Relations and Functions (40)

8/11/2019 Relations and Functions (40)

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nzks.kkpk;ZACADEMY & RESONANT CLASSES

nzks.kkpk;ZAcademy

S.C.F. 57, Sector 7

Kurukshetra

&

Resonant Classes

S.C.O. 53, Sector 17

Kurukshetra

RELATIONS AND FUNCTIONS

IMPORTANT TOOLS / TECHNIQUES TIPS

LetX and Ybe two non empty sets. A function f fromX to Y (written as YXf : ) is an

association or correspondence between X and Y such that corresponding to each Xx there

corresponds a unique Xy .

All elements of set X are independent variables and called domain of the function. All

elements of Y are dependent variables and called co-domain of the function. The set of all values

taen by a function is called range of the function.

(i) !fxin Xcorresponds toyin Y then we write ( )xfy= and say theyy isf-image of xorpre-image ofy.

(ii) The set of allf-images of a function YXf : is called range of the function.

Thus fR " range of ( ){ }Xxxff = : .

(iii) A function ( )xf is said to be onto or sur#ective if YRf = $ i.e.$ range and co-domain

must be same. A function ( )xf is said to be into if it is not onto. %or an into function fR

must be a proper subset of Y. !f nothing is specified an onto function is the one which taes all

real values.

(iv) A function ( )xf is said to be one-one or in#ective if two distinct domain elements have

distinct images$ i.e.$ if &' xx then ( ) ( )&' xfxf for all &' $ xx inX function ( )xf is said

to be one-one if ( ) ( ) &'&' xxxfxf == for all&' $ xx .

(v) A function which is both one-one and onto i.e. in#ective and sur#ective is called bi#ective

such as function are invertible.

Inverse of a functon!

Let ( )xfy= be a bi#ective function fromX to Yand suppose it is possible to write ( )ygx=

then the function ( )xgy= is called inverse of ( )xf whose domain and range are Y and X

respectively note that

(i) !f ( )yx$ is point on the graph of ( )xfy= then ( )xy$ must be a point on the graph of

( )xgy= .

(ii) The graph of a function and its inverse are symmetrical about the liney = x.

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(iii) ( )( ) ( )( ) xxfgxgf == .

RULES TO "IND DOMAIN AND RAN#E O" "UNCTION

To fn$ Do%an!

!f the domain of ( )xf and ( )xg are 'D and &D respectively$ then the domain the

( ) ( )xgxf and ( ) ( )xgxf . is &' DD $ while the domain of( )( )xgxf

is &' DD ecept

those values of x where ( ) =xg . *omain of composite function ( )xfog is allx for which x

domain ( )xg and ( )xg domain of ( )xf .

To fn$ Rane!

(i) To find the range of a function$ put ( )xfy= . Then find the value ofxin terms ofy. The

set of all values ofyfor which function is defined will be the required range of the function. !f

the function is rational function then cross multiply with y and mae quadratic in x and

discuss the nature of roots.

(ii) +ange can also be determined by finding the derivative of the function. !f a function is

increasing in the interval ,a, b then range will be ( ) ( )[ ]bfaf $ . !f the function is decreasing

in the interval ,a, b then range will be ( ) ( )[ ]afbf $ . o range$ (maimum and minimum

value) of the function can be found by maima and minima theory$ i.e.$ +ange of

cxbxa ++ cossin is

[ ]&&&& $ bacbac +++ .

(iii) +ange of odd degree polynomial function is always R. The domain and range of several

functions are truncated to mae them invertible. %or eample domain of tan x is chosen as

&

$&

through the general domain is

( ) ( )

+

&'&$

&'&

nn

zn

.

The range of x'sin for the same reason is chosen as

&$

&

.

(iv) +ange ofbax+

'is all real ecept $ where range of

dcx

bax

++

is all real eceptc

a$ where

bcad .

+ange of x " All non-negative real.

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(v) !f a value is attained at a point which si not there in the domain then this value is to be

deleted from the range of the function. %or eample$ consider ( ) ( )/0

'& ++

=

xx

xxf . *omain of

( )xf " All real ecept ' and / for$x" '$ ( )1

'

/

' =x

xf .

+ange of ( )xf " +ange of

1'

/

'

x.

" All reals ecept and 2'31.

Do%an an$ Rane for Trono%etrc an$ Inverse Trono%etrc functons

S' No' " Do%an Rane

'. xsin Rx ,2'$ '

&. xcos Rx ,2'$ '4. xtan Rx $ ( ) &3'& + nx R5. xcot Rx $ nx R6. xsec Rx $ ( ) &3'& + nx R2 (2'$ ')1. xeccos Rx $ nx R2 (2'$ ')/. x'sin ,2'$ '

&$

&

0. x'cos ,2'$ ' ( )$7. x'tan R

&

$

&

'. x'cot R ( )$''. x'sec R2(2'$ ') [ ] [ ]&3$ '&. xec 'cos R2(2'$ ')

&$

&

2 89

Do%an an$ Rane for ot(er usua) functons

S' No' "uncton Do%an Rane

'.olynomial function

R R if the degree is odd$ subset ofR !f

the degree is even.

&. +ational function All real ecept forwhich ( ) =xQ

*epends on particular rationalfunction

4. $ >aa x R ( )$5. $$log >> axxa

and 'a( )$ R

6. x R [ )$1.

x

x R2 89 82'$ '9

/. ,x R et of integers

0. 8x9 R [ )'$

PERIODIC "UNCTION

4


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A function ( )xf is said to be periodic if there eists a number a such that

( ) ( )xfaxf =+ for all x. The number a is called the period of the function. The least positive a

satisfying ( ) ( )xfaxf =+ for allxis called the fundamental period or the least period of the function

( )xf .

(i) !f period of ( )xf is ( )$ aa then period of ( ) ( )$ xf must be

a.

(ii) !f ( ) ( ) ( )xhbxgaxf += $ where ( )xg and ( )xh are periodic functions with period

'P and &P then ( )xf will also be periodic with period L.;.


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"uncton Pero$xecxxx cos$sec$cos$sin

xx cot$tan

( )xx sincos$sin

$cossin$cossin 55 xxxx ++

( ) ( )xx sincossincos +

[ ]xx

[ ]nxx

&

&3

'

'3n (nis positive)

ODD AND E*EN "UCNTIONS

A function ( )xf is said to be even if ( ) ( )xfxf = $ for allx. Again a function ( )xf is said to be

odd if ( ) ( )xfxf = $ for allx.(i) The sum$ difference$ product of two even functions is even.

(ii) The sum of an even function and odd function is neither even nor odd.

(iii) The product of an even and odd function must be odd.

(iv) The derivative of an odd function is an even function and conversely.

