1. 4.1
4.2
4.3
4.4
4.5
4.1
Inverse Functions
Inverse Operations ▪ One-to-One Functions ▪ Inverse Functions ▪ Equations of Inverses ▪ An Application of Inverse Functions to Cryptography
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2. 4.1 Example 2(a) Using the Horizontal Line Test (page 404)
Determine whether each graph is the graph of a one-to-one function.
Since the horizontal line will intersect the graph at more than one point, the function is not one-to-one.
Passes the vertical and horizontal line test, so the function is one-to-one.
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3. 4.1 Example 1 Deciding Whether Functions are One-to-One (page 403)
Decide whether each function is one-to-one.
(a) f(x) = –3x + 7
For a function to be one-to-one, the graph must pass both the vertical and the horizontal line test.
The graph of f(x) is an oblique line, therefore f(x) is one-to-one
(b)
y2 = 49 – x2 → x2 + y2 = 49
Since this graph is the top half of a circle with center at (0,0) and a radius of 7, it doesn’t pass the horizontal line test. This function is NOT one-to-one
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4. 4.1 Example 3 Deciding Whether Two Functions are Inverses (page 405)
Let and Is g the inverse function of f ?
Find
Since , g is NOT the inverse of f.
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5. 4.1 Example 4(a) Finding Inverses of One-to-One Functions (page 407)
Find the inverse of the function F = {(–2, –8), (–1, –1), (0, 0), (1, 1), (2, 8) .
Reverse the x- and y-values in each ordered pair in order to find the inverse function.
Find the inverse of the function G = {(–2, 5), (–1, 2), (0, 1), (1, 2), (2, 5) .
If you reverse the x- and y-coordinates, G-1 will not be a function. {(5,-2), (2,-1), (1,0), (2,1), (5,2)}
Thus, G is not one-to-one and does not have an inverse.
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6. 4.1 Example 4(c) Finding Inverses of One-to-One Functions (page 407)
Find the inverse of the function h defined by the table.
Each x-value in h corresponds to just one y-value.
However, the y-value 33 corresponds to two x-values, 2002 and 2005.
Thus, h is not one-to-one and does not have an inverse.
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7. 4.1 Example 5(a) Finding Equations of Inverses (page 407)
Is a one-to-one function? If so, find the equation of its inverse.
is a function, but it is not a one-to-one function, therefore it does not have an inverse.
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8. 4.1 Example 5(b) Finding Equations of Inverses (page 407)
Is g(x) = 4x – 7 a one-to-one function? If so, find the equation of its inverse.
x + 7 = 4y
The graph of g is a nonhorizontal line, so by the vertical and horizontal line test, g is a one-to-one function, therefore it will have an inverse.

9. 4.1 Example 5(c) Finding Equations of Inverses (page 407)
Is a one-to-one function? If so, find the equation of its inverse.
x – 5 = y3
The graph of h(x) is the x3 graph shifted up 5 units. Therefore h(x) is a one-to-one function.
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10. 4.1 Example 6 Graphing the Inverse (page 409)
Determine whether functions f and g graphed are inverses of each other.
f and g graphed are not inverses because the graphs are not reflections across the line y = x.
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11. f and f -1 are mirror images with respect to the line y = x.
4.1 Example 7 Finding the Inverse of a Function with a Restricted Domain (page 409)
x – 4 = y2
Because the domain is restricted, the function is one-to-one and therefore has an inverse.
f and f -1 are mirror images with respect to the line y = x.
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12. 4.1 Example 8 Finding the Inverse of a Function
Find the inverse of
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13. 4.1 Add on to notes f-1 f Given: F(x) = x3 – 2 Find f-1(x)
Graph f(x) and f-1(x)
State the D and R for f(x) and f-1(x)
y = x3 – 2
x = y3 – 2
x + 2 = y3
Domain and Range of f:
All real numbers
Domain and range of f-1:
f-1 f
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14. Inverse Functions One-to-One Functions Inverse Functions
4.1 Summary
Inverse Functions
One-to-One Functions
Inverse Functions
Equations of Inverses
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15. Exponential Functions
4.2
Exponential Functions
Exponents and Properties ▪ Exponential Functions ▪ Exponential Equations ▪ Compound Interest ▪ The Number e and Continuous Compounding ▪ Exponential Models and Curve Fitting
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16. 4.2 Example 1 Evaluating an Exponential Expression (page 417)
If , find each of the following.
