Chapter 4 Section 4'1 Inverse Functions

1
Chapter 4 Section 4.1 Inverse Functions Warm-up

2
4.1 Inverse Functions
Chapter 4 Section 4.1 Inverse Functions

• Determine whether a function is one-to-one, and
  if it is, find a formula for its inverse.
• Simplify expressions of the type (f f ?1)(x)
  and (f ?1 f)(x).

3
Inverses
Chapter 4 Section 4.1 Inverse Functions

• When we go from an output of a function back to
  its input or inputs, we get an inverse relation.
  When that relation is a function, we have an
  inverse function.

4
Inverses
Chapter 4 Section 4.1 Inverse Functions

• Interchanging the first and second coordinates of
  each ordered pair in a relation produces the
  inverse relation.
• Example Consider the relation g given by
• g (?2, 4), (3, ?4), (?8,
  ?5).
• Solution The inverse of the relation is
• (4, ?2), (?4, 3), (?5,
  ?8).

5
Inverse Relation
Chapter 4 Section 4.1 Inverse Functions

• If a relation is defined by an equation,
  interchanging the variables produces an equation
  of the inverse relation.

6
Inverse Relation
Chapter 4 Section 4.1 Inverse Functions

• Example Find an equation for the inverse of
  the relation
• y x2 ? 2x.
• Solution We interchange x and y and obtain
  an equation of the inverse x y2 ? 2y.
• Graphs of a relation and its inverse are always
  reflections of each other across the line y x.

7
Inverses of Functions
Chapter 4 Section 4.1 Inverse Functions

• If the inverse of a function f is also a
  function, it is named f ?1 and read f-inverse.
  The negative 1 in f ?1 is not an exponent. This
  does not mean the reciprocal of f. f ?1(x) is
  not equal to .

8
One-to-One Functions
Chapter 4 Section 4.1 Inverse Functions

• A function f is one-to-one if different
  inputs have different outputs.
• That is,
• if a ? b
• then
• f(a) ? f(b).

9
One-to-One Functions
Chapter 4 Section 4.1 Inverse Functions

• A function f is one-to-one if when the
  outputs are the same, the inputs are the same.
• That is,
• if f(a) f(b)
• then
• a b.

10
Properties of One-to-One Functions and Inverses
Chapter 4 Section 4.1 Inverse Functions

• If a function is one-to-one, then its inverse is
  a function.
• The domain of a one-to-one function f is the
  range of the inverse f ?1.
• The range of a one-to-one function f is the
  domain of the inverse f ?1.
• A function that is increasing over its domain or
  is decreasing over its domain is a one-to-one
  function.

11
Horizontal-Line Test
Chapter 4 Section 4.1 Inverse Functions

• If it is possible for a horizontal line to
  intersect the graph of a function more than once,
  then the function is not one-to-one and its
  inverse is not a function.

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Horizontal-Line Test
Chapter 4 Section 4.1 Inverse Functions

• Graph f(x) ?3x 4.

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Horizontal-Line Test
Chapter 4 Section 4.1 Inverse Functions

• Example From the graph at the left,
  determine whether the function is one-to-one and
  thus has an inverse that is a function.
• Solution No horizontal line intersects the
  graph more than once, so the function is
  one-to-one. It has an inverse that is a function.

14
Horizontal-Line Test
Chapter 4 Section 4.1 Inverse Functions

• Graph f(x) x2 ? 2.

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Horizontal-Line Test
Chapter 4 Section 4.1 Inverse Functions

• Example From the graph at the left,
  determine whether the function is one-to-one and
  thus has an inverse that is a function.
• Solution There are many horizontal lines
  that intersect the graph more than once. The
  inputs ?1 and 1 have the same output, ?1. Thus
  the function is not one-to-one. The inverse is
  not a function.

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Obtaining a Formula for an Inverse
Chapter 4 Section 4.1 Inverse Functions

• If a function f is one-to-one, a formula
  for its inverse can generally be found as
  follows
• Replace f(x) with y.
• Interchange x and y.
• Solve for y.
• Replace y with f ?1(x).

17
Example
Chapter 4 Section 4.1 Inverse Functions

• Determine whether the function f(x)
  3x ? 2 is one-to-one, and if it is, find a
  formula for f ?1(x).

