Chapter 12

12-1 Inverse Relations and Functions

Inverse

• Operations– Two operations that undo each other

• Addition/Subtraction• Multiplication/Division• Squares/Square Roots

• Functions– Two functions that undo each other

• F(x) = 2x and G(x) = (½)x•

3 3f(x)= x -1andG(x)= x+1

An inverse relation maps the output values back to their original input values.

The graph of an inverse relation is the reflection of the graph of the original relation over the line y = x.

Sketch the graph of y = x2 and its inverse on the graph provided.

Interchange x and y in the equation

This is not equivalent to the original equation, so the. graph is not symmetric to y = x.

The graphs of a relation and its inverse are always reflections of each other across the line y = x.

Graph the function and its inverse: ƒ(x) = x2

Inverse: x = y2

All functions have inverses, but the inverse is not necessarily a function.

The graphs of a relation and its inverse are always reflections of each other across the line y = x.

Graph the Function and its inverse: g(x) = x3

Inverse: x = y3

All functions have inverses, but the inverse is not necessarily a function.

Note: DS{x | x 0} and RS{y | y 0}

1 1{ | 0} { | 0}S S

D x x and R y y

We note that every real number is in the domain of both f and f--1. Thus using Theorem 12-2, we may immediately write the answers, without calculating.

HW #12.1Pg 519-520

1-35 odd, 37-43 all, 45-55 odd 56, 57, 59

Chapter 12

12-2 Exponential and Logarithmic Functions

Definition Exponential Function

The Function f(x) = ax, where a is some positive real number constant different from 1, is called an exponential function, base a.

Note: In an exponential function the variable is in the exponent

12

x

y

12xy

22 3xy

Definition Logarithmic Function

A logarithmic Function is the inverse of an exponential function.

2 to the power of 5 is 32

The exponent you put on 2 to get 32 is 5

Rewrite the equation in exponential form

Rewrite the equation in exponential form

Define the relationship between exponents and logarithms

What is the exponent you put on 27 to get 3?

27 3x

33 = 27

x = 1/3

What is the exponent you put on 6 to get 216?

6 216x

63 = 216

x = 3

Define the relationship between exponents and logarithms

HW #12.2 Pg 525 1-39 Odd, 40-42

12.3 Exponential and Logarithmic Relationships

log xb b x

logb xb x

For all 0 and 1

logxa

a a

y a x y

HW #12.3 Pg 528 1-43 Odd, 44-46

12.4 Properties of Logarithms

sum

Power

2 32Write log 1 , >1, as a sum of logarithms.

Express all powers as factors.

x x x

4

6 22Write log , 0, as a difference of logarithms.

3

Express all powers as factors.

xx

x

3

5 2

2Write log , 2, as a sum and difference of logarithms.

1Express all powers as factors.

x xx

x

2

2

Write each of the following as a single logarithm.

3log 2 log 2

1log 4 2log 5

2

2log 3 3log 2 log 1

a a

a a

a a a

a x

b

c x

Solve: 3 7x

Solve: 5 2 3x

1 2 3Solve: 2 5x x

Solve: 9 3 6 0x x 2

Hint: If you replace 9 with 3 then the equation is quadratic in form.x x

HW #12.4Pg 532-533 1-31 odd, 33-51

12.5 Logarithmic Function Values

Change of base

HW #12.5pg 538 1-41 odd

12.7 Exponential and Logarithmic Equations

9 92log log9 27 x x

52log5 16x

3 3Solve: log 4 2log x

Reminder: Logarithms of negative numbers are not defined so check for extraneous solutions.

2 2Solve: log 2 log 1 1x x

HW #12.7aPg 547-548 1-25 Odd, 39-55 Odd

12.7 Exponential and Logarithmic Equations

Day 2

COMPOUND INTEREST FORMULA

amount at the end

Principal

annual interest rate

(as a decimal)nt

n

rPA

1

time

number of times per year that interest in

compounded

A is typically referred to as the Future Value of the account.

P is typically referred to as the present value of the account.

nt

n

rPA

1

Find the amount that results from $500 invested at 8% compounded quarterly after a period of 2 years.

500.08

4

4 (2)

83.585$A

Find the principal needed now to get each amount; that is, find the present value.

1. To get $100 after three years at 6% compounded monthly

2. To get $1000 after 4 years at 7% compounded daily

3. To get $400 after two and a half years at 5% compounded daily

4. To get $400 after 1 year at 10% compounded daily

How long will it take for an investment of $1000 to double itself when interest is compounded annually at 6%?

Loudness is measured in bels (after Alexander Graham Bell) or in smaller units called decibels. Loudness in decibels of a sound of intensity I is defined to be

0

10 logI

LI

where I0 is the minimum intensity detectable by the human ear.

Find the loudness in decibels, of the background noise in a radio studio, for which the intensity I is 199 times I0

Find the loudness of the sound of a rock concert, for which the intensity is 1011 times I0

The magnitude R on the Richter scale of an earthquake of intensity I is defined as

0

logI

RI

where I0 is the minimum intensity used for comparison.

An earthquake has intensity 4 x 108 I0. What is its magnitude on the Richter scale?

An earthquake in Anchorage, Alaska on March 27, 1964, had an intensity 2.5 x 108 times I0. What was its magnitude on the Richter scale?

Prove:

HW #12.7bPg 547-548 26-35, 54, 56, 57-62

12.8 Natural Logarithms and the Number e

e

CONTINUOUS INTEREST FORMULA

rtPeA Amount at the end

Principal

annual interest rate

(as a decimal)

time

natural base (on calculator)

rtPeA

Find the amount that results from $40 invested at 7% compounded continuously after a period of 3 years.

40(.07)(3)

A = $49.35

Now punch buttons in your calculator. Make sure you put parenthesis around the entire exponent on e.

Find the amount A that results from investing a principal P of $2000 at an annual rate r of 8% compounded continuously for a time t of 1 year.

HW #12.8Pg 555-556 1-41 Odd, 42-46

Test Review

HW R-12 Pg 560-561 1-40 Skip 31, 32

Two Parts

• Non-Calculator• Definition• Properties of Logs• Change of Base• Inverse Functions• Function Notation• Solving Equations• Proofs

• Calculator• Exponential Growth• Exponential Decay• Compound Interest

– Continuous

Non-Calculator

3 5 1If ( ) 2 ,Find ( ).xf x f x

Solve A = B2Ct + D for t using logarithms with base C.

HW R-12b Study Hard