(a)
(b)
(c)
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17. 4.2 Example 2 Graphing an Exponential Function (page 419) NCY -2 -1 1 2 4 2 1
f(x) is an exponential decay with an
important point at (0,1) and an asymptote at y = 0.
f has domain and range and is one-to-one.

18. 4.2 Example 3(a) Graphing Reflections and Translations NC (page 420)
Give the domain and range.
The graph of f(x) = -3x is obtained by reflecting the graph of f(x) = 3x across the x-axis. Original important pt. (0,1) so the new important pt. is (0,-1) X Y -2 -1 1 2
-1/9
-1/3 -1 -3 -9
Asymptote:
Domain:
Range:
y = 0
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19. 4.2 Example 3(b) Graphing Reflections and Translations NC (page 420)
Give the domain and range.
The graph of is obtained by translating the graph of two units to the right. Original important point (0,1) and asymptote of y = 0 shifted to the right two units becomes (2,1) and y = 0.
Domain: Range:
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20. 4.2 Example 3(c) Graphing Reflections and Translations NC (page 420)
Give the domain and range.
The graph of is obtained by translating the graph of two units down. Original important point (0,1) and asymptote of y = 0 become (0,-1) and y = -2.
Domain: Range:
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21. 4.2 Add on to notes
Graph
The new system has them plot points… not all graphs m/c anymore
Y = (1/7)x flipped over the y-axis and shifted right 5 units X Y 3 4 5 1 6 7 X Y 5 X Y 3 4 5 6 7
1/49
1/7 7 49
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22. 3x+1 = (32)x – 3 x = -3 3x+1 = 32x – 6 x + 1 = 2x – 6 7 = x
4.2 Example 4 Using a Property of Exponents to Solve an Equation (page 420) NO CALCULATOR
3x+1 = (32)x – 3
x = -3
3x+1 = 32x – 6
x + 1 = 2x – 6
7 = x
Solution set: {–3}
Solution set: {7}
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23. b = b = 32 b = 9 Solution set: {9}
4.2 Example 6 Using a Property of Exponents to Solve an Equation (page 421)
It is necessary to check all proposed solutions in the original equation when both sides have been raised to an even power.
Check b = 9.
b =
b = 32
b = 9
Solution set: {9}
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24. Additional Example x = 1 2/3 = 5/3 2.5 = 1.5x Solve (NO CALCULATOR)
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25. 4.2 Example 7(a) Using the Compound Interest Formula WITH A CALCULATOR (page 422)
Suppose $2500 is deposited in an account paying 6% per year compounded semiannually (twice per year). Find the amount in the account after 10 years with no withdrawals. And then find how much interest is earned over the 10-year period?
Compound interest formula
P = 2500, r = .06, n = 2, t = 10
Round to the nearest hundredth.
There is $ in the account after 10 years. The interest earned over the 10 years is $ – $2500 = $

26. 4.2 Example 8(a) Finding Present Value (page 423)
WITH A CALCULATOR
Leah must pay a lump sum of $15,000 in 8 years. What amount deposited today at 4.8% compounded annually will give $15,000 in 8 years?
Compound interest formula
A = 15,000, r = .048, n = 1, t = 8
Simplify, then solve for P.
Round to the nearest hundredth.
If Leah deposits $10, now, she will have $15,000 when she needs it.
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27. 4.2 Example 8(b) Finding Present Value (page 423)
WITH A CALCULATOR
If only $10,000 is available to deposit now, what annual interest rate is necessary for the money to increase to $15,000 in 8 years?
Compound interest formula
A = 15,000, P = 10,000, n = 1, t = 8
3/2 = (1 + r)8
An interest rate of about 5.20% will produce enough interest to increase the $10,000 to $15,000 by the end of 8 years.
(3/2)(1/8) = 1 + r
(3/2)(1/8) – 1 = r
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28. 4.2 Example 9 Solving a Continuous Compounding Problem (page 424)
WITH A CALCULATOR
Suppose $8000 is deposited in an account paying 5% interest compounded continuously for 6 years. Find the total amount on deposit at the end of 6 years.
Continuous compounding formula
P = 8000, r = .05, t = 6
Round to the nearest hundredth.
There will be about $10, in the account at the end of 6 years.
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29. 4.2 Example 10 Comparing Interest Earned as Compounding is More Frequent (page 425)
Suppose $2500 is invested at 6% in an account for 10 years. Find the amounts in the account at the end of 10 years if the interest is compounded quarterly, monthly, daily, and continuously.
