Transcript Chapter 5

Exponential and
Logarithmic
Functions
Chapter 4
Composite
Functions
Section 4.1
Composite Functions


Construct new function from two
given functions f and g
Composite function:
Denoted by f ° g
 Read as “f composed with g”
 Defined by
(f ° g)(x) = f(g(x))
 Domain: The set of all numbers x in the
domain of g such that g(x) is in the
domain of f.

Composite Functions

Note that we perform the inside
function g(x) first.
Composite Functions
Composite Functions

Example. Suppose that f(x) = x3 { 2
and g(x) = 2x2 + 1. Find the values
of the following expressions.
(a) Problem: (f ± g)(1)
Answer:
(b) Problem: (g ± f)(1)
Answer:
(c) Problem: (f ± f)(0)
Answer:
Composite Functions

Example. Suppose that f(x) = 2x2 + 3 and
g(x) = 4x3 + 1.
(a) Problem: Find f ± g.
Answer:
(b) Problem: Find the domain of f ± g.
Answer:
(c) Problem: Find g ± f.
Answer:
(d) Problem: Find the domain of f ± g.
Answer:
Composite Functions

Example. Suppose that f(x) =
g(x) =
and
(a) Problem: Find f ± g.
Answer:
(b) Problem: Find the domain of f ± g.
Answer:
(c) Problem: Find g ± f.
Answer:
(d) Problem: Find the domain of f ± g.
Answer:
Composite Functions

Example.
Problem: If f(x) = 4x + 2 and
g(x) =
show that for all x,
(f ± g)(x) = (g ± f)(x) = x
Decomposing Composite
Functions

Example.
Problem: Find functions f and g such that
f ± g = H if
Answer:
Key Points
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
Composite Functions
Decomposing Composite Functions
One-to-One
Functions;
Inverse Functions
Section 4.2
One-to-One Functions

One-to-one function: Any two
different inputs in the domain
correspond to two different outputs in
the range.

If x1 and x2 are two different inputs of a
function f, then f(x1)  f(x2).
One-to-One Functions

One-to-one
function

Not a one-to-one
function

Not a function
One-to-One Functions

Example.
Problem: Is this function one-to-one?
Answer:
Person
Melissa
John
Jennifer
Patrick
Salary
$45,000
$40,000
$50,000
One-to-One Functions

Example.
Problem: Is this function one-to-one?
Answer:
Person
Alex
Kim
Dana
Pat
ID Number
1451678
1672969
2004783
1914935
One-to-One Functions

Example. Determine whether the
following functions are one-to-one.
(a) Problem: f(x) = x2 + 2
Answer:
(b) Problem: g(x) = x3 { 5
Answer:
One-to-One Functions

Theorem.
A function that is increasing on an
interval I is a one-to-one function on
I.
A function that is decreasing on an
interval I is a one-to-one function on
I.
Horizontal-line Test

If every horizontal line intersects the
graph of a function f in at most one
point, then f is one-to-one.
Horizontal-line Test

Example.
Problem: Use the graph to determine
whether the function is one-to-one.
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Horizontal-line Test

Example.
Problem: Use the graph to determine
whether the function is one-to-one.
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Inverse Functions


Requires f to be a one-to-one function
The inverse function of f
Written f{1
 Defined as the function which takes

f(x) as input
 Returns the output x.


In other words, f{1 undoes the action of
f
f{1(f(x)) = x for all x in the domain of f
 f(f{1(x)) = x for all x in the domain of f{1

Inverse Functions

Example. Find the inverse of the
function shown.
Problem:
Person
Alex
Kim
Dana
Pat
ID Number
1451678
1672969
2004783
1914935
Inverse Functions

Example. (cont.)
Answer:
ID Number
1451678
1672969
2004783
1914935
Person
Alex
Kim
Dana
Pat
Inverse Functions

Example.
Problem: Find the inverse of the function
shown.
f(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)g
Answer:
Domain and Range of
Inverse Functions



If f is one-to-one, its inverse is a
function.
The domain of f{1 is the range of f.
The range of f{1 is the domain of f
Domain and Range of
Inverse Functions

Example.
Problem: Verify that the inverse of
f(x) = 3x { 1 is
Graphs of Inverse Functions

The graph of a function f and its
inverse f{1 are symmetric with respect
to the line y = x.
Graphs of Inverse Functions

Example.
Problem: Find the graph of the inverse
function
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Finding Inverse Functions

If y = f(x),
Inverse if given implicitly by x = f(y).
 Solve for y if possible to get y = f {1(x)


Process
Step 1: Interchange x and y to obtain an
equation x = f(y)
 Step 2: If possible, solve for y in terms of
x.
 Step 3: Check the result.

