECTIVES D Solve applications involving functions.
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DEFINITION The relation obtained by reversing the order of x and y. INVERSE OF A FUNCTION
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FINDING THE EQUATION OF AN INVERSE FUNCTION PROCEDURE 1.Interchange the roles of x and y. 2.Solve for y.
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DEFINITION If y = ƒ(x) is one-to-one, the inverse of ƒ is also a function, denoted by y = ƒ –1 (x).
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Section 10.2 Exercise #6 Chapter 10
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Section 10.2 Exercise #8 Chapter 10
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The inverse is not a function.
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Section 10.3 Exponential Functions
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OBJECTIVES A Graph exponential functions of the form a x or a – x ( a > 0).
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OBJECTIVES B Determine whether an exponential function is increasing or decreasing.
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OBJECTIVES C Solve applications involving exponential functions.
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DEFINITION EXPONENTIAL FUNCTION A function defined for all real values of x by:
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DEFINITION Increasing: rises left to right. Decreasing: falls left to right. INCREASING AND DECREASING FUNCTIONS
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DEFINITION NATURAL EXPONENTIAL FUNCTION, BASE e
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Section 10.3 Exercise #9 Chapter 10
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Yes x y
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increasing
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Section 10.3 Exercise #10 Chapter 10
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Section 10.4 Logarithmic Functions and their Properties
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OBJECTIVES A Graph logarithmic functions.
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OBJECTIVES B Write an exponential equation in logarithmic form and a logarithmic equation in exponential form.
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OBJECTIVES C Solve logarithmic equations.
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OBJECTIVES D Use the properties of logarithms to simplify logarithms of products, quotients, and powers.
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OBJECTIVES E Solve applications involving logarithmic functions.
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DEFINITION Means the exponent to which we raise 3 to get x. LOG 3 x
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DEFINITION LOGARITHMIC FUNCTION ƒ ( x ) = y = log b x is equivalent to: b y = x ( b > 0, b ≠ 1, and x > 0)
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DEFINITION EQUIVALENCE PROPERTY For any b > 0, b ≠ 1, b x = b y is equivalent to x = y.
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DEFINITION PROPERTIES OF LOGARITHMS
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DEFINITION OTHER PROPERTIES OF LOGARITHMS
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Section 10.4 Exercise #11 Chapter 10
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x y
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x y
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Section 10.4 Exercise #12 Chapter 10
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Section 10.4 Exercise #13 Chapter 10
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Section 10.5 Common and Natural Logarithms
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OBJECTIVES A Find logarithms and their inverses base 10.
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OBJECTIVES B Find logarithms and their inverses base e.
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OBJECTIVES C Change the base of a logarithm.
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OBJECTIVES D Graph exponential and logarithmic functions base e.
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OBJECTIVES E Solve applications involving common and natural logarithms.
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DEFINITION NATURAL LOGARITHMIC FUNCTION ƒ ( x ) = ln x, where x means log e x and x > 0
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FORMULA CHANGE-OF-BASE
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Section 10.5 Exercise #18 Chapter 10
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Section 10.5 Exercise #19 Chapter 10
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x y
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Section 10.5 Exercise #20 Chapter 10
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x y
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x y
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Section 10.5 Exercise #21 Chapter 10
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Section 10.5 Exercise #22 Chapter 10
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Section 10.5 Exercise #23 Chapter 10
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or
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Section 10.5 Exercise #24 Chapter 10
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It takes 8.66 years to double the money.
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Section 10.5 Exercise #25 Chapter 10
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1.386 years is the half-life of this substance.
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Section 10.6 Exponential and Logarithmic Equations and Applications
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OBJECTIVES A Solve exponential equations.
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OBJECTIVES B Solve logarithmic equations.
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OBJECTIVES C Solve applications involving exponential or logarithmic equations.
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DEFINITION An equation in which the variable occurs in an exponent. EXPONENTIAL EQUATION
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DEFINITION EQUIVALENCE PROPERTY For any b > 0, b ≠ 1, b x = b y is equivalent to x = y.
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DEFINITION EQUIVALENCE PROPERTY FOR LOGARITHMS log b M = log b N is equivalent to M = N
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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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