Section 10.1 The Algebra of Functions

Section 10.1 Exercise #1 Chapter 10

Section 10.1 Exercise #3 Chapter 10

Section 10.1 Exercise #4 Chapter 10

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

OBJECTIVES B Find the composite of two functions.

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

OBJECTIVES D Solve an application.

DEFINITION OPERATIONS WITH FUNCTIONS

DEFINITION COMPOSITE FUNCTION If ƒ and g are functions:

Section 10.2 Inverse Functions

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

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

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

OBJECTIVES D Solve applications involving functions.

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

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

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

Section 10.2 Exercise #6 Chapter 10

Section 10.2 Exercise #8 Chapter 10

The inverse is not a function.

Section 10.3 Exponential Functions

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

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

OBJECTIVES C Solve applications involving exponential functions.

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

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

DEFINITION NATURAL EXPONENTIAL FUNCTION, BASE e

Section 10.3 Exercise #9 Chapter 10

Yes x y

increasing

Section 10.3 Exercise #10 Chapter 10

Section 10.4 Logarithmic Functions and their Properties

OBJECTIVES A Graph logarithmic functions.

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

OBJECTIVES C Solve logarithmic equations.

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

OBJECTIVES E Solve applications involving logarithmic functions.

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

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

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

DEFINITION PROPERTIES OF LOGARITHMS

DEFINITION OTHER PROPERTIES OF LOGARITHMS

Section 10.4 Exercise #11 Chapter 10

x y

x y

Section 10.4 Exercise #12 Chapter 10

Section 10.4 Exercise #13 Chapter 10

Section 10.5 Common and Natural Logarithms

OBJECTIVES A Find logarithms and their inverses base 10.

OBJECTIVES B Find logarithms and their inverses base e.

OBJECTIVES C Change the base of a logarithm.

OBJECTIVES D Graph exponential and logarithmic functions base e.

OBJECTIVES E Solve applications involving common and natural logarithms.

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

FORMULA CHANGE-OF-BASE

Section 10.5 Exercise #18 Chapter 10

Section 10.5 Exercise #19 Chapter 10

x y

Section 10.5 Exercise #20 Chapter 10

x y

x y

Section 10.5 Exercise #21 Chapter 10

Section 10.5 Exercise #22 Chapter 10

Section 10.5 Exercise #23 Chapter 10

or

Section 10.5 Exercise #24 Chapter 10

It takes 8.66 years to double the money.

Section 10.5 Exercise #25 Chapter 10

1.386 years is the half-life of this substance.

Section 10.6 Exponential and Logarithmic Equations and Applications

OBJECTIVES A Solve exponential equations.

OBJECTIVES B Solve logarithmic equations.

OBJECTIVES C Solve applications involving exponential or logarithmic equations.

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

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

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

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.