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 ) = x 3 { 2 and g ( x ) = 2 x 2 + 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 ) = 2 x 2 + 3 and g ( x ) = 4 x 3 + 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 ) = and g ( x ) =
(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 ) = 4 x + 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
 Composite Functions
 Decomposing Composite 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 x
1 and x
2 function f are two different inputs of a
, then f ( x
1
)
 f ( x
2
).

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 Salary
Melissa
John
Jennifer
Patrick
$45,000
$40,000
$50,000

One-to-One Functions
 Example.
Problem: Is this function one-to-one?
Answer:
Person ID Number
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935

One-to-One Functions
 Example. Determine whether the following functions are one-to-one.
(a) Problem: f ( x ) = x 2 + 2
Answer:
(b) Problem: g ( x ) = x 3 { 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
-4
-6
2 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
-4
-6
2 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 ID Number
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935

Inverse Functions
 Example. (cont.)
Answer:
Person ID Number
1451678
1672969
2004783
1914935
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 ) = 3 x { 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
6
Answer:
4
2
-6 -4 -2
-2
-4
-6
2 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
-4
-6
2 4 6

Key Points
 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

Section 4.3

Exponents
 For negative exponents:
 For fractional exponents:

Exponents
 Example.
Problem: Approximate 3 ¼ 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
 a s ¢ a t = a s + t
 ( a s ) t = a st
 ( ab ) s = a s ¢ b s
 1 s = 1

 a 0 = 1

Exponential Functions


Exponential function : function of the form

 f ( x ) = a x 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 ) = x n )

Exponential Functions
 Theorem.
For an exponential function f ( x ) = a x , a > 0, a

1, if x is any real number, then

Graphing Exponential
Functions
 Example.
Problem: Graph f ( x ) = 3 x
Answer:
6
4
2
2 4 6 -6 -4 -2
-2
-4
-6

Graphing Exponential
Functions

Properties of the
Exponential Function
 Properties of f ( x ) = a x , 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 ) = a x , a > 1

Properties of the
Exponential Function
 Properties of f ( x ) = a x , 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 ) = a x , 0 < a < 1

The Number e
 Number e : the number that the expression approaches as n

1 .
 Use e x 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 a u = a v , then u = v
 Another way of saying that the function f ( x ) = a x is one-to-one.
 Examples.
(a) Problem: Solve 2 3 x {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

Section 4.4

Logarithmic Functions
 Logarithmic function to the base a
 a > 0 and a

1


Denoted by y = log a x
Read “ logarithm to the base a of x ” or
“ base a logarithm of x ”
 Defined: y = log a x if and only if x = a y
 Inverse function of y = a x
 Domain: All positive numbers (0, 1 )

Logarithmic Functions
 Examples. Evaluate the following logarithms
(a) Problem: log
7
49
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:

Logarithmic Functions
 Examples. Change each exponential expression to an equivalent expression involving a logarithm
(a) Problem: 2 ¼ = s
Answer:
(b) Problem: e d = 13
Answer:
(c) Problem: a 5 = 33
Answer:

Logarithmic Functions
 Examples. Change each logarithmic expression to an equivalent expression involving an exponent.
(a) Problem: log a
10 = 7
Answer:
(b) Problem: log e t = 4
Answer:
(c) Problem: log
5
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 ) = log
9
(4 { x 2 )
Answer:
(b) Problem:
Answer:

Graphing Logarithmic
Functions
 Example. Graph the function
Problem: f ( x ) = log
3 x
Answer:
6
4
2
2 4 6 -6 -4 -2
-2
-4
-6

Properties of the
Logarithmic Function
 Properties of f ( x ) = log a
 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 ) = log a
 x , 0 < a <
Domain: Positive real numbers; (0, 1 )
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 = e y

 ln x = log e x
Common logarithm :
 y = log x if and only if x = 10 y
 log x = log
10 x

Special Logarithm Functions
 Example. Graph the function
Problem: f ( x ) = ln (3{ x )
Answer: 6
4
2
-6 -4 -2
-2
-4
-6
2 4 6

