functions and limit. a function is a rule or correspondence which associates to each number x in a...

Functions and Limit

• A function is a rule or correspondence which

associates to each number x in a set A a unique

number f(x) in a set B.

•The set A is called the domain of f and the set

of all f(x)'s is called the range of f.

Definition of Functions

• Let A and B be any two non-empty sets. Then a

function f is a rule from A to B which associate to

each element ‘a’ of A to a unique element ‘b’ of B.

• The element ‘b’ is called IMAGE of ‘a’ under f.

• The element ‘a’ is called PRE-IMAGE of ‘b’.

• Set A is called DOMAIN of f and B is called

CODOMAIN of f.

DOMAIN RANGE

X Y

f

x

x

x

y

y

M: Mother Function

Joe

Samantha

Anna

Ian

Chelsea

George

Laura

Julie

Hilary

Barbara

Sue

Humans Mothers

M: Mother function

Domain of M = {Joe, Samantha, Anna, Ian, Chelsea,

George}

Range of M = {Laura, Julie, Hilary, Barbara}

In function notation we write

M(Anna) = Julie

M(George) = Barbara

M(x)=Hilary indicates that x = Chelsea

For the function f below , evaluate f at the indicated values and find the domain and range of f

1

2

3

4

5

6

7

10

11

12

13

14

15

16

f(1) f(2)

f(3) f(4)

f(5) f(6)

f(7)

Domain of f

Range of f

Example

• Let A = {a,b,c} and B ={x,y,z,t}. The diagrams in

the figures given below show correspondence by

which elements of A are associated to elements of B

A

a

b

c

B

x

y

Z

t

(Fig.1)In Fig.1, each element of A is associated to a

unique element of B.

A

a

b

c

B

x

y

Z

t

(Fig2)

In Fig.2, element ‘a’ of A is mapped to two elements x and y

of B which is against the definition of a function (as per

definition, to each element of A there must be a unique

element of B.) So Fig.2 doesn’t define a function.

A

a

b

c

B

x

y

Z

t

A

a

b

c

B

x

y

Z

t

(Fig.3)

Fig.4)

In Fig.3, element ‘c’ is not mapped to any

element of B. So this does not define a function.

In Fig.4, Each element of A is

mapped to a unique element of B.

(This defines a function A to B)

Find Range of f from the following figure?

A

1

2

3

B

a

b

c

d

Domain f = set of elements of A = {1,2,3}

Co domain f = Set of elements of B={a, b, c, d}

Range f = {a, c}

f(1) = a

f(2) = a

f(3) = c

Algebra of Functions

• If ‘f’ and ‘g’ are two functions then

(1) (f+g)(x) = f(x) +g(x)

(2) (fg)(x) = f(x) g(x)

(3) (f-g)(x) = f(x)-g(x)

(4) (f/g)(x) = f(x) / g(x), where g(x) # 0

Evaluating functions

)5(

)0(

)3(

12)(

f

f

f

xxf

)3(

)7(

)1(3

15)(

g

g

gx

xg

)5(

)1(

)0(

4)(

h

h

h

xxh

Graph of a function

The graph of the function f(x) is the set of points (x,y) in the xy-plane that satisfy the relation y = f(x).

Example: The graph of the function f(x) = 2x – 1 is the graph of the equation y = 2x – 1, which is a line.

3210-1-2-3

5

4

3

2

1

0

-1x

y

x

y

Domain and Range from the Graph of a function

Domain = {x / or }

Range = {y / or }

13 x 31 x

21 y 53 y

Types of Functions or Mapping f: A to B

INTO Function or Mapping: If at least one element

of B is not the image of an element of A, i.e. all the

elements of B are not the images, then function f is

called INTO function.

ONTO Function or Mapping: If every element of B

is image of some element of A, then f is called an

ONTO function.

NOTE:- EVERY MAPPING IS EITHER INTO

OR ONTO.

Types of Functions or Mapping f: A to B

MANY-ONE Mapping or Function: If more than

one different elements of A have the same image in

B, then the mapping or function f is called many

one function or mapping.

ONE-ONE Mapping or Function: If different

elements of set A have different images, i.e. no two

different elements have the same images, then

function or mapping is called a one-one mapping.

Types of Functions or Mapping f: A to BConstant Function or mapping: A mapping of

function f: A to B is said to be a constant mapping if

all the elements of A have same image.

Even and Odd Functions: A function f: A to B is

said to be an even if f(-x) = f(x) and odd if

f(-x) = -f(x).

Equal Functions: Two functions or mappings f & g

having the same domain A are said to be equal if

under both the functions all the elements have same

images, i.e. f(x)=g(x).

Various Functions

• Constant Function: A function of the form f(x) = c,

where c is a constant is called a constant function.

• Linear function: A function of the form f(x) = ax+b,

where a & b are constants and a#0 is called a linear

function.