(v) Any function ( )xf can be epressed as sum of an even and odd functions observe that

( ) ( ) ( ) ( ) ( )&&

xfxfxfxfxf

++= .

?ote that ( ) ( )&

xfxf +is even while ( ) ( )

&

xfxf is odd.

(vi) @very constant function is an even function.

(vii) ( ) =xf is even as well as odd function x .

COM+INATORICS IN "UNCTIONS

!fAandBare two finite sects with mand nelements respectively then

(i) Total number of functions fromAtom

nB=

(ii) Total number of one-one functions from

>

=

nm

nmPBA m

n

if

ifto

(iii) ?umber of functions satisfying ( ) ( ) '...$4$&$'$...$$&$' ==== mnnjmijif

(iv) ?umber of functions in which elements nbbb $...$$ &' of B have nrrr $...$$ &'

pre-images in ( )...

....&'

&'

n

nrrr

mmrrrA ==+++

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(v) The number of onto functions fromAtoB'

( ')n

n r n m

rr

C r

==

(vi) The number of ob#ective functionAto nB= (when m = n).

STANDARD RESULT AND IMPORTANT TIPS

(i) The graph of a function will be met by any vertical line at most at one point.

(ii) The graph of a one-one function will be met by any hori>ontal line at most at one point.

(iii) !f a function is non-in#ective$ it can be easily proved by taing numerical values. =ut =x

and then solve ( ) ( )fxf = . incefis not in#ective$ we may get other value ofx.

(iv) !ffis a differentiable one-one function then either ( ) xf or ( ) xf for all values

ofxin the domain.

(v) The functions xx elog$tan and odd degree polynomials are essentially onto.

(vi) !f for a differentiable function ( )xf whose inverse is ( )xg then ( )xg can be found by

differentiating ( )( ) xxfg = and replacingxby ( )xg .(vii) !f a function has at least one local maima or local minima then it is always many-one

function.

(viii) !n the case of composite functions$ if gof is one-one function then fmust be one-one

function and ifgofis onto function thengmust be onto function.

SPECIAL POINTS

(i) !f ncccc xP fro allxin ( ) ( ) ( ) $...$$ 4&' ncccc

( ) xP fro allxin ( ) ( ) ( ) $...$$ 54&' nccccc

( ) a then yxa >log

yxax ay log$

yax


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(iv) Let cbxax ++& be a quadratic polynomial then min ( )a

baccbxax

5

5 &

& =++ if >a $

ma ( )a

baccbxax

5

5 && =++ if


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Q' ' how that the relation + in the set Rof real numbers$ defined as 9:)$8(+ &baba = is

neither refleive nor symmetric nor transitive.

Q' 0' ;hec whether the relation + defined in the set 8'$ &$ 4$ 5$ 6$ 19 as + " 8( a$ b) : b" aC '9 is

refleive$ symmetric or transitive.

Q' 1' how that the relation + in Rdefined as 9:)$8(+ baba = $ is refleive and transitive

but not symmetric.

Q'2' ;hec whether the relation + in R defined by 9:)$8(+ 4baba = is refleive$

symmetric or transitive.

Q'4' how that the relation + in the set 8'$ &$ 49 given by + " 8('$ &)$ (&$ ') is symmetric but

neither refleive nor transitive.

Q'5' how that the relation + in the set A of all the boos in a library of a college$ given by

9pagesofnumbersamehaveand:)$8( yxyxR= is an equivalence relation.

Q'6' how that the relation + in the set A " 8'$ &$ 4$ 5$ 69 given by

9evenis:)$8( babaR = $ is an equivalence relation. how that all the elements of

96$4$'8 are related to each other and all the elements of 95$&8 are related to each other.

=ut no element of 8'$ 4$ 69 is related to any element of 8&$ 59.

Q'7' how that each of the relation + in the set 9'&:8A = xx 3 $ given by

(i) 95ofmultipleais:)$8( babaR =

(ii) 9:)$8( babaR ==

is an equivalence relation. %ind the set of all elements related to ' in each case.

Q' .8' Eive an eample of a relation. Fhich is

(i) ymmetric but neither refleive nor transitive.

(ii) Transitive but neither refleive nor symmetric.

(iii) +efleive and symmetric but not transitive.

(iv) ymmetric and transitive but not refleive.

Q' ..' how that the relation + in the set A of points in a plane given by + " 8($ G) : distance of the

point from the origin is same as the distance of the point G from the origin9$ is an

equivalence relation. %urther$ show that the set of all points related to a point )($ is

the circle passing through with origin as centre.

Q' .' how that the relation + defined in the set A of all triangles as similarisT:)T$8(T+ '&'=

9Tto & $ is equivalence relation. ;onsider three right angle triangles 'T with sides 4$ 5$ 6$

&T with sides 6$ '&$ '4 and 4T with sides 1$ 0$ '. Fhich triangles among

4&' TandT$T are relatedH

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Q' .0' how that the relation + defined in the set A of all polygons as haveand:)$8(+ &'&'=

sides9ofnumbersame $ is an equivalence relation. Fhat is the set of all elements in Arelated to the right angle triangle T with sides 4$ 5 and 6H

Q' .1' Let L be the set of all lines in IJ plane and + be the relation in L defined as

9LtoparallelisL:)L$8(L+ &'&'= . how that + is an equivalence relation. %ind the set

of all lines related to the liney" &xC 5.

Q' .2' Let + be the relation in the set 8'$ &$ 4$ 59 given by + " 8('$ &) (&$ &) ('$ ') (5$ 5)$ ('$ 4)$ (4$ 4)

(4$ &)9. choose the correct answer.

(A) + is refleive and symmetric but not transitive.

(=) + is refleive and transitive but not symmetric.

(;) + is symmetric and transitive but not refleive.

(*) + is an equivalence relation.

Q' .4' Let + be the relation in the set Ngiven by + " 8(a$ b) : a" b2 &$ bK 19. ;hoose the correct

answer.

(A) +5)(&$ (=) +0)(4$ (;) +0)(1$ (*)

+/)(0$

E,ERCISE - .'

E' .' Let A be the set of all 6 students of ;lass I in a chool. Let NA:f be function defined

byf(x) " roll number of the I. how thatfis one-one but not onto.

E' ' how that the functionf : ?$? given byf(x) " &x, is one but not onto.

E' 0' rove that the functionf: R9R given byf(x) " &x$ is one-one and onto.

E' 1' how that the function f : ?? given byf (') "f(&) " ' andf (x) " x2 '$ for every

$&>x is onto but one-one.

E'2' hot that the functionf:$RR

defined asf(x) " $

&

x is neither one-one nor onto.

E'4' how thatf: $NN given byf(x) "

+

evenisif$'

oddisif$'

xx

xxis both one-one and onto.