18
Example
Chapter 4 Section 4.1 Inverse Functions

• Solution The graph is that of a line and
  passes the horizontal-line test. Thus it is
  one-to-one and its inverse is a function.
• 1. Replace f(x) with y y 3x ? 2
• 2. Interchange x and y x 3y ? 2
• 3. Solve for y x 2 3y
• 4. Replace y with f ?1(x)
• f ?1(x)

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Example
Chapter 4 Section 4.1 Inverse Functions

• Graph f(x) 3x ? 2 and
• f ?1(x)
• using the same set of axes. Then compare the
  two graphs.

20
Example
Chapter 4 Section 4.1 Inverse Functions

• Solution The solutions of the inverse
  function can be found from those of the original
  function by interchanging the first and second
  coordinates of each ordered pair. The graph f ?1
  is a reflection of the graph f across the line y
  x.

21
Solution
Chapter 4 Section 4.1 Inverse Functions
22
Inverse Functions and Composition
Chapter 4 Section 4.1 Inverse Functions

• If a function f is one-to-one, then f ?1 is
  the unique function such that each of the
  following holds
• 
  
• for each x in the domain of f, and for each x in
  the domain of f ?1.

23
Example
Chapter 4 Section 4.1 Inverse Functions

• Given that f(x) 7x ? 2, use composition of
  functions to show that
  f ?1(x) (x 2)/7.
• Solution

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Example
Chapter 4 Section 4.1 Inverse Functions

• Solution

25
Restricting a Domain
Chapter 4 Section 4.1 Inverse Functions

• When the inverse of a function is not a function,
  the domain of the function can be restricted to
  allow the inverse to be a function. In such
  cases, it is convenient to consider part of the
  function by restricting the domain of f(x). If
  the domain is restricted, then its inverse is a
  function.

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Chapter 4 Section 4.1 Inverse Functions

25 / 356-357 / 1-31 Odd
27
4.2 Exponential Functions and Graphs
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph exponential equations and functions.
• Solve applied problems involving exponential
  functions and their graphs.

28
Exponential Function
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• The function f(x) ax, where x is a real number,
  a gt 0 and a ? 1, is called the exponential
  function, base a.
• The base needs to be positive in order to avoid
  the complex numbers that would occur by taking
  even roots of negative numbers.
• Examples

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Graphing Exponential Functions
Chapter Section Exponential Functions and
Graphs

• To graph an exponential function, follow the
  steps listed
• 1. Compute some function values and list the
  results in a table.
• 2. Plot the points and connect them with a
  smooth curve. Be sure to plot enough points to
  determine how steeply the curve rises.

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Example
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph the exponential function y f(x) 3x.

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Example
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph the exponential function
  .

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Example
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph y 3x 2.
• The graph is the graph of y 3x shifted to left
  2 units.

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Example
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph y 4 ? 3?x
• The graph is a reflection of the graph of y 3x
  across the x-axis, followed by a reflection
  across the y-axis and then a shift up of 4
  units.

34
The Number e
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• e ? 2.7182818284
• Find each value of ex, to four decimal places,
  using the ex key on a calculator.
• a) e4 b) e?0.25
• c) e2 d) e?1

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Graphs of Exponential Functions, Base e
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph f(x) ex.

36
Example
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph f(x) 2 ? e?3x.

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Example
Chapter 4 Section 4.2 - Exponential Functions
and Graphs

• Graph f(x) ex2.

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Chapter 4 Section 4.1 Inverse Functions

26 / 370 / 2 40 Even
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4.3 Logarithmic Functions and Graphs
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• Graph logarithmic functions.
• Convert between exponential and logarithmic
  equations.
• Find common and natural logarithms with and
  without using a calculator.
• Change logarithm bases

40
Logarithmic Functions
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• These functions are inverses of exponential
  functions.
• Graph x 3y.
• 1. Choose values for y.
• 2. Compute values for x.
• 3. Plot the points and connect them with a
  smooth curve.
• Note that the curve does not touch or cross
  the y-axis.

