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Exponential and Logarithmic Functions
Chapter 4
Composite Functions
Section 4.1
Composite FunctionsConstruct new function from
two given functions f and gComposite function:
Denoted by f ° gRead 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) =
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) = 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
Composite FunctionsDecomposing 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:
Melissa
John
Jennifer
Patrick
$45,000
$40,000
$50,000
Person
Salary
One-to-One Functions
Example.Problem: Is this function one-to-one?Answer:
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935
Person
ID Number
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 2 4 6
-6
-4
-2
2
4
6
Horizontal-line Test
Example. Problem: Use the graph to
determine whether the function is one-to-one.
Answer:
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Inverse FunctionsRequires f to be a one-to-one
functionThe inverse function of f
Written f{1
Defined as the function which takes f(x) as inputReturns 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 ff(f{1(x)) = x for all x in the domain of f{1
Inverse Functions
Example. Find the inverse of the function shown.Problem:
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935
Person
ID Number
Inverse Functions
Example. (cont.)Answer:
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935
Person
ID Number
Inverse Functions
Example. Problem: Find the inverse of the
function shown.f(0, 0), (1, 1), (2, 4), (3, 9), (4,
16)gAnswer:
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.
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Graphs of Inverse Functions
Example. Problem: Find the graph of the
inverse functionAnswer:
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.
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Restricting the Domain
Example. Problem: Find the inverse of
if the domain of f is x ¸ 0.Answer:
Key Points
One-to-One FunctionsHorizontal-line Test Inverse FunctionsDomain and Range of
Inverse FunctionsGraphs of Inverse FunctionsFinding Inverse FunctionsRestricting 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
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Graphing Exponential Functions
Example. Problem: Graph f(x) = 3x
Answer:
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 = vAnother 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
ExponentsLaws of ExponentsExponential FunctionsGraphing Exponential
FunctionsProperties of the Exponential
FunctionThe Number eExponential Equations
Logarithmic Functions
Section 4.4
Logarithmic Functions
Logarithmic function to the base aa > 0 and a 1Denoted by y = logaxRead “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
functionSame as range of the exponential
functionAll positive real numbers, (0, 1)
Range of the logarithmic functionSame as domain of the exponential
functionAll 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:
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Graphing Logarithmic Functions
Example. Graph the function
Problem: f(x) = log3 x
Answer:
Properties of the Logarithmic Function
Properties of f(x) = loga x, a > 1Domain: Positive real numbers;
(0, 1)Range: All real numbersIntercepts:
x-intercept of x = 1No y-intercepts
y-axis is horizontal asymptoteIncreasing and one-to-one.Smooth and continuousContains points (1,0), (a, 1) and
Properties of the Logarithmic Function
Properties of the Logarithmic Function
Properties of f(x) = loga x, 0 < a < 1Domain: Positive real numbers; (0,
1)Range: All real numbers Intercepts:
x-intercept of x = 1No y-intercepts
y-axis is horizontal asymptoteDecreasing and one-to-one.Smooth and continuousContains 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
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Special Logarithm Functions
Example. Graph the function
Problem: f(x) = ln (3{x)
Answer:
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
Logarithmic FunctionsDomain and Range of
Logarithmic FunctionsGraphing Logarithmic
FunctionsProperties of the Logarithmic
FunctionSpecial Logarithm FunctionsLogarithmic 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 10Example.
Problem: Approximate log3 19 rounded to four decimal places
Answer:
Logarithms with Bases Other than e
and 10Theorem. [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 LogarithmsProperties of LogarithmsLogarithms 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 FormulaPrincipal of P dollars borrowed
for t years at per annum interest rate r
Interest is I = Prtr must be expressed as
decimal
Compound Interest
Payment periodAnnually: Once per yearSemiannually: Twice per yearQuarterly: Four times per yearMonthly: 12 times per yearDaily: 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
continuouslyAnswer:
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 InterestCompound InterestEffective Rates of InterestPresent ValueTime 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 growthExamples
Cell division (early in process)Compound Interest
Uninhibited DecayExamples
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 DecayA(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 CoolingThe temperature u of a heated object at a given time t can be modeled by
u(t) = T + (u0 { T)ekt, k < 0where 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 growthPopulation approaches
carrying capacity
Logistic ModelLogistic 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 FunctionDomain is set of all real numbersRange is interval (0, c)Intercepts:
no x-intercepty-intercept is P(0).
Increasing if b > 0, decreasing if b < 0
Inflection point when P(t) = 0.5cGraph 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 CoolingLogistic 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 ten-year 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 20-24 year olds in the US between 1985-1993 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