Finding Inverse Functions

Example.
Problem: Find the inverse of the function
Answer:
Restricting the Domain

If a function is not one-to-one, we can
often restrict its domain so that the
new function is one-to-one.
Restricting the Domain

Example.
Problem: Find the inverse of
if the domain of f is x ¸ 0.
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Key Points
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One-to-One Functions
Horizontal-line Test
Inverse Functions
Domain and Range of Inverse
Functions
Graphs of Inverse Functions
Finding Inverse Functions
Restricting the Domain
Exponential
Functions
Section 4.3
Exponents

For negative exponents:

For fractional exponents:
Exponents

Example.
Problem: Approximate 3¼ to five decimal
places.
Answer:
Laws of Exponents

Theorem. [Laws of Exponents]
If s, t, a and b are real numbers with a > 0
and b > 0, then

as ¢ at = as+t

(as)t = ast

(ab)s = as ¢ bs

1s = 1


a0 = 1
Exponential Functions

Exponential function: function of the
form
f(x) = ax



where a is a positive real number (a > 0)
a  1.
Domain of f: Set of all real numbers.
Warning! This is not the same as a power
function.
(A function of the form f(x) = xn)
Exponential Functions

Theorem.
For an exponential function
f(x) = ax, a > 0, a  1, if x is any
real number, then
Graphing Exponential
Functions

Example.
Problem: Graph f(x) = 3x
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Graphing Exponential
Functions
Properties of the
Exponential Function

Properties of f(x) = ax, a > 1



Domain: All real numbers
Range: Positive real numbers; (0, 1)
Intercepts:






No x-intercepts
y-intercept of y = 1
x-axis is horizontal asymptote as x  {1
Increasing and one-to-one.
Smooth and continuous
Contains points (0,1), (1, a) and
Properties of the
Exponential Function
f(x) = ax, a > 1
Properties of the
Exponential Function

Properties of f(x) = ax, 0 < a < 1



Domain: All real numbers
Range: Positive real numbers; (0, 1)
Intercepts:






No x-intercepts
y-intercept of y = 1
x-axis is horizontal asymptote as x  1
Decreasing and one-to-one.
Smooth and continuous
Contains points (0,1), (1, a) and
Properties of the
Exponential Function
f(x) = ax, 0 < a < 1
The Number e


Number e: the number that the
expression
approaches as n  1.
Use ex or exp(x) on your calculator.
The Number e

Estimating value of e








n = 1: 2
n = 2: 2.25
n = 5: 2.488 32
n = 10: 2.593 742 460 1
n = 100: 2.704 813 829 42
n = 1000: 2.716 923 932 24
n = 1,000,000,000: 2.718 281 827 10
n = 1,000,000,000,000: 2.718 281 828 46
Exponential Equations



If au = av, then u = v
Another way of saying that the
function f(x) = ax is one-to-one.
Examples.
(a) Problem: Solve 23x {1 = 32
Answer:
(b) Problem: Solve
Answer:
Key Points







Exponents
Laws of Exponents
Exponential Functions
Graphing Exponential Functions
Properties of the Exponential
Function
The Number e
Exponential Equations
Logarithmic
Functions
Section 4.4
Logarithmic Functions

Logarithmic function to the base a
a > 0 and a  1
 Denoted by y = logax
 Read “logarithm to the base a of x” or
“base a logarithm of x”
 Defined: y = logax if and only if x = ay



Inverse function of y = ax
Domain: All positive numbers (0,1)
Logarithmic Functions

Examples. Evaluate the following
logarithms
(a) Problem: log7 49
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Logarithmic Functions

Examples. Change each exponential
expression to an equivalent expression
involving a logarithm
(a) Problem: 2¼ = s
Answer:
(b) Problem: ed = 13
Answer:
(c) Problem: a5 = 33
Answer:
Logarithmic Functions

Examples. Change each logarithmic
expression to an equivalent expression
involving an exponent.
(a) Problem: loga 10 = 7
Answer:
(b) Problem: loge t = 4
Answer:
(c) Problem: log5 17 = z
Answer:
Domain and Range of
Logarithmic Functions


