Math Analysis I - Limits of Functions Part 1 (Functions)

LIMITS OF FUNCTIONS

Definition function

of

Let X and Y be two sets. A function f defined on X with values in Y is a

correspondence associating to each element x∈X at most one element y∈Y .

This is often shortened to ‘a function from X to Y’. A synonym for function is map or

application. The symbol for an element of X (to which one refers as the independent

variable) and the symbol for an element in Y (dependent variable), are completely

arbitary. What really determines the function is the way of associating each element of

the domain to its corresponding image.

The set of x∈X to which f associates an element in Y is the domain of f; the domain is

a subset of X, indicated by dom f. Y

X TA R G E T

d o m

f : L E S E T

→

if Y

d o m y - X , j : X ' →

The element y∈Y associated to an element x∈dom f is called the image of x by or

under f and denoted y = f(x).

jcxa@A.E

f : X A

The set of images y = f(x) of all points in the domain constitutes the range of f,

a subset of Y indicated by im f. {ye forty}

F x

l

-Ryu EX,

R e n t -T I X ) (X) Y

- through

(x) t h e f

where f image ×

o f

is

The graph of f is the subset Γ(f) of the Cartesian product X × Y made of pairs (x, f(x))

when x varies in the domain of f, i.e.,

{(x. doms}

/

Text) X x

Act) Y

c- x e

-

2. Math Analysis Limits of Functions 1 of 36

Rouge end pre-image

Let A be a subset of X. The image of A under f is the set of all the images of

elements of A. EA}

{ICH

f l a t I x im t

E

Notice that f(A) is empty if and only if A contains no elements of the domain of f.

The image f(X) of the whole set X is the range of f, already defined by im f.

Let y be any element of Y ; the pre-image of y by f is the set of elements in X

whose image is y.

{X y}

j' I

(y) domf

E fail =

-

This set is empty precisely when y does not belong to the range of f.

If B is a subset of Y , the pre-image of B under f is defined as the set union of all

pre-images of elements of B.

' Ix B}

f - (B) x x )

domy: E

E

=

let (flat)

dem f '

A

A E f then E

,

let Y,

B (B)) - B

' B

n

f l y -

then

E E

i m y

R a IR,

f: x2

C . f (x)

g. = [in]

[1,2] B -

A i m (A)

y

=

= {X z}

- I ] [1/2]

I-2,

' I

IR

U

(B)

f - e x t pre-image

l

E =

-

=

¥±÷÷÷i." ÷÷÷t÷÷.....

.

2. Math Analysis Limits of Functions 2 of 36

Definition ( r e a c t i o n s )

If X = R, the function is of one real variable.

If Y = R, the function f is said real or real-valued. 1122

Therefore the graph of a real function is a subset of the Cartesian plane .

reel

a r e a l

function Va r i a b l e

of is map

a a

IR X#far) R

such f:

that, E E c

where

→

let f i t # R be function

r e e l

a {ye SH-y}

R e a ( s )

then, - R e n E t R

ICE) I I x

= EE,

f through

jet the

where y

is image (orb

R constants)

IR, b

f u k

e . ' → e x t

j :

g. . T line

straight

is a ¥

R IR, t

x2

j : berebole

f a t is a

• ¥

2. Math Analysis Limits of Functions 3 of 36

R ,

{03

R t LAI-±

s : →

• the

is re c ta n g u l a r in

h y p e r b o l e

a

coordinate a s y m p to te s

i t s

system o f

# defined

f u n c t i o n b r a n c h e s

b y

r e a l

a

• function

is called piecewise

a

sant"

3X X I I

O f

'÷¥÷,

X-l

^

H'

x

-

I i ,

O

2. Math Analysis Limits of Functions 4 of 36

R # I R ¥

f:

•

sa-KI-II ¥8 {I,' II}

I R A Z , (x)

f :

S I G N f a t - s i g n

FUNCTION =

. c o

X

-

¥

- ya-Lx]

Z,

R

f: →

. [x) "integer

IR,

let then × "

par t o f

x E -

×}

ft E Z , I t

2-

m e ←

x

= [Elt

[x] i

I

such x

t h a t l

[3) [ T I - 3, f t ]

e . = 3 ,

g. =-4

I

§ €

x x

x x

-

x x

X - D -

2. Math Analysis Limits of Functions 5 of 36

the mantissa

• R # R , [x]

y a - M a - x -

f : M a l e

then I

0 1

¥f÷±*.

1122#R, F E

fa r,

f: g) =

• play) t o i t s

point

mops a

distance 0

from

1122 1122, l y, x )

y)

fix,

j : → =

• P'

P point

p o i n t

t o t h e

associates a (bisectrix)

w i t h respect t o y - x

2. Math Analysis Limits of Functions 6 of 36

Injection

A reid its

f

function injective

i s o n

domain if

domf and if

only

FX,,Xz domf #xz Tex,)ffCxz)

t h a t X,

s o

E

then, if f(x,) - f i x z ) , ONE

O N E

X, TO

- X z

Graphically this means that the graph of an injective function cannot have more

than one point at fixed height. t h e r a u l t )

Namely, if we consider any horizontal line y = k, , then the graph

of f can intersect the line at most once.

F r i R e a c t )

If, such that the graph has two or more ponts in comon with the

E-

line y = k, then f is not injective.

# ^

y - U z #

y-U,

÷

Y y. I R ? y # y = k g ( n o intersection)

I N J E CT I V E

y¥¥µ

NOT INJECTIVE

2. Math Analysis Limits of Functions 7 of 36

Inversefunction

If a map f is injective (one to one), we can associate to each element y in the

range the unique x in the domain with f(x) = y. Such correspondence

determines a function defined on Y and with values in X, called inverse function

f - i .

of f and denoted by the symbol

' (y) y-fix)

x - f - j '

The inverse function has the image of f as its domain, and the domain

of f as range: -Ren

dom j' (ft)

(f), Ren f

dom

=

f-i

Q.fi#.@..-

FYEfCXz)

f

Given R e n (J)

d o

1:

function

a m y i →

Fg:

it invertible if

said ( f )

Reen domf

is →

(f (x)) Fx E d o m

T h e t :

s u c h j

F x

=

(y))

Cg Hye

j Ren

y f

=

t h e inverse

is

y function

A one-to-one map is therefore invertible; the two notions (injectivity and

invertibility) coincide.

2. Math Analysis Limits of Functions 8 of 36

Grophoftheinversefunction

{ l y, (y)) j'}

l

f '

(j')

I X

Y dorn

y

E x E

= {(je), my}

X

Y

x ) / do

E × E

x

=

Therefore, the graph of the inverse map may be obtained from the graph of f

by swapping the components in each pair.