2. Chapter 12 12-1 Inverse Relations and Functions

5. 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

6. 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.

7. Sketch the graph of y = x 2 and its inverse on the graph provided.

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

13. 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) = x 2 Inverse: x = y 2 All functions have inverses, but the inverse is not necessarily a function.

14. 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) = x 3 Inverse: x = y 3 All functions have inverses, but the inverse is not necessarily a function.

16. Note: D S {x | x  0} and R S {y | y  0}

18. 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.

20. HW #12.1 Pg 519-520 1-35 odd, 37-43 all, 45-55 odd 56, 57, 59

21. Chapter 12 12-2 Exponential and Logarithmic Functions

22. Definition Exponential Function The Function f(x) = a x, 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

28. Definition Logarithmic Function A logarithmic Function is the inverse of an exponential function.

29. 2 to the power of 5 is 32 The exponent you put on 2 to get 32 is 5

30. Rewrite the equation in exponential form

32. Define the relationship between exponents and logarithms What is the exponent you put on 27 to get 3? 3 3 = 27 x = 1/3 What is the exponent you put on 6 to get 216? 6 3 = 216 x = 3

33. Define the relationship between exponents and logarithms

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

37. 12.3 Exponential and Logarithmic Relationships

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

42. 12.4 Properties of Logarithms

43. sum

44. Power

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

54. 12.5 Logarithmic Function Values

55. Change of base

56. HW #12.5 pg 538 1-41 odd

57. 12.7 Exponential and Logarithmic Equations

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

76. HW #12.7a Pg 547-548 1-25 Odd, 39-55 Odd

77. 12.7 Exponential and Logarithmic Equations Day 2

79. COMPOUND INTEREST FORMULA amount at the end Principal annual interest rate (as a decimal) 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.

80. Find the amount that results from $500 invested at 8% compounded quarterly after a period of 2 years. 500.08 4 4 (2)

81. 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

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

83. 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 where I 0 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 I 0 Find the loudness of the sound of a rock concert, for which the intensity is 10 11 times I 0

84. The magnitude R on the Richter scale of an earthquake of intensity I is defined as where I 0 is the minimum intensity used for comparison. An earthquake has intensity 4 x 10 8 I 0. What is its magnitude on the Richter scale? An earthquake in Anchorage, Alaska on March 27, 1964, had an intensity 2.5 x 10 8 times I 0. What was its magnitude on the Richter scale?

85. Prove:

86. HW #12.7b Pg 547-548 26-35, 54, 56, 57-62

87. 12.8 Natural Logarithms and the Number e

90. CONTINUOUS INTEREST FORMULA Amount at the end Principal annual interest rate (as a decimal) time natural base (on calculator)

91. 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.

92. 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.

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

101. Test Review

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

103. 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

104. Non-Calculator

113. Solve A = B  2C t + D for t using logarithms with base C.

114. HW R-12b Study Hard