Logarithmic Equations
 Examples. Solve the logarithmic equations. Give exact answers.
(a) Problem: log
4 x = 3
Answer:
(b) Problem: log
6
( 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: 3 1+2 x = 243
Answer:
(b) Problem: e x +8 = 3
Answer:

Key Points
 Logarithmic Functions
 Domain and Range of Logarithmic
Functions
 Graphing Logarithmic Functions
 Properties of the Logarithmic
Function
 Special Logarithm Functions
 Logarithmic Equations

Section 4.5

Properties of Logarithms
 Theorem. [Properties of Logarithms]
For a > 0, a

1, and r some real number:
 log a
1 = 0
 log a a = 1

 log a a r = r

Properties of Logarithms
 Theorem. [Properties of Logarithms]
For M , N , a > 0, a

1, and r some real number:
 log a
( MN ) = log a
M + log a
N

 log a
M r = r log a
M

Properties of Logarithms
 Examples. Evaluate the following expressions.
(a) Problem:
Answer:
(b) Problem: log
140
10 + log
140
14
Answer:
(c) Problem: 2 ln e 2.42
Answer:

Properties of Logarithms
 Examples. Evaluate the following expressions if log log b
B = {4.
b
A = 5 and
(a) Problem: log b
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: log a q { log a r + 6 log a p
Answer:

Properties of Logarithms
 Theorem. [Properties of Logarithms]
For M , N , a > 0, a

1,



If M
If log a
= N
M
, then log
= log a a
M = log a
N
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 log
3 four decimal places
19 rounded to
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: log
6.32
65.16
Answer:
(b) Problem:
Answer:

Key Points
 Properties of Logarithms
 Properties of Logarithms
 Logarithms with Bases Other than e and 10

Section 4.6

Solving Logarithmic Equations
 Example.
Problem: Solve log
3 algebraically.
4 = 2 log
3 x
Answer:

Solving Logarithmic Equations
 Example.
Problem: Solve log
3 graphically.
4 = 2 log
3 x
Answer:

Solving Logarithmic Equations
 Example.
Problem: Solve log
2
( x +2) + log
2
(1{ x ) = 1 algebraically.
Answer:

Solving Logarithmic Equations
 Example.
Problem: Solve log
2
( x +2) + log
2
(1{ x ) = 1 graphically.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve 9 x { 3 x { 6 = 0 algebraically.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve 9 x { 3 x { 6 = 0 graphically.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve 3 x = 7 algebraically. Give an exact answer, then approximate your answer to four decimal places.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve 3 x = 7 graphically.
Approximate your answer to four decimal places.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve 5 ¢ 2 x = 3 algebraically.
Give an exact answer, then approximate your answer to four decimal places.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve 5 ¢ 2 x = 3 graphically.
Approximate your answer to four decimal places.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve 2 x {1 = 5 2 x +3 algebraically.
Give an exact answer, then approximate your answer to four decimal places.
Answer:

Solving Exponential Equations
 Example.
Problem: Solve e 2 x { x 2 = 3 graphically.
Approximate your answer to four decimal places.
Answer:

Key Points
 Solving Logarithmic Equations
 Solving Exponential Equations

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
 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 ) = N
0
N
0
: initial population k : positive constant e kt , k > 0
 t : time
 Uninhibited Decay


A ( t ) = A
0
N
0
: initial amount k : negative constant e kt , k < 0
 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 ) = 600 e 0.24
t 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 + ( u
0
{ T ) e kt , k < 0 where T is the constant temperature of the surrounding medium, u
0 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 ) = 0.5
c
 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

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
1
251
332
4
5
2
3
6
7
8
466
629
906
1171
1594
2148
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
Year Rate of Death, r
 Example. The death rate (in deaths per 100,000 population) for 20-
24 year olds in the
US between 1985-
1993 are given in the following table.
(Source: NCHS
Data Warehouse)
1985
1987
1989
1991
1992
134.9
154.7
162.9
174.5
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.)
5
10
15
20
25
Temperature T
( ± F)
100
180
270
300
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