E' 5' how that an onto function 94$&$'894$&$'8: f is always one-one.

E' 6' how that a one-one function 94$&$'894$&$'8: f must be onto.

Q' .' how that the function +: Rf defined byx

xf ')( = is one-one and onto where R8

is the set of all non->ero real numbers. !s the result true$ if the domain R8 is replaced by N

with co-domain being same as R8H

Q' ' ;hec the in#ectivity and sur#ectivity of the following function:

(i) NN:f given by &)( xxf =

7


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(ii) 33:f given by &)( xxf =

(iii) RR:f given by 2= xxf )(

(iv) NN:f given by 4)( xxf =

(v)33:f

given by

4

)( xxf =Q' 0' rove that the Ereatest integer function $: RRf given by $,)( xxf = is neither one-

one nor onto$ where ,x denotes the greatest integer less than or equal to x.

Q' 1' how that the

=if'$-

if$

if$'

)(

x

x

x

xf is neither one-one nor onto.

Q'4' Let A " 94$&$'8 $ = " 9/$1$6$58 and let 9)1$4()$6$&()$5$'(8=f be a function

from A to =. how thatfis one-one.

Q' 5' !n each of the following cases$ state whether the function is one-one$ onto or bi#ective$ ustify

your answer.

(i) RR:f defined by xxf 54)( =

(ii) RR:f defined by &')( xxf +=

Q' 6' Let A and = be sets. how that ABBAf : such that )$()$( abbaf = is

bi#ective function.

Q' 7' Let NN :f be defined by

+

=evenisif$

&

n

oddisif$&

'

)(

n

nn

nf for all Nn .

tate whether the function is bi#ective. ustify your answer.

Q' .8' Let A " R2 849 and = " R2 8'9. ;onsider the function =A: f defined by

=

4

&)(

x

xxf . !sfone-one and ontoH ustify your answer.

Q' ..' Let RR:f be defined by 5)( xxf = . ;hose the correct answer.

(A) fis one-one onto (=) fis many-one onto

(;) fis one-one but not onto (*) fis neither one-one nor onto

Q' .' Let RR:f be defind as xxf 4)( = . ;hoose the correct answer.

(A) fis one-one onto (=) fis many-one onto

(;) fis one-one but not onto (*) fis neither one-one nor onto

E,ERCISE - .'0

E' .' Let 97$6$5$4896$5$4$&8: f and 9'6$''$/897$6$5$48: g be

function defined as6)6()5($5)4($4)&(

==== ffff

and/)5()4(

== gg

and '')7()6( == gg . %indgof.

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E' ' %ind gof and fog$ if RRRR :and: gf are given by xxf cos)( = and&4)( xxg = . how that foggof .

E' 0' how that if

6

4

6

/: RRf is defined by

/6

54)(

+

=x

xxf and

6

/

6

4: RRg is defined by

46

5/)(

+

=x

xxg $ then AIfog= and

BIgof = $ where $

BxxxIAxxxIBA BA ==

=

= $)($$)(M

6

/$

6

4RR are

called identity function on setsAandB$ respectively.

E' 1' how that if BAf : and CBg : are one-one$ then CAgof : is also one-

one.E' 2' how that if BAf : and CBg : are onto$ then CAgof : is also onto.

E' 4' ;onsider functionfandgsuch that compositegofis defined and is one-one. Arefandgboth

necessarily one-one.

E' 5' Arefandgboth necessarily onto$ ifgofis ontoH

E' 6' Let 9$$894$&$'8: cbaf be one-one and onto function given by

cfbfaf === )4(and)&($)'( . how that there eists a function

94$&$'89$$8: cbag such that XIgof= and YIfog= $ where$ I " 8'$ &$ 49

and J " 8a$ b$ c9.

E' 7' Let Yf N: be a function defined as 45)( += xxf $ where$

9somefor45:8 N+== xxyNyY . how that f is invertible. %ind the inverse.

E' .8' Let NN = 9:8 & nnY . ;onsider Yf N: as &)( nnf = . how that f is

invertible. %ind the inverse off.

E' ..' ;onsider NNNN :$: gf and RN:h defined as

54)($&)( +== yygxxf and zyxzzh and$$sin)( = in ?. how that

ofhoggofho )()( = .

E' .' ;onsider 9$$894$&$'8: cbaf and 9catball$apple$89$$8: cbag defined as

ball)($apple)($)4($)&($)'( ===== bgagcfbfaf and cat)( =cg . how

that gf$ and gof are invertible. %ind out ''$ gf and ')( gof and show that

''')( = ogfgof .

E' .0' Let " 8'$ &$ 49. *etermine whether the function f : defined as below have

inverses. %ind 'f $ if it eists.

(a) )94$4()$&$&()$'$'8(=f

(b) )9'$4()$'$&()$&$'8(=f(c) )9'$&()$&$4()$4$'8(=f

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Q' .' Let 96$&$'895$4$'8: f and 94$'896$&$'8: g be given by

9)'$5()$6$4()$&$'(8=f and 9)'$6()$4$&()$4$'(8=g . Frite down

gof .

Q' ' Letf$gand hbe functions from Rto R. how that

gohfohohgf +=+ )( )(.)().( gohfohohgf

Q' 0' %ind gof and fog $ if

(i) xxf =)( and &6)( = xxg

(ii) 40)( xxf = and 4'

)( xxg = .

Q' 1' !f( )( ) 4

&$

51

45)(

+= x

x

xxf $ show that xxfof =)( $ for all

4

&x . Fhat is the inverse

offH

Q' 2' tate with reason whether following functions have inverse

(i) 9'895$4$&$'8: f with

9)'$5()$'$4()$'$&()$'$'(8=f

(ii) 95$4$&$'890$/$1$68: g with

9)&$0()$5$/()$4$1()$5$6(8=g

(iii) 9'4$''$7$/896$5$4$&8: h with

9)'4$6()$''$5()$7$4()$/$&(8=h

Q' 4' how that R '$',:f $ given by( )&

)(+

=x

xxf is one-one. %ind the inverse of the

function ff +ange'$',: .

(Nint : %or&

)($+ange+

==x

xxfyfy $ for some x in -'$', $ i.e.$

)'(

&

y

yx

= )

Q' 5' ;onsider RR:f given by 45)( += xxf . how thatf is invertible. %ind the inverse

off.

Q' 6' ;onsider )$5,: +Rf given by 5)( & += xxf . how thatfis invertible with the

inverse 'f of fgiven by 5)(' = yyf $ where +R is the set of all non-negative real

numbers.

Q' 7' ;onsider )$6,: +Rf given by 617)( & += xxxf . how that f is invertible

with( )

+=

4

'1)(' y

yf .

Q' .8' Let JI: f be an invertible function. how thatf has unique inverse.