Logarithmic function is inverse of the
exponential function.
Domain of the logarithmic function
Same as range of the exponential
function
 All positive real numbers, (0, 1)


Range of the logarithmic function
Same as domain of the exponential
function
 All real numbers, ({1, 1)

Domain and Range of
Logarithmic Functions

Examples. Find the domain of each
function
(a) Problem: f(x) = log9(4 { x2)
Answer:
(b) Problem:
Answer:
Graphing Logarithmic
Functions

Example. Graph the function
Problem: f(x) = log3 x
6
Answer:
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Properties of the
Logarithmic Function

Properties of f(x) = loga x, a > 1
Domain: Positive real numbers; (0, 1)
 Range: All real numbers
 Intercepts:

x-intercept of x = 1
 No y-intercepts

y-axis is horizontal asymptote
 Increasing and one-to-one.
 Smooth and continuous
 Contains points (1,0), (a, 1) and

Properties of the
Logarithmic Function
Properties of the
Logarithmic Function

Properties of f(x) = loga x, 0 < a < 1
Domain: Positive real numbers; (0, 1)
 Range: All real numbers
 Intercepts:

x-intercept of x = 1
 No y-intercepts

y-axis is horizontal asymptote
 Decreasing and one-to-one.
 Smooth and continuous
 Contains points (1,0), (a, 1) and

Properties of the
Logarithmic Function
Special Logarithm Functions

Natural logarithm:
y = ln x if and only if x = ey
 ln x = loge x


Common logarithm:
y = log x if and only if x = 10y
 log x = log10 x

Special Logarithm Functions

Example. Graph the function
Problem: f(x) = ln (3{x)
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Logarithmic Equations

Examples. Solve the logarithmic
equations. Give exact answers.
(a) Problem: log4 x = 3
Answer:
(b) Problem: log6(x{4) = 3
Answer:
(c) Problem: 2 + 4 ln x = 10
Answer:
Logarithmic Equations

Examples. Solve the exponential
equations using logarithms. Give
exact answers.
(a) Problem: 31+2x= 243
Answer:
(b) Problem: ex+8 = 3
Answer:
Key Points
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

Logarithmic Functions
Domain and Range of Logarithmic
Functions
Graphing Logarithmic Functions
Properties of the Logarithmic
Function
Special Logarithm Functions
Logarithmic Equations
Properties of
Logarithms
Section 4.5
Properties of Logarithms

Theorem. [Properties of Logarithms]
For a > 0, a  1, and r some real
number:

loga 1 = 0

loga a = 1


loga ar = r
Properties of Logarithms

Theorem. [Properties of Logarithms]
For M, N, a > 0, a  1, and r some
real number:

loga (MN) = loga M + loga N


loga Mr = r loga M
Properties of Logarithms

Examples. Evaluate the following
expressions.
(a) Problem:
Answer:
(b) Problem: log140 10 + log140 14
Answer:
(c) Problem: 2 ln e2.42
Answer:
Properties of Logarithms

Examples. Evaluate the following
expressions if logb A = 5 and
logbB = {4.
(a) Problem: logb AB
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Properties of Logarithms

Example. Write the following
expression as a sum of logarithms.
Express all powers as factors.
Problem:
Answer:
Properties of Logarithms

Example. Write the following
expression as a single logarithm.
Problem: loga q { loga r + 6 loga p
Answer:
Properties of Logarithms

Theorem. [Properties of Logarithms]
For M, N, a > 0, a  1,
If M = N, then loga M = loga N
 If loga M = loga N, then M = N


Comes from fact that exponential and
logarithmic functions are inverses.
Logarithms with Bases
Other than e and 10

Example.
Problem: Approximate log3 19 rounded to
four decimal places
Answer:
Logarithms with Bases
Other than e and 10

Theorem. [Change-of-Base Formula]
If a  1, b  1 and M are all positive
real numbers, then

In particular,
Logarithms with Bases Other
than e and 10

Examples. Approximate the following
logarithms to four decimal places
(a) Problem: log6.32 65.16
Answer:
(b) Problem:
Answer:
Key Points



Properties of Logarithms
Properties of Logarithms
Logarithms with Bases Other than e
and 10
Logarithmic and
Exponential
Equations
Section 4.6
Solving Logarithmic Equations

