THE FUNCTION OF COMPOSITION AND INVERSE

7/30/2019 THE FUNCTION OF COMPOSITION AND INVERSE

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MATHEMATICS 2FOR SENIOR HIGH SCHOOL

CHAPTER 6

THE FUNCTION OF COMPOSITION AND

INVERSE


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Function1. Definition of function

Definition

A relation between the set A to set B is called a function or a mapping if and only if

ever member of the set A corresponds to exactly one member of the set B.

The function offmaps the set A set to the set B is symbolized as

The set A is called the domain, symbolized by

The set B is called the codomain, symbolized by

is called the range, symbolized by

BAf :

fD

fK

fR

AxRyxBy ,),(

A


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2. Characteristic Of Function

a. Surjective Function

The range of the function offisSo, .

It means that the range of function off

Is the set of Q.

Definition

A function of with the range of the function f is equal to the set of Q called

surjective function or onto function.

QPf :

.9,4,1RQRf


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b. Injective Function

A function where each member of the domain

relates to exactly one member of codomain is

called an injective function (an 1-1 function).

Definition

The function of is called an injective function if each and

apply

QPf : PPP 21,

21 PP ).()( 21 PfPf


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Example

Given that function of below, investigate if it is an injective

function!

Answer :

Take where , then

and

Thus, the function of is an

Injective function.

The diagram of in the right shows that

For any the domain of thenTherefore, the function of is an

Injective function.

Rxxxfy ,32)(

Rxx 21 , 21 xx

32)(11 xxf 32)( 22 xxf

)()(

3232

21

21

xfxf

xx

32)(11 xxf

32)( 11 xxf

21 ,xx 21 xx )()(

21 xfxf 32)( xxfy


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Definition

The function of is called a bijective function if it is a surjective and aninjective function.

c. Bijective Function

BAf :


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Composition Function

1. The Rules of Composition from several function

Using the functions offandgcan be composed a function called the composition of

function yaitu ( read fdot g or f noktah g )

The relation among the sets of A, B, and C

Can be expressed in the following notation.

or

or

and

gf

,:,:,: CAhCBgBAf

zxhzygyxf :,:,:)(xhz )(ygz

B


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The equation of is equivalent to , whereas

, then it can be concluded that

The domain of function is the domain of function f. in order that then

the range of the functionfis the subset of the domain of the function g or

).(xhz )()()( xfgygxh ))(()( xfgxh

))(())(( xfgxfg

fg fg

. gf DR


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2. Value of Composition Functions

To determine the value of the compositions can be done by first determining the rules

for the composition function or can also be directly calculated step by step.

Example

The function of and are determined by and

. Determine the value of :

a. b.

Answer :

a. b.

RRf : RRg : 63)( xxf2

5)( xxg )2)(( fg )1)(( gf

)2)(( fg

0

0.5

)0(

)6)2(3(

))2((

2

g

g

fg )1)(( gf

21

65.3

)5(

)1.5(

))1((

2

f

f

gf


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3. Determining the Component Formed of the Composition Function

If the rules for functions offand are given, then the rule for

function of g can be determined. In order to make your understand, let us

consider some of the following examples.

Example

Given that and . Determine

Answers :

Thus,

)( forggf

35)( xxg 710))(( xxgf ).(xf

710))(( xxgf

1)35(2)35(

710)35(

xxf

xxf

12)( xxf

710))(( xxgf


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4. Charateristics of the Composition Function

Suppose determined the rules for funtions of from

a. In general, the operations on the compositions is not commutative,

means

b. On the composition functions holds the charateristic of associative, that

is

c. Suppose I is the function of and it fulfill then I

is the identity function.

hgf ,, RR

)()( fggf

).()( hgfhgf

)(xI ffIIf


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C. Inverse Function

1. Definition of Inverse Function

If the function of then its inverse is

The inverse function offis symbolized by

(read inversef) And the inverse function

of g is symbolized by (read inverse of g).

So, and

The inverse of a function can be a function (called the inverse function) or a

simply relation.

Definition

The functio of have inverse function of if f is a

bijective function or the set A and B is one-to-one corresponding.

BAf : ABg :

1f

1g1 fg 1 gf

BAf : ABf :1


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2. The Rules of Inverse Function

If is a bijective funtion where and

Then the inverse of f is function where

and

BAf : Axxfyyxf ),(),( By

ABf :1

Axyfxxyf ),(),( 11 By


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Example

the function of is determined by . Determine rule

for

Answer :

Hence, the rules for is

RRf : 54)( xxf1

f

54)( xxf

)5(4

1

54

yx

xy

)5(4

1)(

1 yyf

)5(

4

1)(

1 xxf

1f )5(

4

1)(

1 xxf


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3. Determining the Domain and the Range of a Function

Iffis a bijective function, then or , where is

the domain of the function offand is the range of the functionsf.

ExampleGiven that function offwith the rule of . Determine the

domain function offso that the functionfhas an inverse function, and then

determine the formula of for the domain!

Answer :

1ff

RD 1ff

DR fD

4)(2 xxfy

)(1xf

4)(4)(44)(112 xxfyyfyxxxfy


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In figure (a), it seems that the function of f is not bijective so the function of f does not

have an inverse function of f will have an inverse function. In order that f has an

inverse function, the domain of can be given, for instance :

1) If then (Figure (b))

2) If then (Figure (c))

fDf

RxxxDf ,0 4)(1 xxf RxxxDf ,0 4)(1 xxf


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Example

Given that

a. Determine

b. Show that where

Answer :

a.

So,

b.

So,

).3(2

1)(

1 xxf

)(xf

)())(())((11

xIxffxff .)( xxI

).3(2

1)(

1 xxf

).3(2

1)(

1 xxf

32

32

)3(

2

1

xy

yx

yx

32)( xxf

))(())((11

xffxff

x

x

x

xf

)2(2

1

3)32(2

1

)32(1

)())(())((11

xIxffxff

)3(2

1))((

1

xfxff

x

x

x

33

3)3(212


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Based on example we have the following characteristics

a.

b.

Derived from the characteristics of and

we have another characteristics, that is

ff 11 )(

Iffff )()( 11

ffIIf Iffff 11

.: 11 fggggff


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4. Drawing the Graph of an Inverse Function From the Origin of Function Graph

If the graph of the function and graph of the function are drawn in a

coordinate system.

Example

The function of is determined by . Determine , then drawthe graph of the function and in one system of coordinates!

Answers :

It is clear that the graph of

and the graph of

are symmetric to the line ofThus, the graph of inverse function can be

drawn by reflecting the graph of the function

on the line of

)(xfy )(1 xfy

RRf : 62)( xxf)(xfy )(

1xfy

32

1)(

3

2

1)(

1

1

xxf

yyf

62)( xxf

32

1

62

62

yx

yx

xy

62)( xxfy

32

1)(

1 xxfy

.xy )(

1xf

)(xfxy


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5. Inverse Function from the Composition Function

Consider picture

The function of is a composition functions,

Then is the inverse functionFrom the composition functions.

It is shown in picture that :

or

or

or

Because and then

. (1)

Even . (2)Based on (1) and (2) can be obtained :

For any z, it can be concluded that :

or

fgh 11

)( fgh

xyf :1 )(1 yfx

yzg :1

xzh :1

)(1zgy

)(1zhx

)(1yfx

)(1zgy

))(())((1111zgfzgfx

)()()(11

zfgzhx

)()())((111 zfgzgf

))(()()(111 xgfxfg ))(()()( 111 xfgxgf

THE FUNCTION OF COMPOSITION AND INVERSE

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