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(Nint : suppose 'g and &g are two inverses of f. Then for all Jy $

)()(')( &' yfogyyfog Y == . Ose one-one ness off ).

Q' ..' ;onsider 9$$894$&$'8: cbaf given by bfaf == )&($)'( and cf =)4( .

%ind 'f and show that ff = '')( .

Q' .' Let JI: f be an invertible function. how that the inverse of 'f is f$ i.e.$

ff = '')( .

Q' .0' !f RR:f be given by 4'

4 )4()( xxf = $ then )(xfof is

(A) 4'

x(=) 4x (;) x (*) )4( 4x .

Q' .1' Let RR

4

5:f be a function defined as

54

5)(

+=

x

xxf . The inverse of f is the

map

4

5+ange: Rfg given by

(A)y

yyg

54

4)(

= (=)

y

yyg

45

5)(

=

(;)y

yyg

54

5)(

= (*)

y

yyg

45

4)(

=

E,ERCISE - .'1

E' .' how that addition$ subtraction and multiplication are binary operations on R$ but division is

not a binary operation on R. %urther$ show that division is binary operation on the set :R of

non>ero real numbers.

E'' how that subtraction and division are not binary operations on N.

E' 0' how that RRR :P given by &5)$( baba + is a binary operation.

E' 1' Let be the set of all subsets of a given set I. how that : given by

=A=)(A$ and : given by =A=)(A$ are binary

operations on the set .

E' 2' how that the RRR : given by 9$8ma)$( baba and the

RRR : given by 9$8min)$( baba are binary operations.

E' 4' how that RRR + : and RRR : are commutative binary operations$

but RRR : and RRR : are not commutative.

E'/. how that RRR :P defined by baba &P += is not commutative.

E'0. how that addition and multiplication are associative binary operation on R. =ut subtraction

is not associative on R. *ivision is not associative on :R .

E'7. how that RRR :P given by baba &P += is not associative.

'4


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E''. how that >ero is the identity for addition on + and ' is the identity for multiplication on +.

=ut there is not identity element for the operations

:R R R and :R R R

E'..' how that 2ais the inverse of aof for the addition operation QCR on Rand

a

'is the inverse

of a for the multiplication operation QR on R.

E'.' how that 2ais not the inverse of Na for the addition operation C on Nanda

'is not

the inverse of Na for multiplication operation on$ N$ for a '.Q' .' *etermine whether or not each of the definition of P given below given a binary operation. !n

the event that P is not a binary operation$ give #ustification for this.

(i) Sn +3 $ define P by baba =P

(ii) Sn +3 $ define P by abba =P

(iii) Sn R$ define P &P abba =

(iv) Sn +3 $ define P by baba =P

(v) Sn +3 $ define P by aba =P

Q' %or each binary operation P defined below$ determine whether P is commutative or

associative.

(i) Sn 3 $ define baba =P

(ii) Sn Q $ define 'P += abba

(iii) Sn Q $ define&

P ab

ba =

(iv) Sn +3 $ defineabba &P =

(v) Sn +3 $ definebaba =P

(vi) Sn 9'8R $ define'

P+

=b

aba

Q' 0' ;onsider the binary operation on the set 96$5$4$&$'8 defined by =ba min

9$8 ba . Frite the operation table of the operation Q' 1' ;onsider a binary operation P on the set 96$5$4$&$'8 given by the following

multiplication table (Table '.&).

(i) ;ompute P)4P&( 5 and )5P4(P&

(ii) !s P ;ommutativeH

;< ;ompute baP )6P5(P)$4$&( .

(Nint: use the following table)

Ta=)e .'

P ' & 4 5 6

' ' ' ' ' '

& ' & ' & '

'5


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4 ' ' 4 ' '

5 ' & ' 5 '

6 ' ' ' ' 6

Q' 2' Let P be the binary operation on the set 96$5$4$&$'8 defined by ba P " N.;.%. of

a and b. !s the operation P same as the operation Pdefined in eercise 5 aboveH

ustify your answer.

Q' 4' Let P be the binary operation on Ngiven by baP " L.;.


16/55


E'' Let + be a relation on the set A of ordered pairs of positive integers defined by

)$(+)$( y!yx if and only if y!x"= . how that + is an equivalence relation.

E'0' Let 97$0$/$1$6$5$4$&$'8I = . Let '+ be a relation in I given by

yxyx =

:$(8+'

is divisible by 49 and &+ be another relation on I given by 9$0$6$&89$8:)$(8+& = yxyx

or 997$1$489$8 yx . how that &' ++ = .

E'1' Let JI: f be a function. *efine a relation + in I given by9)()(:)$(8+ bfafba == . @amine if + is an equivalence relation.

E'2' *etermine which of the following binary operations on the set ? are associative and which

are commutative.

(a) N= baba $'P (b) N=+= bababa $&

)(P

E' 4' %ind the number of all one-one function from set 94$&$'8A = to itself.

E' 5' Let 94$&$'8A = . Then show that the number of relations containing )&$'( and )4$&(

which are refleive and transitive but not symmetric is four.

E'6' how that number of equivalence relation in the set 94$&$'8 containing )&$'( and

)'$&( is two.

E'7' how that the number of binary operation on 9&$'8 having ! as identity and having & as the

inverse of & is eactly one.

E'.8' ;onsider the identity function NN!IN defined as ?)( = xxxNI show that

although ?! is onto but NN+ :!! ?? defined as

xxxxxx &)(!)(!)()!!( ???? =+=+=+ is not onto.

E' ..' ;onsider a function +&

$:

f given by =)(xf sinxand R

&$:

g given

by .cos)( xxg = show thatfandgare one-one$ but gf+ is not one-one.

Q' .' Let RR:f be defined as /')( += xxf . find the function RR:g such that

R'== goffog .

Q'' Let ##f : be defined as ')( = nnf $ if nis odd and ')( +=nnf $ if nis even.

how thatfis invertible. %ind the inverse off. Nere$ F is the set of all whole numbers.

Q' 0' !f RR:f is defined by &4)( & += xxxf $ find ))(( xff .

'1


17/55


Q'1' how that the function 9'':8:

='if'

'if')(

x

xxxg ).

Q' 6' Eiven a non empty set I$ consider (I) which is the set of all subsets of I.*efine the relation + in (I) as follows :

%or subsets A$ = in (I)$ A+= if and only if =A . !s + an equivalence relation on (I)H

ustify your answer.

Q' 7' Eiven a non-empty set I$ consider the binary operation P : (I)(I)(I)

given by =A$=A=PA = in (I) $ where (I) is the power set of I.

how that I is the identity element for this operation and I is the only invertible element in

(I) with respect to the operation P.

Q' .8' %ind the number of all onto functions from the set 9....$$4$&$'8 n to itself.