Example.
Problem: Solve log3 4 = 2 log3 x
algebraically.
Answer:
Solving Logarithmic Equations

Example.
Problem: Solve log3 4 = 2 log3 x
graphically.
Answer:
Solving Logarithmic Equations

Example.
Problem: Solve log2(x+2) + log2(1{x) = 1
algebraically.
Answer:
Solving Logarithmic Equations

Example.
Problem: Solve log2(x+2) + log2(1{x) = 1
graphically.
Answer:
Solving Exponential Equations

Example.
Problem: Solve 9x { 3x { 6 = 0
algebraically.
Answer:
Solving Exponential Equations

Example.
Problem: Solve 9x { 3x { 6 = 0 graphically.
Answer:
Solving Exponential Equations

Example.
Problem: Solve 3x = 7 algebraically. Give
an exact answer, then approximate your
answer to four decimal places.
Answer:
Solving Exponential Equations

Example.
Problem: Solve 3x = 7 graphically.
Approximate your answer to four
decimal places.
Answer:
Solving Exponential Equations

Example.
Problem: Solve 5 ¢ 2x = 3 algebraically.
Give an exact answer, then approximate
your answer to four decimal places.
Answer:
Solving Exponential Equations

Example.
Problem: Solve 5 ¢ 2x = 3 graphically.
Approximate your answer to four
decimal places.
Answer:
Solving Exponential Equations

Example.
Problem: Solve 2x{1 = 52x+3 algebraically.
Give an exact answer, then approximate
your answer to four decimal places.
Answer:
Solving Exponential Equations

Example.
Problem: Solve e2x { x2 = 3 graphically.
Approximate your answer to four
decimal places.
Answer:
Key Points


Solving Logarithmic Equations
Solving Exponential Equations
Compound
Interest
Section 4.7
Simple Interest

Simple Interest Formula
Principal of P dollars borrowed for t
years at per annum interest rate r
 Interest is I = Prt
 r must be expressed as decimal

Compound Interest

Payment period
Annually: Once per year
 Semiannually: Twice per year
 Quarterly: Four times per year
 Monthly: 12 times per year
 Daily: 365 times per year

Compound Interest

Theorem. [Compound Interest
Formula]
The amount A after t years due to a
principal P invested at an annual
interest rate r compounded n times
per year is
Compound Interest

Example. Find the amount that
results from the investment of $1000
at 8% after a period of 8 years.
(a) Problem: Compounded annually
Answer:
(b) Problem: Compounded quarterly
Answer:
(c) Problem: Compounded daily
Answer:
Compound Interest

Theorem. [Continuous Compounding]
The amount A after t years due to a
principal P invested at an annual
interest rate r compounded
continuously is
Compound Interest

Example. Find the amount that
results from the investment of $1000
at 8% after a period of 8 years.
Problem: Compounded continuously
Answer:
Effective Rates of Interest

Effective Rate of Interest:
Equivalent annual simple interest rate
that yields same amount as
compounding after 1 year.
Effective Rates of Interest

Example. Find the effective rate of
interest on an investment at 8%
(a) Problem: Compounded monthly
Answer:
(a) Problem: Compounded daily
Answer:
(a) Problem: Compounded continuously
Answer:
Present Value

Present value: amount needed to
invest now to receive A dollars at a
specified future time.
Present Value

Theorem. [Present Value Formulas]
The present value P of A dollars to
be received after t years, assuming a
per annum interest rate r
compounded n times per year, is
if the interest is compounded
continuously, then
Present Value

Example.
Problem: Find the present value of $5600
after 4 years at 10% compounded
semiannually. Round to the nearest cent.
Answer:
Time to Double an Investment

Example.
Problem: What annual rate of interest is
required to double an investment in 8
years?
Answer:
Key Points
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


Simple Interest
Compound Interest
Effective Rates of Interest
Present Value
Time to Double an Investment
Exponential Growth and
Decay;
Newton’s Law;
Logistic Growth and
Decay
Section 4.8
Uninhibited Growth and Decay

Uninhibited Growth:
No restriction to growth
 Examples

Cell division (early in process)
 Compound Interest


Uninhibited Decay

Examples

Radioactive decay

Compute half-life
Uninhibited Growth and Decay

Uninhibited Growth:
N(t) = N0 ekt, k > 0




N0: initial population
k: positive constant
t: time
Uninhibited Decay
A(t) = A0 ekt, k < 0