41
Logarithmic Functions continued
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs
Graph x 3y
42
Logarithmic Function, Base a
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• We define y loga x as that number y such that x
  ay, where x gt 0 and a is a positive constant
  other than 1.
• We read loga x as the logarithm, base a, of x.

43
Finding Certain Logarithms
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• Find each of the following logarithms.
• a) log2 16
• The exponent to which we raise 2 to obtain 16
  is 4 thus log2 16 4.
• b) log10 1000
• The exponent to which we raise 10 to
• obtain 1000 is 3 thus
  log10 1000 3.

44
Finding Certain Logarithms
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• c) log16 4
• The exponent we raise 16 to get 4 is ½, so
  log16 4 ½.
• d) log10 0.001
• We have The exponent
  to
• which we raise 10 to get 0.001 is ?3, so
  log10 0.001 ?3.

45
Logarithms
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• loga 1 0 and loga a 1, for any logarithmic
  base a.
• Convert each of the following to a logarithmic
  equation.
• a) 25 5x
• log5 25 x
• b) ew 30
• loge 30 w

46
Example
Chapter 4 Section 4.3-Logarithmic Functions
and Graphs

• Convert each of the following to an exponential
  equation.
• a) log7 343 3 log7 343 3
• 73 343
• b) logb R 12
• logb R 12
• b12 R

47
Example
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• Find each of the following common logarithms on a
  calculator. Round to four decimal places.
• a) log 723,456
• b) log 0.0000245
• c) log (?4)

48
Natural Logarithms
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• Logarithms, base e, are called natural
  logarithms. The abbreviation ln is generally
  used for natural logarithms. Thus,
• ln x means loge x.
• ln 1 0 and ln e 1, for the logarithmic base
  e.

49
Example
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• Find each of the following natural logarithms on
  a calculator. Round to four decimal places.
• a) ln 723,456
• b) ln 0.0000245
• c) ln (?4)

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Changing Logarithmic Bases
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• The Change-of-Base Formula
• For any logarithmic bases a and b, and any
  positive number M,

51
Example
Chapter 4 Section 4.3 Logarithmic Functions and
Graphs

• Find log6 8 using common logarithms.
• Solution First, we let a 10, b 6, and M
  8. Then we substitute into the
  change-of-base formula

52
Example
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• We can also use base e for a conversion.
• Find log6 8 using natural logarithms.
• Solution Substituting e for a, 6 for b and 8
  for M, we have

53
Graphs of Logarithmic Functions
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• Graph y f(x) log6 x.
• Select y.
• Compute x.

54
Example
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• Graph each of the following. Describe how each
  graph can be obtained from the graph of y ln
  x. Give the domain and the vertical asymptote of
  each function.
• a) f(x) ln (x ? 2)
• b) f(x) 2 ? ln x
• c) f(x) ln (x 1)

55
Graph f(x) ln (x ? 2)
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• The graph is a shift 2 units right. The domain is
  the set of all real numbers greater than 2. The
  line x 2 is the vertical asymptote.

56
Graph f(x) 2 ? ln x
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• The graph is a vertical shrinking, followed by a
  reflection across the x-axis, and then a
  translation up 2 units. The domain is the set of
  all positive real numbers. The y-axis is the
  vertical asymptote.

57
Graph f(x) ln (x 1)
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• The graph is a translation 1 unit to the left.
  Then the absolute value has the effect of
  reflecting negative outputs across the x-axis.
  The domain is the set of all real numbers greater
  than ?1. The line x ?1 is the vertical
  asymptote.

58
Application Walking Speed
Chapter 4 Section 4.3 Logarithmic Functions and
Graphs

• In a study by psychologists Bornstein and
  Bornstein, it was found that the average
  walking speed w, In feet per second, of a person
  living in a city of population P, in thousands,
  is given by the function
• w(P) 0.37 ln P 0.05.

59
Application Walking Speed continued
Chapter 4 Section 4.3- Logarithmic Functions
and Graphs

• The population of Philadelphia, Pennsylvania, is
  1,517,600. Find the average walking speed of
  people living in Philadelphia.
• Since 1,517,600 1517.6 thousand, we substitute
  1517.6 for P, since P is in thousands
• w(1517.6) 0.37 ln 1517.6 0.05
• ? 2.8 ft/sec.
• The average walking speed of people living in
  Philadelphia is about 2.8 ft/sec.