Q' ..' Let 98 cb,a,= and 94$&$'8T = . %ind -'% of the following function % from to

T$ if it eists.

(i) 9)'$()$&$()$4$(8% cba= (ii)

9')(c$$')(b$)$&(8% a,=

Q' .' ;onsider the binary operation P : RRR and o : RRR defined as

baba =P and R= baaboa $$ . how that P is commutative but not

associative$ o is associative but not commutative. %urther$ show that R cba $$ $

).P()P()(P baobacoba = ,if it is so$ we say that the operation P distributes

over the operation o. *oes o distribute over PH ustify your answer.

Q' .0' Eiven a non-empty set I$ let P : (I)(I))I( be defined as

(I)=A$A)$-(==)-(A=PA = . how that the empty set is the

identity for the operation P and all the elements A of (I) are inverible with AA -' = .

(Nint : A)A()A( = and )AA)AA()AA( == $ .

'/


18/55


Q' .1' *efine a binary operation P on the set 9$6$5$4$&$'$8 as

++ero is the identity for this operation and

each element aof the sea is invertible with a1 being the inverse of a.Q' .2' Let 9$&$'$$'8A = $ 9&$$&$58= = and =A:$ gf be function

define by A$)( & = xxxxf and A$'&

'&)( = xxxg . Aref andg

equalH ustify your answer.

(Nint: one may not that two function =A: f and =A: g such that

A)()( = aagaf $ are called equal functions).

Q' .4' Let 94$&$'8A = . Than number of relations containing )&$'( and )4$'( which are

refleive and symmetric but not transitive is

(A) ' (=) & (;) 4 (*) 5

Q' .5' Let A " 8'$ &$ 49. Then number of equivalence relations containing ('$ &) is

(A) ' (=) & (;) 4 (*) 5

Q' .6' Let RR:f be the ignum %unction defined as

=$'

$

$'

)(

x

x

x

xf

and ++: g be the Ereatest !nteger %unction given by ,)( xxg = $ where -, x is

greatest integer less than or equal to x . Then$ does fog and gof coincide in ($ 'H

Q' .7' ?umber of binary operations on the set 8a$ b9 are

(A) ' (=) '1 (;) & (*) 0

TYPE 2 (EXTRA PRACTICE QUESTIONS)

.' Let 91$6$5$4$&$'8=A . Let + be the relation on A defined by

9bydivisibleeactlyis$$:)$8( abAbaba

(i) Frite + in roster form (ii) 91$6$5$4$&$'8=A

(iii) %ind the range of +.

' Let 96$5$4$&$'8=A and 91/.....$..........$4$&$'8=B . !f + be a relation from

the set A to the set = defined by (i) is square root of (ii) is cube root of$ find + and also its

domain and range.

0' Let + be the relation on 3defined by + " 9$$)$$8( && bababa = 3 . %ind (i) + (ii)

domain of + (iii) range of +.

1' *etermine the domain and range of the following relations on R:

'0


19/55


(i)


20/55


.' ;onsider the relation (less than or equal) on the set 3of integers. how that this relation isrefleive$ transitive but not symmetric.

.0' ;onsider the relation (perpendicular) on a set L of lines in a plane. show that this relation

is symmetric and neither refleive nor transitive.

.1' Let + be the relation on the set Rof real numbers defined by Rba )$( iff ' >+ab .

how that the relation + is refleive symmetric but not transitive.

.2' Test whether the following relations are refleive$ symmetric$ transitive :

(i) 'R on 3defined by ')$( Rba iff / ba

(ii) &R on Qdefined by &)$( Rba iff 5=ab .

(iii) 4R on Rdefined by 4)$( Rba iff 45 && =+ baba

(iv) Rdefined on 8'$ &$ 4$ 5$ 6$ 19 by Rba )$( iff '+=ab

.4' Eiven the relation 94)(&$')$('$&)$('$8+= on the set A " 8'$ &$ 49$ add the

minimum number of elements of AA to + so that the enlarged relation is refleive$symmetric and transitive.

.5' Let A " 8'$ &$ 49. Then show that the number of relations containing ('$ &) and (&$ 4) which

are refleive and transitive but not symmetric is four.

.6' Let A " 8'$ &$ 49. how that none of the following relations on A is an equivalence relation :

(i) 9)&$4()$4$&()$&(&$')$('$8+'=

(ii) 9)&$4()$4$'()$4$4()$&(&$')$('$8+& =(iii) 9)&$4()$4$&()$'$&()$&$'()$4$4()$&(&$')$('$8+4 =

.7' Let YXf : be a function. *efine a relation + on I given by

)9()(:)$8( bfafbaR == . @amine if + is an equivalence relation.

8' Let + be the relation of congruency on the set A of all triangles in a plane. how that the

relation + is an equivalence relation.

.' how that the relation + defined on the set A of all triangles as

9similar tois:)$8(&'&' %%%%R=

is an equivalence relation. ;onsider three right angle triangle '% with sides 4$ 5$ 6$ &% with

sides 6$ '&$ '4 and 4% with sides 1$ 0$ '. Fhich triangles among &'$ %% and 4% are

relatedH

' how that the relation + on the set A of points in a plane given by )$8( QPR = : distance of thepoint from the origin is same as the distance of the point G from the origin9$ is an

equivalence relation. %urther$ show that the set of all points related to a point )$(P is

the circle passing through P with origin as centre.

&


21/55


0' Let integer mbe related to another integer niff mis a multiple of n. how that this relation is

not an equivalence relation.

1' Let mbe a fied positive integer. Two integers aand bare said to be congruent modulo m$

written ba (mod m) if mdivides ba . how that hte relation of congruent modulo m is

an equivalence relation.

2' how that the relation + on the set A " 8'$ &$ 4$ 5$ 6) give by

9evenis:)$8( babaR = $ is an equivalence relation. show that all the elements of


=ut no element of 96$4$'8 is related to any element of 95$&8 .

4' how that the relation + on the set NN defined by )$()$( dcRba iff cbda +=+

is an equivalence relation.

5' how that the relation R on the set NN defined by )$()$( dcRba iff)()( dabccbad +=+ is an equivalence relation.

6' !fRis an equivalence relation on a set A then show that the inverse relation 'R of R on A

is also an equivalence relation.

7' !f + and are equivalence relations on a set A then show that the relation R on A is also

an equivalence relation.

08' how that the number of equivalence relations on the set 94$&$'8 containing ('$ &) and (&$

') is two.

0.' Let 95$4$&$'8=A . ;onsider the equivalence relation

)9'$&()$&$'()$5$5()$4$4()$&$&()$'$'8(=R on A.

%ind the equivalence classes of the elements of A.

