Math 1F03 – Lecture 2

1.5 Exponential Functions1.6 Inverse Functions and Logarithms

2.2 Introduction to Limits

1.5 Exponential Functions

An exponential function is a function of the form

where is a positive real number called the base and is a variable called the exponent.

Domain:Range:

f (x) ax

a

x

x R

y 0

Graphs of Exponential Functions

f (x) 3x

f (x) 12 x

When a>1, the function is increasing. When a<1, the function is decreasing.

y=0 is a horizontal asymptote

Transformation of an Exponential Function

Graph

Recall: is a special irrational number between 2 and 3 that is commonly used in calculus

Approximation:

f (x) e2x 1.

e 2.718

e

Laws of Exponents

Examples:

1. ax ay axy

2.ax

ayax y

3. (ax )y axy

4. (ab)x axbx

a x 1

ax

ax

y axy ( ay )x

a1 a

a0 1

Exponential Models

P. 57 #24.Suppose you are offered a job that lasts one

month. Which of the following methods of payment do you prefer?

I. One million dollars at the end of the month.II. One cent on the first day of the month, two

cents on the second day, four cents on the third day, and, in general, cents on the nth day.

2n 1

1.6 Inverse Functions

The function is the inverse of if and .

Each of and undoes the action of the other.

Diagram:

Some simple examples:

f

f 1

f 1( f (x)) x

f ( f 1(x)) x

f 1

f

What Functions Have Inverses?

A function has an inverse if and only if it is a one-to-one function.

A function f is one-to-one if for every y-value in the range of f, there is exactly one x-value in the domain of f such that y=f(x).

Examples:

Horizontal Line Test

If every horizontal line intersects the graph of a function in at most one point, then the graph represents a one-to-one function.

Finding the Inverse of a Function

Algorithm:

1.Write the equation y=f(x).2.Solve for x in terms of y.3.Replace x by (x) and y by x.

Note: The domain and range are interchanged

Example: Find the inverse of the following functions.State domain and range.

f 1

Graphs of and

The graph of is the graph of reflected in the line

Points (x,y) on become the points (y,x) on

Example: Given sketch and .

f 1

f

f 1

f

y x.

f

f 1.

f (x) (x 2)2,x 2

f 1

f

f

1.6 Logarithmic Functions

The inverse of an exponential function is a logarithmic function, i.e.

Cancellation equations:In general: For exponentials & logarithms:

alog a x x

loga ax x

f ( f 1(x)) x

f 1( f (x)) x

If f (x) ax, then f 1(x) loga x.

e ln x x

lnex x

Understanding Logarithm Notation

Examples:

Graphs of Logarithmic Functions

Recall: For inverse functions, the domain and range are interchanged and their graphs are reflections in the line

Example: Graph

f (x) ln x.

y x.

Graphs of Logarithmic Functions

f 1(x) ex

( 1,e 1)

(0,1)

(1,e)

e 2.7

The Natural Logarithm

Domain: Range: Graph: The graph increases from negative infinity near x=0 (vertical asymptote) and rises more and more slowly as x becomes larger.

Note:

and

f (x) ln x

x 0

y R

lne1

ln10

Transformation of a Log Function

Example:Graph State the domain.

f (x) log3(2 x).

Laws of Logs

For x,y>0 and p any real number:

Example:Simplify, if possible.(a) (b)(c) (d)

ln(xy) ln x ln y

ln(x / y) ln x ln y

ln(x p ) p ln x

ln 1e

log2 4 log216 log2 2

log5 58

ln(1 e)

Some Exercises

Solve the following equations for x.

(a) (b)

(c) (d)

23x 8

ln x ln(x 1) 12 ln 36

ex1 8

log2(x 2 1) 3

2.2 The Limit of a Function

Notations:

means that the y-value of the function AT x=2 is 5

means that the y-value of the function NEAR x=2 is NEAR 4

f (2) 5

limx 2f (x) 4

2 if 5

2 if 2

4)(

2

x

xx

xxf

The Limit of a Function

Definition:

“the limit of as approaches , equals ”

means that the values of (y-values) approach the number more and more closely as approaches more and more closely (from both sides of ), but ignoring when .

limx af (x) L

ax

Limit of a Function

Some examples:

Note: f may or may not be defined AT x=a. Limits are only concerned with how f is defined NEAR a.

Left-Hand and Right-Hand Limits

means as from the left

means as from the right

** The full limit exists if and only if the left and right limits both exist (equal a real number) and are the same value.

limx a

f (x) L

f (x) L

x a

(x a).

limx a

f (x) L

f (x) L

x a

(x a).

Left-Hand and Right-Hand Limits

For each function below, determine the value of the limit or state that it does not exist.

limx 4f (x)

limx 0g(x)

limx 3h(x)

Evaluating Limits

We can evaluate the limit of a function in 3 ways:

1. Graphically2. Numerically 3. Algebraically

Evaluating Limits

Example:Evaluate graphically.

limx1

(x 2 2x)

Evaluating Limits

Example:Use a table of values to estimate the value of

x f(x)

3.5

3.9

3.99

4 undefined

4.01

4.1

4.5

limx 4

x 2 16

x 4

Evaluating Limits

Example:Use a table of values to estimate the value of

x f(x)

0.1

0.01

0.001

0 undefined

-0.001

-0.01

-0.1

limx 0

1

x

Infinite Limits

Definition:

“the limit of , as approaches , is infinity”

means that the values of (y-values) increase without bound as becomes closer and closer to (from either side of ), but

Definition:

“the limit of , as approaches , is negative infinity”

means that the values of (y-values) decrease without bound as becomes closer and closer to (from either side of ), but

limx af (x)

limx af (x)

ax ax

Infinite Limits

Example:Determine the infinite limit.

#30.

#34.

limx 3

x 2

x 3

limx

cot x

Note:Since the values of these functionsdo not approach a real number L, these limits do not exist.

Vertical Asymptotes

Definition:The line is called a vertical asymptote of the curve y=f(x) if either

Example:Basic functions we know that have VAs:

limx a

f (x)

or

limx a

f (x)