60
4.4 Properties of Logarithmic Functions
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Convert from logarithms of products, powers, and
  quotients to expressions in terms of individual
  logarithms, and conversely.
• Simplify expressions of the type loga ax and
  .

61
Logarithms of Products
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• The Product Rule
• For any positive numbers M and N and any
  logarithmic base a,
• loga MN loga M loga N.
• (The logarithm of a product is the sum of the
  logarithms of the factors.)

62
Example
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Express as a single logarithm
  .
• Solution

63
Logarithms of Powers
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• The Power Rule
• For any positive number M, any logarithmic base
  a, and any real number p,
• loga Mp p loga M.
• (The logarithm of a power of M is the exponent
  times the logarithm of M.)

64
Examples
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Express as a product.

65
Examples
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Express as a product.

66
Logarithms of Quotients
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• The Quotient Rule
• For any positive numbers M and N, and any
  logarithmic base a,
• 
  .
• (The logarithm of a quotient is the logarithm of
  the numerator minus the logarithm of the
  denominator.)

67
Examples
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Express as a difference of logarithms.

68
Examples
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Express as a single logarithm.

69
Applying the Properties
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Express in terms of sums and differences of
  logarithms.

70
Example
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Express as a single logarithm.

71
Final Properties
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• The Logarithm of a Base to a Power
• For any base a and any real number x,
• loga ax x.
• (The logarithm, base a, of a to a power is the
  power.)
• A Base to a Logarithmic Power
• For any base a and any positive real number x,
• (The number a raised to the power loga x is x.)

72
Examples
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Simplify.
• a) loga a6
• b) ln e?8
• Solution
• a) loga a6 6
• b) ln e?8 ?8

73
Examples
Chapter 4 Section 4.4- Properties of
Logarithmic Functions

• Simplify.
• a)
• b)
• Solution
• a)
• b)

74
4.5 Solving Exponential and Logarithmic Equations
Chapter 4 Section 4.5- Solving Exponential and
Logarithmic Equations

• Solve exponential and logarithmic equations.

75
Solving Exponential Equations
Chapter 4 Section 4.5- Solving Exponential and
Logarithmic Equations

• Equations with variables in the exponents, such
  as
• 3x 40 and 53x 25, are called exponential
  equations.
• Base-Exponent Property
• For any a gt 0, a ? 1,
• ax ay ? x y.

76
Example
Chapter 4 Section 4.5- Solving Exponential and
Logarithmic Equations

• Solve .
• Write each side with the same base.
• Since the bases are the same number, 5, we can
  use the base-exponent property and set the
  exponents equal
• Check 52x ? 3 125
• 52(3) ? 3 ? 125
• 53 ? 125
• 125 125 True
  
  The solution is 3.

77
Graphical Solution
Chapter 4 Section 4.5- Solving Exponential and
Logarithmic Equations

• We will use the Intersect method. We graph y1
  and y2 53

78
Another Property
Chapter 4 Section 4.5- Solving Exponential and
Logarithmic Equations

• Property of Logarithmic Equality
• For any M gt 0, N gt 0, a gt 0, and a ? 1,
• loga M loga N ? M N.
• Solve 2x 50

This is an exact answer. We cannot simplify
further, but we can approximate using a
calculator. x ? 5.6439 We can check by finding
25.6439 ? 50.
79
Example
Chapter 4 Section 4.5- Solving Exponential and
Logarithmic Equations

• Solve

• Check For x 3
• For x ?3

Negative numbers do not have real-number
logarithms. The solution is 3.
80
Example
Chapter 4 Section 4.5- Solving Exponential and
Logarithmic Equations

• Solve

The value 6 checks and is the solution.
81
4.6 Applications and Models Growth and Decay
Chapter 4 Section 4.6 Applications and Models
Growth and Decay

• Solve applied problems involving exponential
  growth and decay and compound interest.
• Find models involving exponential and
  logarithmic functions.

82
Population Growth
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• The function P(t) P0ekt, k gt 0 can model many
  kinds of population growths.
• In this function
• P0 population at time 0,
• P population after time,
• t amount of time,
• k exponential growth rate.
• The growth rate unit must be the same as the
  time unit.