0' Let 9&$'7......$$4$&$'8=A . Let R be the equivalence relation on AA defined

by )$()$( dcRba iff dcad= . %ind the equivalence classes of )4$'( and )'$5( .

00' Let R be the equivalence relation on the set 3defined by yRx iff )5(modyx .

%ind the equivalence classes of $ '$ & and 4. Also show that the union of these equivalence

classes is U.

01' Let 91$6$5$4$&$'8=A . ;onsider the equivalence relation

)$&$1()$6$6()$'$6()$5$5()$1$4()$4$4()$&$4()$1$&()$4$&()$&$&()$6$'()$'$'8(=R

%ind the equivalence classes of + and verify that :

(i) Aaaa everyfor, (ii) Rbaba = )$(iff,,

(iii) %or Aba $ $ either == ,,or,, baba

02' Let 95$4$&$'8=A and 9'6$''$0$1$'8=B . Fhich of the following are functions

fromBAto

H(i) BAf : defined by 0)5($0)4($1)&($')'( ==== ffff .

(ii) BAf : defined by '6)4($1)&($')'( === fff .

&'


22/55


(iii) BAf : defined by 1)5($1)4($1)&($1)'( ==== ffff .

(iv) BAf : defined by '')5(0)4($0)&($1)&($')'( ===== fffff .

(v) BAf : defined by '6)5($'')4($0)&($')'( ==== ffff .

04' !f&

)( xxf = $ find ''.')'()'.'(

ff

.

05' Fhich one of the following graphs represent the function of x H FhyH

06' The function t which maps temperature in ;elsius in temperature in %ahrenheit is defined by

4&6

7)( +=

cc& .

%ind (i) )(& (ii) )&0(& (iii) )'(& (iv). The value of c when &'&)( =c& .

07' Let RR:f be defined as

++

=55

5'&)(

xx

xxxf . how thatfis not a function.

18' !f RR:f is defined by &4)( & += xxxf $ find ))(( xff . Also evaluate ))6((ff .

1.' Let 9&$$&$589$&$'$$'8 == BA and BAgf :$$ be functions defined

by ( ) Axxxxf = $& and Axxxf = $'&

'&)( . Are f and g equalH ustify

your answer.

1' %ind the domain and range of the following functions :

(i)&

'

x(ii)

6+xx

(iii)&& x

x

+(iv) 4

/

x

x P

10' %ind the domain and range of the following functions :

(i) &+x (ii)5

'

x

(iii))'()4&(

'

+ xx (iv) )4()'( xx

11' !f ')( & += xxxf and /5)( = xxg be real functions then find :

(i) )&()( gf + (ii) )/()( gf (iii) )6()( gf (iv) )5(

g

f

12' Let f and g be real functions defined by 5$5)( += xxxf and

5$5)( = xxxg . %ind the functions gffggfgf $$$ + .

y

S

(i)

y

S

(ii)

&&


23/55


14' Let 95$4$&$'8=A and 90$/$5$'8=B . %ind which of the following functions from A

and = are '-' :

(i) BAf : defined as 0)5($5)4($5)&($')'( ==== ffff

(ii) BAf : defined as 0)5($')4($/)&($5)'( ==== ffff

15' ;onsider a function R&$,: f given by xxf sin)( = and R&$,: g

given by xxg cos)( = . how that f and g are one-one$ but gf+ is not one-one.

16' Fhich of the following functions are one-oneH

(i) RR:f $ defined by R= xxf $5)(

(ii) RR:f $ defined by R= xxxf $'1)(

(iii) RR:f $ defined by R+= xxxf $/)( &

(iv) RR:f $ defined by R= xxxf $)( 4

(v) RR 9/8: $ defined by 9/8$/

'&)(

+

= Rxx

xx .

17' Let 95$4$&$'8=A and 9/$5$48=B . find which of the following functions are onto :

(i) BAf : defined as 4)5($4)4($5)&($4)'( ==== ffff

(ii) BAf : defined as /)5($5)4($4)&($4)'( ==== ffff

28' Fhich of the following functions are ontoH

(i) RR:f defined as R+= xxxf $57''6)(

(ii) RR:f defined as R= xxxf $)(

(iii) RR:f defined as R= xxxf $)( &

(iv) RR:f defined as R+= xxxf $5)( &

2.' Let A be a finite set. !f AAf : is onto$ show that f is one-one.

2' Let 94$&$'894$&$'8: f be a function. how that :

(i) f is onto if f is one-one (ii) f is one-one if f is onto.

20' %ind the number of all onto functions from the set 9...$$4$&$'8 n to itself.

21' how that the signum function RR:f $ given by

=

if'

if

if'

)(

x

x

x

xf is neither

one-one nor onto.

22' rove that the Ereatest integer function RR:f $ given by ,)( xxf = is neither one-one

nor onto$ where ,x denotes the greatest integer less than or equal to x.

24' Let RR:f be a function defined by )4&(cos)( += xxf . how that this function is

neither one-one nor onto.

25' Let 948=RA and 9'8=RB . ;onsider the function BAf : defined by4&

xx . !s

f one-one and ontoH ustify your answer.

&4


24/55


26' !f RR:f be a function defined by /5)( 4 = xxf $ show that the function f is a

bi#ective function.

27' A function 3N:f is defined by

= evenisif&

oddisif&

'

)(n

n

nn

nf

how that this function is a bi#ection.

48' Let '$',=A and AAf : be a function defined by xxxf =)( . how that f

is a bi#ection.

4.' Let 94$&$'8=A and 91$6$58=B . BAf : is a function defined as

$6)&($5)'( == ff 1)4( =f . Frite down 'f as a set of ordered pairs.

4' tate with reason whether following functions have inverse :(i) )9'$5()$'$4()$'$&()$'$'8(with9'895$4$&$'8: = ff

(ii) )9&$0()$5$/()$4$1()$5$68(with95$4$&$'890$/$1$68: = gg

(iii)

)9'4$6()$''$5()$7$4()$/$&8(with9'4$''$7$/896$5$4$&8: = hh

40' %ind the inverse of the function R= xxxf $/5)( .

41' how that the function RR:f defined by4

'&)( = x

xf $ Rx is one-one and

onto function. Also$ find the inverse of the functionf.

42' Let

6

&

6

4: RRf be a function defined as

46

&)(

+=

x

xxf . %ind the inverse of

the function f .

44' !f { } { }6464: RRf be a function defined by46

&4)(

+

=x

xxf $ { }64Rx .

how that )()(' xfxf = $ { }64Rx .

45' Let NN = 9:8 & nnY . ;onsider Yf N: as &)( nnf = . how that f is

invertible. %ind the inverse of f .

46' how that the function NN:f defined by N++= xxxxf $')( & is not invertible.