• Quadratic function: A function of the form

, where

a, b & c are constants and a#0 is called a quadratic

function.

cbxaxxf 2)(

Various Functions

• Polynomial function: A function of the form

, where a0, a1,….an are constants and n is

non- negative integer, a0#0, is called a polynomial function

of x of degree n.

• Rational function: A function of the form f(x) = p(x) /

q(x), q(x) #0 & where p(x) & q(x) are polynomialis called

a Rational function.

• Exponential function: A function of the form

, where a>0 &

a#1, x is any real number is called exponential function.

n

nnn axaxaxaxf .....)( 2

2

1

10

xaxf )(

x

0 4 0

1 3 -1

-1 3 1

2 0 4

-2 0 -4

24)( xxf xxxg 2)( 3

• Let the demand be 40 units of a product when the

price is Rs. 10 per unit and 20 units when price is

Rs. 15 per unit. Find the demand function assuming

it to be linear. Also determine the price when

demand is 16 units.

Solution:

As Demand is a linear function

P = a + b X

where, P is price per unit &

X is quantity demand

Now P = 10 when X = 40

implies that 10 = a + 40 b …….(1)

Also, at X= 20 , P = 15

we have 15 = a + 20 b………. (2)

Solving (1) and (2), we get

a= 20, b = -1/4

Therefore Demand function is

p = 20 – (¼ )X …………………… (3)

By putting X = 16 in (3), We get

P = 16 ( Price is Rs. 16 per unit when demand is 16 units)

1. One-one functionThere is one-one correspondence between the elements of the set A and the set B.

2. Many-one functionThere is many-one correspondence between the elements of the set A and the set B.

3. Onto functionEvery element of the set B has at least one pre-image. In above fig.(i), the function is one-one and onto, while in fig.(ii) the function is many-one and onto.

4. Into functionThere is at least one element of B which has no pre-image.

One to one

MANY TO ONE

ONTO

INTO

LIMITS

Introduction to Limits

The student will learn about:

Functions and graphs,

limits from a graphic approach, limits from an algebraic approach, and limits of difference quotients.

Functions and GraphsA Brief Review

The graph of a function is the graph of the set of all ordered pairs that satisfy the function. As an example the following graph and table represent the same function f (x) = 2x – 1.

x f(x)

-2 -5

-1 -3

0 -1

1 1

2 ?

4 ?

We will use this point on the next slide.

Limits(THIS IS IMPORTANT)

Analyzing a limit. We can also examine what occurs at a particular point by the limit ideas presented in the previous chapter. Using the above function, f (x) = 2x – 1, let’s examine what happens when x = 2 through the following chart:

x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5

f (x) 2 4

2

3

2.8 3.2 2.98 3.02 2.998 3.002?

Note; As x approaches 2, f (x) approaches 3. This is a dynamic situation

Limits IMPORTANT!This table shows what f (x) is doing as x approaches 2. Or we have the limit of the function as x approaches 2. We have written this procedure with the following notation.

x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5

f (x) 2 2.8 2.98 2.998 ? 3.002 3.02 3.2 4

31x2lim2x

Def: We writeL)x(flim

cx

if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c).

or as x → c, then f (x) → L 2

3

One-Sided Limit

This idea introduces the idea of one-sided limits. We write

and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line.

Kxfcx

)(lim

One-Sided Limit

We write

and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.

L)x(flimcx

The Limit

K)x(flimcx

L)x(flimcx

Thus we have a left-sided limit:

And a right-sided limit:

And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

Example 1Example from my graphing calculator.

4)x(flim4x

4)x(flim2x

2)x(flim2x

The limit does not exist at 2!

4)x(flim4x

4)x(flim4x

The limit exists at 4.

2 42

4

Limit Properties

Let f and g be two functions, and assume that the following two limits are real and exist.

• the limit of the sum of the functions is equal to the sum of the limits.

Then:

• the limit of the difference of the functions is equal to the difference of the limits.

M)x(glimandL)x(flimcxcx

Limit Properties

If:

• the limit of the nth root of a function is the nth root of the limit of that function.

• the limit of the quotient of the functions is the quotient of the limits of the functions.

• The limit of a constant times a function is equal to the constant times the limit of the function.

Then:

• the limit of the product of the functions is the product of the limits of the functions.

M)x(glimandL)x(flimcxcx

Example 2

x2 – 3x =

cxlim

2xlim 2x

lim 2x

lim

x2 - 3x = 4 -2.

From this example we can see that for a polynomial function

f (x) = f (c)

– 6 =

Indeterminate Form

The term indeterminate is used because the limit may or may not exist.

If and , then

is said to be indeterminate.

0)x(flimcx

0)x(glimcx

)x(g

)x(flim

cx

The Limit of a Difference Quotient.