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30. 4.2 Example 11(a) Using Data to Model Exponential Growth (page 426)
If current trends of burning fossil fuels and deforestation continue, then future amounts of atmospheric carbon dioxide in parts per million (ppm) will increase as shown in the table.
The data can be modeled by the function
What will be the atmospheric carbon dioxide level in 2015?
≈ ppm
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31. Exponential Functions
4.2 Summary
Exponential Functions
Exponents and Properties
Exponential Functions
Exponential Equations
Compound Interest
The Number e and Continuous Compounding
Exponential Models and Curve Fitting
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32. Logarithmic Functions
4.3
Logarithmic Functions
Logarithms ▪ Logarithmic Equations ▪ Properties of Logarithms
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33. Log Rules to Memorize
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34. 4.3 Example 1 Solving Logarithmic Equations (page 433)
Solve:
(a)
(b)
(c)
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35. 4.3 Example 2(a) Graphing Logarithmic Functions (cont.) NC
The graph of is the reflection of the graph of across the line y = x.
Therefore the two functions are inverses of each other.
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36. 4.3 Example 2(b) Graphing Logarithmic Functions (cont.) NC
The graph of is the reflection of the graph of across the line y = x.
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37. 4.3 Example 2(a) Graphing Logarithmic Functions (page 436) NC
The graph of f(x) is a logarithmic decay.
Important point at (1,0) and asymptote at x = 0.
The graph of f(x) is a logarithmic growth.
Important point at (1,0) and asymptote at x = 0.
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38. The graph of f(x) is g(x) = log3(x) transformed: Reflection
4.3 Example 3(a) Graphing Translated Logarithmic Functions (page 437) NC
Parent function:
g(x) = log3(x)
The graph of f(x) is g(x) = log3(x) transformed:
Reflection
Horizontal stretch
Vertical stretch
Horizontal shift
Vertical shift
None –2
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39. The graph of f(x) is g(x) = log4(x) transformed: Reflection
4.3 Example 3(b) Graphing Translated Logarithmic Functions (page 437) NC
Parent function:
g(x) = log4(x)
The graph of f(x) is g(x) = log4(x) transformed:
Reflection
Horizontal stretch
Vertical stretch
Horizontal shift
Vertical shift
None +2
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40. Graph f(x) = log⅓(3 – x) 4.3 Add on (no calculator) Parent function:
g(x) = log⅓(x)
The graph of f(x) is g(x) = log⅓(x) transformed:
Reflection
Horizontal stretch
Vertical stretch
Horizontal shift
Vertical shift
Y-axis
None +3
Asy: x = 3
D: (-∞,3) R: (-∞,∞)
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41. 4.3 Example 4 Using the Properties of Logarithms (page 438)
Rewrite each expression using the Properties of Logarithms. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
(a)
(b)
(c)
(d)

42. 4.3 Example 4 Using the Properties of Logarithms (cont.)
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
(e)
(f)
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43. 4.3 Example 5 Using the Properties of Logarithms (page 439)
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
(a)
(b)
(c)
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44. Additional Example
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers
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45. 4.3 Example 6 Using the Properties of Logarithms (page 440)
Assume that log107 = Find each logarithm.
(a)
(b)
(c)

46. Additional Example
Suppose f(x) = loga(x) and f(5) = 2. Determine the function value of each:
f(1/25)
f(125)
f(25)
f(√5/5)
f(5) = 2 means loga(5) = 2
= loga(1/25)
= loga(5-2)
= -2loga(5)
= -4
= loga(125)
= loga(53)
= 3loga(5)
= 6
= loga(25)
= loga(52)
= 2loga(5)
= 4
= loga(5-1/2)
= -1/2 loga(5)
= -1
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47. Logarithms • Logarithmic Equations Properties of Logarithms
4.3 Summary
Logarithms • Logarithmic Equations
Properties of Logarithms
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48. Evaluating Logarithms and the Change-of-Base Theorem
4.4
Evaluating Logarithms and the Change-of-Base Theorem
Common Logarithms ▪ Applications and Modeling with Common Logarithms ▪ Applications and Modeling with Natural Logarithms ▪ Logarithms with Other Bases
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49. 4.4 Example 1(a) Finding pH given the hydronium ion concentration
Find the pH of a solution with
Substitute.