47' ;onsider )$6,: +Rf given by 617)( & += xxxf . how that f is invertible.

%ind the inverse of f .

58' Let 9898: NNf be defined by +

=oddisif'-n

evenisif')(

n

nnnf . how that f

is invertible and'= ff .

5.' Let 96$5$4$&$'8=A and let AAf : and AAg : be defined as

')6($6)5($5)4($4)&($&)'( ===== fffff

4)6($&)5($')4($')&($5)'( ===== ggggg

&5


25/55


%ind the graphs of functionsfogandgof.

5' Let gf$ be real valued functions defined as R++= xxxxf $/)( & and

R= xxxg $46)( . %indfogandgof. Also find )&()(fog and )'()(gof .

50' !f f be the greatest integer function andgbe the absolute value function. %ind the value of

+

4

5)(

&

4)( goffog .

51' Let gf$ be real valued functions defined as :

++


26/55


(v) P on 3defined by 4P += baba Pfor ePvery 3ba$

(vi) P on Qdefined byba

baba

+=P Pfor ePvery Qba$

60' Let 91$6$5$4$&$'8=A and P be an operationAdefined by rba =P $ where r is the

least non-negative remainder when the product ab is divided by / . how that P is a binary

operation on A .

61' Let P be a binary operation on 3.

(i) %ind 6 P 4 if 3+= bababa $M)(P &4& .

(ii) %ind '6 P (2 '5) if 3= bababa $M&P .

(iii) %ind P 6 if 3+= bababa $M)&(P 5 .

(iv) %ind '7 P'1 if 3= baabba $M&P .

62' Let P be a binary operation on the set Rdefined by R++= baabbaba $$P . olve the

equation : /)P4(P& =x

64' Let 9$8 baA = . %ind the number of binary operations on A .

65' Let P be the binary operation addition on the set of integers. how that :

(i) (3$ P) is commutative (ii) (3$ P) is associative

66' ;onsider the binary operation P on Qdefined by abbaba ++= '&P for Qba$ .

67' Let A be the set of all subsets of a non-empty set . how that the binary operation union

on A is left distributive over the binary operation intersection on A . Also show that

intersection is left distributive over union.

78' Let P be a binary operation on N defined by N= baba ab $M&P 6 . *iscuss the

commutativity and associativity of this binary operation.

7.' Let P be a binary operation on NN defined by )$()$(P)$( bdbcaddcba += for

NN)$()$$( dcba . *iscuss the commutativity and associativity of this binary

operation.

7' Let A be the set of all functions from a non-empty set to itself. *iscuss the

commutativity and associativity of the binary operation composition of functions on the set

A .

70' ;onsider the binary operation RRR!: and RRR :o defined as

baba =P and R= baaboa $$ . how that P is commutative but not

associative. o is associative but not commutative. %urther$ show that

)P()P()(P$$$ baobacobacba =R . ,!f it is so$ we say that the operation P

distributes over the operation o *oes odistribute over PH ustify your answer.

71' Let P be the binary operation addition on the set of integers. how that :

(i) is identity element of (3$ P)

(ii) !nverse of the element eist in (3$ P)

&1


27/55


28/55


.80' Let 9$$$8 54&' ffffA = be teh set of four functions from 8R onto itself and defined as

follows:

xxfxxf

xxfxxf

')($)($

')($)(

54&' ==== $ where 8Rx .

*raw the composition table corresponding to the binary operation composition of functions.

.81' Three +elations &' $ RR and 4R are defined on set 9$$8 cbaA = as follows :

(i) 9)$()$$()$$()$$()$$()$$()$$()$$8(' ccbcaccbbbcabaaaR =

(ii) )9$()$$()$$()$$8(& accaabbaR =

(iii) )9$()$$()$$8(4 accbbaR =

%ind whether each of 'R $ &R and 4R is refleive$ symmetric and transitive.

.82' how that the relation R on the set 94$&$'8=A given by

)94$&()$&$'()$4$4()$&$&()$'$'8(=R is refleive but neither symmetric nor

transitive.

.84' how that the relation R on the set 94$&$'8=A given by )9'$&()$&$'8(=R is

symmetric but neither refleive nor transitive.

.85' ;hec the following relations R and for refleivity$ symmetry and transitivity :

(i) aRb iff b is divisible by Nbaa $M

(ii) &' '' iff &' '' $ where '' and &' are straight lines in a plane.

.86' Let a relation 'R on the set R of real numbers be defined as ')$( ' >+ abRba for

all Rba $ . how that 'R is refleive and symmetric but not transitive.

.87' Let 97$0$/$1$6$5$4$&$'8=X . Let 'R be a relation on I given by

94bydivisibleis:)$8('

yxyxR = and &R be another relation on I given by

97$1$489$890$6$&89$89/$5$'89$8:)$8(& = yxoryxoryxyxR

. how that

&' RR = .

..8' *etermine whether each of the following relations are refleive$ symmetric and transitive :

(i) +elation R on the set 9'5$'4...$$4$&$'8=A defined as

94:)$8( == yxyxR

(ii) +elation R on the set N of all natural number defined as

95and6:)$8(


29/55


9integeranis:)$8( yxyxR =

...' how that the relation R on the set R of all real numbers$ defined as

9:)$8( &babaR = is neither refleive nor symmetric not transitive.

..' how that the relation R on Rdefined as9:)$8( babaR

= $ is refleive and transitivebut not symmetric.

..0' Let 94$&$'8=A . Then$ show that the number of relations containing )&$'( and )4$&(

which are refleive and transitive but not symmetric is four.

..1' Let be the set of all points in a plane and R be a relation on defined as

9units&thanlessisGandbetween*istance:)$8( QPR= .

how that R is refleive and symmetric but not transitive.

..2' Let R be a relation on the set of all lines in a plane defined by

&'&' linetoparallelisline)$( ''R'' .

how that R is an equivalence relation.

..4' how that the relation is congruent to on the set of all triangles in a plane is an equivalence

relation.

..5' how that the relation R defined on the set A of all triangles in a plane as

9similar tois:)$8(&'&'

%%%%R= is an equivalence relation.

;onsider three right angle triangles '% with sides &M6$5$4 % with sides '4$'&$6 and

4% with sides '$0$1 . Fhich triangles among &'$ %% and 4% are relatedH

..6' rove that the relation R on the set ( of all integers numbers defined by

yxRyx )$( is divisible by nis an equivalence relation on U.

..7' how that the relation + on the set A of all the boos in a library of a college$ given by

9pagesofnumbersame$thehaveand:)$8( yxyxR= $ is an equivalent relation.

.8' how that the relation R on the set 96$5$4$&$'8=A $ given by

9evenis:)$8( babaR = $ is an equivalent relation. how that all the elements of


=ut no element of 96$4$'8 is related to any element of 95$&8 .