Let f (x) = 3x –1. Findh

)a(f)ha(flim

0h

f (a + h) = 3 (a + h) - 1 = 3a + 3h - 1

f ( a ) = 3 ( a ) – 1 = 3a - 1

f (a + h) – f (a) = 3h

3

h

)a(f)ha(flim

0h

h

h3lim

0h

3lim

0h

Limit Theorems

If is any number, lim ( ) and lim ( ) , thenx a x a

c f x L g x M

a) lim ( ) ( )x a

f x g x L M

b) lim ( ) ( ) x a

f x g x L M

c) lim ( ) ( )x a

f x g x L M

( )d) lim , ( 0)( )x a

f x L Mg x M

e) lim ( )x a

c f x c L

f) lim ( ) n n

x af x L

g) lim x a

c c

h) lim x a

x a

i) lim n n

x ax a

j) lim ( ) , ( 0)

x af x L L

Examples Using Limit RuleEx. 2

3lim 1x

x

2

3 3lim lim1x x

x

2

3 3

2

lim lim1

3 1 10

x xx

Ex.1

2 1lim

3 5x

x

x

1

1

lim 2 1

lim 3 5x

x

x

x

1 1

1 1

2lim lim1

3lim lim5x x

x x

x

x

2 1 1

3 5 8

More Examples

3 31. Suppose lim ( ) 4 and lim ( ) 2. Find

x xf x g x

3

a) lim ( ) ( ) x

f x g x

3 3 lim ( ) lim ( )

x xf x g x

4 ( 2) 2

3

b) lim ( ) ( ) x

f x g x

3 3 lim ( ) lim ( )

x xf x g x

4 ( 2) 6

3

2 ( ) ( )c) lim

( ) ( )x

f x g x

f x g x

3 3

3 3

lim 2 ( ) lim ( )

lim ( ) lim ( )x x

x x

f x g x

f x g x

2 4 ( 2) 5

4 ( 2) 4

Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions.

Ex.25

5lim

25x

x

x

Notice form0

0

5

5lim

5 5x

x

x x

Factor and cancel common factors

5

1 1lim

5 10x x

Indeterminate Forms

9

3a) lim

9x

x

x

9

( 3)

( 3)

( 3) = lim

( 9)x

x

x

x

x

9

9 lim

( 9)( 3)x

x

x x

9

1 1 lim

63x x

2

2 3 2

4b) lim

2x

x

x x

2 2

(2 )(2 )= lim

(2 )x

x x

x x

2 2

2 = lim

x

x

x

2

2 ( 2) 41

( 2) 4

More Examples

Summary.

We now have some very powerful tools for dealing with limits and can go on to our study of calculus. Please pay special attention to the practice problems.

• We started by using a table to see how a limit would react. This was an intuitive way to approach limits. That is they behaved just like we would expect.

• We saw that if the left and right limits were the same we had a limit.

• We saw that we could add, subtract, multiply, and divide limits.

Limits at Infinity

For all n > 0,1 1

lim lim 0n nx xx x

provided that is defined.1nx

Ex.2

2

3 5 1lim

2 4x

x x

x

2

2

5 13lim

2 4x

x x

x

3 0 0 3

0 4 4

Divide by 2x

2

2

5 1lim 3 lim lim

2lim lim 4

x x x

x x

x x

x

More Examples

3 2

3 2

2 3 21. lim

100 1x

x x

x x x

3 2

3 3 3

3 2

3 3 3 3

2 3 2

lim100 1x

x xx x x

x x xx x x x

3

2 3

3 22

lim1 100 1

1x

x x

x x x

22

1

0

2

3 2

4 5 212. lim

7 5 10 1x

x x

x x x

2

3 3 3

3 2

3 3 3 3

4 5 21

lim7 5 10 1x

x xx x x

x x xx x x x

2 3

2 3

4 5 21

lim5 10 1

7x

x x x

x x x

0

7

2 2 43. lim

12 31x

x x

x

2 2 4

lim12 31x

x xx x x

xx x

42

lim31

12x

xx

x

2

12

24. lim 1x

x x

22

2

1 1 lim

1 1x

x x x x

x x

2 2

2

1lim

1x

x x

x x

2

1 lim

1x x x

1 1

0

Practice Problems

Evaluate the following limits

.2

2x

1x

5x

lim .1

Def.: A function f(x) is said to tend to a number L as

x approaches to c if to a given there exist a

Such that

and we write

Limit of a function exists if Right Hand Limit =

Left Hand Limit

0ε

0δ εf(x) δcx L

Lf(x)cx

lim

f(x)cx

lim f(x)

cx

lim

Algebra of Limits

g(x)ax

lim f(x)

ax

lim g(x) f(x)

ax

lim

g(x)ax

lim . f(x)

ax

lim g(x). f(x)

ax

lim

g(x)#0 where,g(x)

ax

lim

f(x)ax

lim

g(x)

f(x)

ax

lim

f(x)ax

lim c f(x) c

ax

lim

1.

2.

3.

4.