Product property
log 10–8 = –8
pH ≈ 7.2
W/ Calc
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50. 4.4 Example 1(b) Finding pH (page 448)
Find the hydronium ion concentration of a solution with pH = 4.3
Substitute.
Multiply by –1.
Write in exponential form.
Use a calculator.
W/ Calc
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51. 4.4 Example 2 Using pH in an Application (page 448)
Wetlands are classified as shown in the table. ≤
The hydronium ion concentration of a water sample from a wetland is 4.5 x 10–3. Classify this wetland.
The wetland is a bog.
W/ Calc
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52. 4.4 Example 3 Measuring the Loudness of Sound (page 449)
Find the decibel rating of a sound with intensity 10,000,000I0.
Let I = 10,000,000I0.
log 10,000,000 = log 107 = 7.
The sound has a decibel rating of 70.
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53. 4.4 Example 6 Using the Change-of-Base Theorem (page 451)
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm.
(a)
(b)
(a)
(b)
W/ Calc
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54. 4.4 additional example Find the value of each:
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55. 4.4 additional example Find the value of each:
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56. Evaluating Logarithms and the Change-of-Base Theorem
4.4 Summary
Evaluating Logarithms and the Change-of-Base Theorem
Common Logarithms
Applications and Modeling with Common Logarithms
Applications and Modeling with Natural Logarithms
Logarithms with Other Bases
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57. Exponential and Logarithmic Equations
4.5
Exponential and Logarithmic Equations
Exponential Equations ▪ Logarithmic Equations ▪ Applications and Modeling
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58. 4.5 Example 1 Solving an Exponential Equation (page 458)
Give the solution to the nearest thousandth.
Solve 8x = 21
Solve 52x–3 = 8x+1
( )
( ) ( )
( )
( ) ( )
Solution set: {1.464}
Solution set: {6.062}
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59. 4.5 Example 3(a) Solving Base e Exponential Equations (page 460)
Give the solution to the nearest thousandth.
Solve e│x│ = 50
Solve e4x ∙ ex – 1 = 5e
Solution set: {±3.912}
( )
( )
Side note: x6∙x5 = x11 or
Solution set: {.722}
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60. 4.5 Example 4 Solving an Logarithmic Equation (page 460)
Solve log(2x + 1) + log x = log(x + 8). Give the exact value(s) of the solution(s).
The negative solution is not in the domain of log x in the original equation, so the only valid solution is x = 2.
Solution set: {2}
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61. 4.5 Example 5 Solving a Logarithmic Equation (page 461)
Solve Give the exact value(s) of the solution(s).
The (11-√65)/4 makes 2x – 5 in the original equation negative, so reject that solution.
Use the quadratic formula with a = 2, b = –11, and c = 7.
Solution set:
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62. 4.5 Example 6 Solving a Logarithmic Equation (page 461)
Solve Give the exact value(s) of the solution(s).
Solution set: {5}
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63. Add on to notes Solve Let u = ex e2x – 11ex + 10 = 0 u2 – 11u + 10 = 0
u = or u = 1
ex = or ex = 1
xlne = ln10 or xlne = ln1
X = ln or x = ln1
x = ln or x = 0
ln 1 = ?
e? = 1
? = 0
ln 1 = 0
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64. Add on to notes 52x - 15(5x) + 56 = 0 Let u = 5x u2 – 15u + 56 = 0
52x - 15(5x) + 56 = 0
Let u = 5x
u2 – 15u + 56 = 0
(u – 7)(u – 8) = 0
u = or u = 8
5x = or x = 8
xln5 = ln7 or xln5 = ln8
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65. Exponential and Logarithmic Equations
4.5 Summary
Exponential and Logarithmic Equations
Exponential Equations
Logarithmic Equations
Applications and Modeling
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66. Applications and Models of Exponential Growth and Decay
4.6
Applications and Models of Exponential Growth and Decay
The Exponential Growth or Decay Function ▪ Growth Function Models ▪ Decay Function Models
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67. The table shows the growth of atmospheric carbon dioxide over time.
4.6 Example 1 Determining an Exponential Function to Model the Increase of Carbon Dioxide (page 469)
The table shows the growth of atmospheric carbon dioxide over time.
(a) Find an exponential model using the data for 2000 and Let the year 2000 correspond to t = 0.
