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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) = 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 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 function

Not a one-to-one function

Not a function


Example.Problem: Is this function one-to-one?

Answer:

Melissa

John

Jennifer

Patrick

$45,000

$40,000

$50,000

Person Salary


Example.Problem: Is this function one-to-one?

Answer:

Alex

Kim

Dana

Pat

1451678

1672969

2004783

1914935

Person ID Number


Example. Determine whether the following functions are one-to-one.

(a) Problem: f(x) = x2 + 2

Answer:

(b) Problem: g(x) = x3 — 5

Answer:


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.


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 Functions

Requires f to be a one-to-one function

The inverse function of fWritten 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:

Alex

Kim

Dana

Pat

1451678

1672969

2004783

1914935

Person ID Number

Inverse Functions

Example. (cont.)

Answer:

Alex

Kim

Dana

Pat

1451678

1672969

2004783

1914935

PersonID Number

Inverse Functions

Example.

Problem: Find the inverse of the function shown.

{(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)}

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


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


Example.

Problem: Find the graph of the inverse function

Answer:

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.


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


Example.

Problem: Find the inverse of if the domain of f is x ≥ 0.

Answer:

Key Points



Inverse Functions





Exponential Functions

Section 4.3

Exponents

For negative exponents:

For fractional exponents:

Exponents

Example.

Problem: Approximate 3π 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 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)


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:

Properties of the Exponential Function

Properties of f(x) = ax, a > 1

Domain: All real numbers

Range: Positive real numbers; (0, ∞)

Intercepts:No x-intercepts

y-intercept of y = 1

x-axis is horizontal asymptote as x → —∞

Increasing and one-to-one.

Smooth and continuous

Contains points (0,1), (1, a) and


f(x) = ax, a > 1


Properties of f(x) = ax, 0 < a < 1

Domain: All real numbers

Range: Positive real numbers; (0, ∞)

Intercepts:No x-intercepts

y-intercept of y = 1

x-axis is horizontal asymptote as x → ∞

Decreasing and one-to-one.


Contains points (0,1), (1, a) and


f(x) = ax, 0 < a < 1

The Number e

Number e: the number that the expression

approaches as n → ∞.

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




The Number e

Exponential Equations

Logarithmic Functions

Section 4.4


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,∞)


Examples. Evaluate the following logarithms

(a) Problem: log7 49

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:


Examples. Change each exponential expression to an equivalent expression involving a logarithm

(a) Problem: 2π = s

Answer:

(b) Problem: ed = 13

Answer:

(c) Problem: a5 = 33

Answer:


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 function

All positive real numbers, (0, ∞)

Range of the logarithmic functionSame as domain of the exponential function

All real numbers, (—∞, ∞)


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, ∞)

Range: All real numbers

Intercepts:x-intercept of x = 1

No y-intercepts

y-axis is horizontal asymptote

Increasing and one-to-one.


Contains points (1,0), (a, 1) and


Properties of f(x) = loga x, 0 < a < 1Domain: Positive real numbers; (0, ∞)

Range: All real numbers

Intercepts:x-intercept of x = 1

No y-intercepts

y-axis is horizontal asymptote

Decreasing and one-to-one.


Contains points (1,0), (a, 1) and

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


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:


Examples. Solve the exponential equations using logarithms. Give exact answers.

(a) Problem: 31+2x= 243

Answer:

(b) Problem: ex+8 = 3

Answer:

Key Points



Graphing Logarithmic Functions




Properties of Logarithms

Section 4.5


Theorem. [Properties of Logarithms]

For a > 0, a ≠ 1, and r some real

number:

loga 1 = 0

loga a = 1

loga ar = r



For M, N, a > 0, a ≠ 1, and r some real number:

loga (MN) = loga M + loga N

loga Mr = r loga M


Examples. Evaluate the following expressions.

(a) Problem:

Answer:

(b) Problem: log140 10 + log140 14

Answer:

(c) Problem: 2 ln e2.42

Answer:


Examples. Evaluate the following expressions if logb A = 5 and logbB = —4.

(a) Problem: logb AB

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:


Example. Write the following expression as a sum of logarithms. Express all powers as factors.

Problem:

Answer:


Example. Write the following expression as a single logarithm.

Problem: loga q — loga r + 6 loga p

Answer:



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:


Theorem. [Change-of-Base Formula]

If a ≠ 1, b ≠ 1 and M are all positive real numbers, then

In particular,


Examples. Approximate the following logarithms to four decimal places

(a) Problem: log6.32 65.16

Answer:

(b) Problem:

Answer:

Key Points



Logarithms with Bases Other than eand 10

Logarithmic and Exponential Equations

Section 4.6

Solving Logarithmic Equations

Example. Problem: Solve log3 4 = 2 log3 x

algebraically.

Answer:


Example.

Problem: Solve log3 4 = 2 log3 xgraphically.

Answer:


Example. Problem: Solve log2(x+2) + log2(1—x) = 1

algebraically.

Answer:


Example.

Problem: Solve log2(x+2) + log2(1—x) = 1 graphically.

Answer:

Solving Exponential Equations

Example. Problem: Solve 9x — 3x — 6 = 0

algebraically.

Answer:


Example.

Problem: Solve 9x — 3x — 6 = 0 graphically.

Answer:


Example. Problem: Solve 3x = 7 algebraically. Give

an exact answer, then approximate your answer to four decimal places.

Answer:


Example.

Problem: Solve 3x = 7 graphically. Approximate your answer to four decimal places.

Answer:


Example. Problem: Solve 5 · 2x = 3 algebraically.

Give an exact answer, then approximate your answer to four decimal places.

Answer:


Example.

Problem: Solve 5 · 2x = 3 graphically. Approximate your answer to four decimal places.

Answer:


Example. Problem: Solve 2x—1 = 52x+3 algebraically.

Give an exact answer, then approximate your answer to four decimal places.

Answer:


Example.

Problem: Solve e2x — x2 = 3 graphically. Approximate your answer to four decimal places.

Answer:

Key Points



Compound Interest

Section 4.7

Simple Interest

Simple Interest Formula

Principal of P dollars borrowed for tyears 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.


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 rcompounded 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


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 decayCompute half-life


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


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:


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 < 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.


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 ModelIn 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 tyears.(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



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 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


Example. (cont.)

(a) Problem: State whether the data can be more accurately modeled using an exponential or logarithmic function.

Answer:


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


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


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




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