The Algebra of Functions

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Section 10.1
The Algebra of Functions
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Section 10.1Exercise 1
Chapter 10
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Section 10.1Exercise 3
Chapter 10
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Section 10.1Exercise 4
Chapter 10
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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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
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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DEFINITION
INVERSE OF A FUNCTION
The relation obtained by reversing the order of x
and y.
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PROCEDURE
FINDING THE EQUATION OF AN INVERSE FUNCTION

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 10.2Exercise 6
Chapter 10
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Section 10.2Exercise 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
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OBJECTIVES
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OBJECTIVES
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DEFINITION
EXPONENTIAL FUNCTION

A function defined for all real values of x by
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DEFINITION
INCREASING AND DECREASING FUNCTIONS
Increasing rises left to right. Decreasing
falls left to right.
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DEFINITION
NATURAL EXPONENTIAL FUNCTION, BASE e

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Section 10.3Exercise 9
Chapter 10
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Yes
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increasing
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Section 10.3Exercise 10
Chapter 10
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Section 10.4
Logarithmic Functions and their Properties
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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DEFINITION
LOG3x

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

ƒ(x) y logbx is
equivalent to by x (b gt 0, b ? 1, and x gt 0)
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DEFINITION
EQUIVALENCE PROPERTY
For any b gt 0, b ? 1, bx by 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 10.4Exercise 11
Chapter 10
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Section 10.4Exercise 12
Chapter 10
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Section 10.4Exercise 13
Chapter 10
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Section 10.5
Common and Natural Logarithms
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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OBJECTIVES
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DEFINITION
NATURAL LOGARITHMIC FUNCTION

ƒ(x) ln x, where x means loge x and x gt 0
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FORMULA
CHANGE-OF-BASE

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Section 10.5Exercise 18
Chapter 10
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Section 10.5Exercise 19
Chapter 10
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Section 10.5Exercise 20
Chapter 10
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Section 10.5Exercise 21
Chapter 10
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Section 10.5Exercise 22
Chapter 10
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Section 10.5Exercise 23
Chapter 10
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or
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Section 10.5Exercise 24
Chapter 10
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It takes 8.66 years to double the money.
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Section 10.5Exercise 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
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OBJECTIVES
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OBJECTIVES
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DEFINITION
EXPONENTIAL EQUATION
An equation in which the variable occurs in an
exponent.
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DEFINITION
EQUIVALENCE PROPERTY

For any b gt 0, b ? 1, bx by is equivalent to
x y.
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DEFINITION
EQUIVALENCE PROPERTY FOR LOGARITHMS
logbM logbN is
equivalent to M N

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PROCEDURE
SOLVING LOGARITHMIC EQUATIONS

1. Write equation logbM N
2. Write equivalent exponential equation. Solve.
3. Check answer and discard values for M 0.