83
Example
Chapter Section Applications and Models
Growth and Decay

• Population Growth of the United States. In 1990
  the population in the United States was about 249
  million and the exponential growth rate was 8
  per decade. (Source U.S. Census Bureau)
• Find the exponential growth function.
• What will the population be in 2020?
• After how long will the population be double what
  it was in 1990?

84
Solution
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• At t 0 (1990), the population was about 249
  million. We substitute 249 for P0 and 0.08 for k
  to obtain the exponential growth function.
• P(t) 249e0.08t
• In 2020, 3 decades later, t 3. To find the
  population in 2020 we substitute 3 for t
• P(3) 249e0.08(3) 249e0.24 ? 317.
• The population will be approximately 317
  million in 2020.

85
Solution continued
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• We are looking for the doubling time T.
• 498 249e0.08T
• 2 e0.08T
• ln 2 ln e0.08T (Taking the
  natural logarithm on both sides)
• ln 2 0.08T (ln ex x)
• T
• 8.7? T
  
• The population of the U.S. will double in about
  8.7 decades or 87 years. This will be
  approximately in 2077.

86
Interest Compound Continuously
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• The function P(t) P0ekt can be used to
  calculate interest that is compounded
  continuously.
• In this function
• P0 amount of money invested,
• P balance of the account,
• t years,
• k interest rate compounded continuously.

87
Example
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• Suppose that 2000 is deposited into an IRA at an
  interest rate k, and grows to 5889.36 after 12
  years.
• What is the interest rate?
• Find the exponential growth function.
• What will the balance be after the first 5 years?
• How long did it take the 2000 to double?

88
Solution
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• At t 0, P(0) P0 2000. Thus the exponential
  growth function is
• P(t) 2000ekt. We know that
  P(12) 5889.36. We then substitute and
  solve for k
• 5889.36
  2000e12k
• 
  
• 
  The interest rate is about 9.

89
Solution continued
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• The exponential growth function is
• P(t) 2000e0.09t.
• The balance after 5 years is
• P(5) 2000e0.09(5)
• 2000e0.45
• ? 3136.62

90
Solution continued
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• To find the doubling time T, set P(T) 2 ? P0
  4000 and solve for T.
• 4000 2000e0.09T
• 2 e0.09T
• ln 2 ln e0.09T
• ln 2 0.09T
• T 7.7 ? T
• The original investment of 2000 doubled in
  about 7.7 years.

91
Growth Rate and Doubling Time
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• The growth rate k and the doubling time T are
  related by
• kT ln 2
• or
• or
• The relationship between k and T does not
  depend on P0.

92
Example
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• A certain towns population is doubling every
  37.4 years. What is the exponential growth rate?
• Solution

93
Models of Limited Growth
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• In previous examples, we have modeled
  population growth. However, in some populations,
  there can be factors that prevent a population
  from exceeding some limiting value.
• One model of such growth is
  
• which is called a logistic function. This
  function increases toward a limiting value a as t
  approaches infinity. Thus, y a is the
  horizontal asymptote of the graph.

94
Exponential Decay
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• Decay, or decline, of a population is represented
  by the function P(t)
  P0e?kt, k gt 0.
• In this function
• P0 initial amount of the substance,
• P amount of the substance left after time,
• t time,
• k decay rate.
• The half-life is the amount of time it takes for
  half of an amount of substance to decay.

95
Graphs
Chapter 4 Section 4.6- Applications and Models
Growth and Decay
96
Example
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• Carbon Dating. The radioactive element
  carbon-14 has a half-life of 5750 years. If a
  piece of charcoal that had lost 7.3 of its
  original amount of carbon, was discovered from an
  ancient campsite, how could the age of the
  charcoal be determined?

97
Example
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• Solution We know (from Example 5 in our
  book), that the function for carbon dating is
• P(t) P0e-0.00012t.
• If the charcoal has lost 7.3 of its
  carbon-14 from its initial amount P0, then
  92.7P0 is the amount present.

98
Example continued
Chapter 4 Section 4.6- Applications and Models
Growth and Decay

• To find the age of the charcoal, we solve the
  equation for t
• The charcoal was about 632 years old.