Use the points: (0, 375) and (175, 1090)
y0 = 375
y = 375ekt
1090 = 375e175k
1090/375 = e175k
The equation of the model is
ln(1090/375) = 175k
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68. Estimate the carbon dioxide level will be 840 ppm in 2132.
4.6 Example 1 Determining an Exponential Function to Model the Increase of Carbon Dioxide (cont.)
(b) Use the model to estimate when future levels of carbon dioxide will be 840 ppm.
Since t = 0 corresponds to the year 2000, t = 132 corresponds to the year 2132.
Estimate the carbon dioxide level will be 840 ppm in 2132.
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69. 4.6 Example 2 Finding Doubling Time for Money (page 470)
How long will it take for the money in an account that is compounded continuously at 5.75% to double?
Use the continuously compounding formula with A = 2P and r = to solve for t.
It will take about 12 years for the money to double.
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70. The year 2015 corresponds to t = 15.
4.6 Example 3 Determining an Exponential Function to Model Population Growth (page 471)
The projected world population (in billions of people) t years after 2000, is given by the function
(a) What will the world population be at the end of 2015?
The year 2015 corresponds to t = 15.
The world population will be about billion at the end of 2015.
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71. (b) In what year will the world population reach 8 billion?
4.6 Ex 3 Determining an Exponential Function to Model Population Growth (cont.)
(b) In what year will the world population reach 8 billion?
Given equation:
t represents the # of years after 2000
The world population reach 8 billion during the year 2022.
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72. The equation becomes Now, let y = 400 and t = 2.5, then solve for k.
4.6 Example 4 Determining an Exponential Function to Model Radioactive Decay (page 471)
If 800 grams of a radioactive substance are present initially, and 2.5 years later only 400 grams remain, how much of the substance will be present after 4 years?
The equation will take the form Then y0 = 800, the initial amount at t = 0.
The equation becomes Now, let y = 400 and t = 2.5, then solve for k.
Used the stored value for k in the calculator
The exponential equation is
Letting t = 4, we have
263.90
There will be about grams of the substance left after 4 years.

73. 4.6 Example 5 Solving a Carbon Dating Problem (page 472)
Suppose that the skeleton of a woman who lived in the Classical Greek period was discovered in Carbon 14 testing at that time determined that the skeleton contained ¾ of the carbon 14 of a living woman of the same size. Estimate the year in which the Greek woman died.
The amount of radiocarbon present after t years is given by
The woman died approximately 2366 years before 2005, in 361 B.C.
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74. Applications and Models of Exponential Growth and Decay
4.6 Summary
Applications and Models of Exponential Growth and Decay
The Exponential Growth or Decay Function
Growth Function Models
Decay Function Models
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75. Example 4.2 example 1d
(d)
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76. 4.2 Add on to notes
And the system gives a graph and they have to find the equation.
Given the graph passing through (4,-1), (5,0), and (7,1) find the logarithmic equation.
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77. 4.2 Example 11(b) Using Data to Model Exponential Growth (page 426)
Use a graph of this model to estimate when the carbon dioxide level will be double the level that it was in 2000.
In 2000, the carbon dioxide level was 375 ppm, so we want to know when the carbon dioxide level will be 750 ppm.
The carbon dioxide level will be double the level in 2000, 750 ppm, by 2113.
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78. 4.4 Additional Example W/ Calc
The average global temperature increase T (in ˚F) is expressed
by , where C0 is the preindustrial amount
of carbon dioxide, C is the current carbon dioxide level, and k is a constant. Suppose 12 < k < 20 where C is double the value of C0. Use T(k) to find the range of the rise in global temperature T.
C = 2C0
Plug in 12 and then 20 for k
The range is temperature is between 8.57 ˚F and 14.28˚F
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79. 4.5 Example 7 Applying an Exponential Equation to the Strength of a Habit (page 462)
The equation can be used to describe Newton’s law of cooling. Solve this equation for k.
Advanced class only
Take the natural logarithm of both sides.
ln e–kt = –kt
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80. 4.4 Example 4(a) Measuring the Age of Rocks (page 450)
By measuring the amounts of potassium 40, K, and argon 40, A, in a rock, the age t of the rock in years can be found with the formula
How old is a rock sample in which A = K?
Since A = K,
The rock sample is about 4.06 billion years old.
W/ Calc
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81. 4.4 Example 4(b) Measuring the Age of Rocks (page 450)
How old is a rock sample in which
The rock sample is about 2.38 billion years old.
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