N0: initial amount
k: negative constant
t: time
Uninhibited Growth and Decay

Example.
Problem: The size P of a small herbivore
population at time t (in years) obeys the
function P(t) = 600e0.24t if they have
enough food and the predator population
stays constant. After how many years
will the population reach 1800?
Answer:
Uninhibited Growth and Decay

Example.
Problem: The half-life of carbon 14 is 5600
years. A fossilized leaf contains 12% of
its normal amount of carbon 14. How
old is the fossil (to the nearest year)?
Answer:
Newton’s Law of Cooling


Temperature of a heated object decreases
exponentially toward temperature of
surrounding medium
Newton’s Law of Cooling
The temperature u of a heated object at a
given time t can be modeled by
u(t) = T + (u0 { T)ekt, k < 0
where T is the constant temperature of the
surrounding medium, u0 is the initial
temperature of the heated object, and k is
a negative constant.
Newton’s Law of Cooling

Example.
Problem: The temperature of a dead body
that has been cooling in a room set at
70±F is measured as 88±F. One hour
later, the body temperature is 87.5±F.
How long (to the nearest hour) before
the first measurement was the time of
death, assuming that body temperature
at the time of death was 98.6±F?
Answer:
Logistic Model




Uninhibited growth is limited in
actuality
Growth starts off like exponential,
then levels off
This is logistic growth
Population approaches carrying
capacity
Logistic Model

Logistic Model
In a logistic growth model, the
population P after time t obeys the
equation
where a, b and c are constants with
c > 0 (c is the carrying capacity).
The model is a growth model if b > 0;
the model is a decay model if b < 0.
Logistic Model
Logistic Model

Properties of Logistic Function
Domain is set of all real numbers
 Range is interval (0, c)
 Intercepts:

no x-intercept
 y-intercept is P(0).

Increasing if b > 0, decreasing if b < 0
 Inflection point when P(t) = 0.5c
 Graph is smooth and continuous

Logistic Model

Example. The logistic growth model
represents the population of a species
introduced into a new territory after t
years.
(a) Problem: What was the initial population
introduced?
Answer:
(b) Problem: When will the population reach 80?
Answer:
(c) Problem: What is the carrying capacity?
Answer:
Key Points



Uninhibited Growth and Decay
Newton’s Law of Cooling
Logistic Model
Building Exponential,
Logarithmic, and
Logistic Models from
Data
Section 4.9
Fitting an Exponential
Function to Data

Example. The
population (in
hundred
thousands) for the
Colonial US in tenyear increments for
the years 1700-1780
is given in the
following table.
(Source: 1998
Information Please
Almanac)
Decade, x
Population, P
0
251
1
332
2
466
3
629
4
906
5
1171
6
1594
7
2148
8
2780
Fitting an Exponential
Function to Data

Example. (cont.)
(a) Problem: State whether the data can
be more accurately modeled using an
exponential or logarithmic function.
Answer:
Fitting an Exponential
Function to Data

Example. (cont.)
(b) Problem: Find a model for population
(in hundred thousands) as a function of
decades since 1700.
Answer:
Fitting a Logarithmic Function
to Data

Example. The
death rate (in
deaths per 100,000
population) for 2024 year olds in the
US between 19851993 are given in
the following table.
(Source: NCHS
Data Warehouse)
Year
Rate of Death, r
1985
134.9
1987
154.7
1989
162.9
1991
174.5
1992
182.2
Fitting a Logarithmic Function
to Data

Example. (cont.)
(a) Problem: Find a model for death rate
in terms of x, where x denotes the
number of years since 1980.
Answer:
(b) Problem: Predict the year in which the
death rate first exceeded 200.
Answer:
Fitting a Logistic Function to
Data

Example. A
mechanic is testing
the cooling system
of a boat engine.
He measures the
engine’s
temperature over
time.
Time t
(min.)
Temperature T
(±F)
5
100
10
180
15
270
20
300
25
305
Fitting a Logistic Function to
Data

Example. (cont.)
(a) Problem: Find a model for the
temperature T in terms of t, time in
minutes.
Answer:
(b) Problem: What does the model imply
will happen to the temperature as time
passes?
Answer:
Key Points



Fitting an Exponential Function to
Data
Fitting a Logarithmic Function to
Data
Fitting a Logistic Function to Data