..' how that the relation R on the set 9'&:8 = x(xA $ given by

95ofmultipleais:)$8( babaR = is an equivalence relation. %ind the set of all

elements related to '.

.' how that the relation R on the set A or points in a plane$ given by :)$8( QPR = *istance ofthe point from the origin is same as the distance of the point G from the origin9$ is an

equivalence relation.

&7


30/55


31/55


.07' Let 9&8= RA and ',= RB . !f BAf : is a mapping defined by&

')(

=x

xxf $

show that f is bi#ective.

.18' Let A and Bbe two sets. how that ABBAf : defined by )$()$( abbaf = is

a bi#ection.

.1.' Let A be any non-empty set. Then$ prove that the identity function on set A is a bi#ection.

.1' how that the function RRf : given by xxxf += 4)( is a bi#ection.

.10' how that NNf : defined by

+

=evenisif$

&

n

oddisif$&

'

)(

n

nn

xf is many-one onto

function.

.11' how that the function NNf : given by nnnf )'()( = for all Nn . i s a

bi#ection.

.12' Let NNf 9'8: be defined by =)(nf the highest prime factor of n. how that f is

neither one-one nor onto. %ind the range of f .

.14' Let 9898: NNf be defined by

+

=oddisif'

evenisif$')(

nn

nnnf . how that f is

a bi#ection.

.15' Let A be a finite set. !f AAf : is a one-one function$ show that f is onto also.

.16' Let A be a finite set. !f AAf : is an onto function$ show that f is one-one also.

.17' Let 9&$'8=A . %ind all one-to-one functions fromAtoA.

.28' ;onsider the identity function NNIN : defined as xxIN =)( for all Nx . how that

although NI is onto but NNII NN + : defined as

xxxxIxIxIINNNN

&)()()()( =+=+=+ is not onto.

.2.' ;onsider the function Rf &$,: given by xxf sin)( = and Rg &$,:

given by xxg cos)( = . how that f and g are one-one$ but gf+ is not one-one..2' Let YXf : be a function. *efine a relation R on X given by

)9()(:)$8( bfafbaR == . how that R is an equivalence relation on X .

.20' Let + be the set of real numbers. !f&

)(M: xxfRRf = and '&)(M: += xxgRRg

. Then$ find fog and gof . Also$ show that goffog .

.21' Let xxfRRf sin)(M: = and &)(M: xxgRRg = find fog and gof .

.22' Let 97$6$5$4896$5$4$&8: f and 9'6$''$/897$6$5$48: g be functions

defined as 6)6()5($5)4($4)&( ==== ffff and'')7()6(and/)5()4( ==== gggg . %ind gof .

4'


32/55


.24' Let 96$&$'895$4$'8: f and 94$'896$&$'8: g be given by

)9'$5()$6$4()$&$'8(=f and )9'$6()$4$&()$4$'8(=g . Frite down .gof

.25' %ind gof and fog $ if RRf : and RRg : are given by xxf =)( and

&6)( = xxg

..26' !f the functions f and g are given by )9'$5()$6$4()$&$'8(=f and

)94$'()$'$6()$4$&8(=g $ find range of f and g . Also$ write down fog and gof

as sets of ordered pairs.

.27' !f the function RRf : be given by &)( & += xxf and RRg : be given by

')(

=x

xxg . %ind fog and gof .

.48' !f 64

6

/: RRf be defined as /6

54)(

+= x

xxf and 6

/

6

4:Rg be

defined as46

5/)(

+=

x

xxg . how that AIgof = and BIfog= $ where

=

6

4RB and

=

6

/RA .

.4.' !f RRf : is defined by &4)( & += xxxf $ find ( ))(xff .

.4' !f RRgf :$ are defined respectively by 4&)($'4)( &

=++= xxgxxxf $ find (i)fog (ii) gof (iii) fof (iv) gog .

.40' Let 9':8 = xRxA . !f AAf : is defined by

=Qxx

Qxxxf

if'

if$)( then

prove that xxfof =)( for all Ax .

.41' Let RRf : and RRg : be two functions such that xxfog &sin)( = and

xxgof &sin)( = . The$ find )(xf and )(xg .

.42' !f RRf : be given by )4cos(cos)4(sinsin)( &&

++++= xxxxxf for allRx $ and RRg : be such that ')56( =g $ then prove that RRgof : is a

constant function.

.44' Let ((f : be defined by nnf 4)( = for all (n and ((g : be defined by

=4ofmultiplenotisif$

4ofmultipleaisif$4)(

n

nn

ng for all (n . how that (Igof = and (Ifog

.

.45' Let RRf : be a function given by baxxf +=)( for all Rx . %ind the constants a

and b such that RIfof= .

4&


33/55


.46' Let ((f : be defined by &)( += xxf . %ind ((g : such that (Igof = .

.47' !f ((f : be defined by xxf &)( = for all (x . %ind ((g : such that

(Igof = .

.58' Let AAf : be a function such that ffof= . how that f is onto if and only if f isone-one. *escribe f in this case.

.5.' Let gf$ and h be functions from R to R . how that :

(i) gohfohohgf +=+ )( (ii) )()()( gohfohohfg =

.5' Let RRf : be the signum function defined as

=$'

$

$'

)(

x

x

x

xf and RRg : be

the greatest integer function given by ,)( xxg = . then$ prove that fog and gof coincide

in )$', .

.50' !f RRf : and RRg : be functions defined by ')( & += xxf and xxg sin)( = $

them find fog and gof .

.51' !f Rf )$,: and RRg : be defined as xxf =)( and ')( & = xxg $ then

find gof and fog .

.52' !fxexf =)( and )(log)( >= xxxg e $ find fog and gof $ !s goffog= H

.54' !f )()( = xxxf and ')( & = xxg are two real functions$ find fog and gof . !s

goffog= H

.55' !fx

xf ')( = and )( =xg are two real functions$ show that fog is not defined.

.56' Let ,)( xxf = and xxg =)( $ %ind

(i)

4

6)(

4

6)( foggof (ii)

4

6)(

4

6)( foggof (iii) )'()&( + gf

.57' Let f and g be real functions defined by'

)(+

=x

xxf and

4

')(

+=x

xg *escribe the

functions gof and fog (if they eist).

.68' !f4&

&4)(

=x

xxf $ prove that xxff =))(( for all

&

4Rx .

.6.' !f&

'$

'&

')(

+= x

xxf $ then show that

4&

'&))((

++

=x

xxff $ provided that

&

4$

&

'x .

.6' Let &')( x

x

xf += . Then$ show that &4')()( x

x

xfofof +=

44


34/55


.60' Let f be a real function defined by ')( = xxf . %ind ).()( xfofof Also$ show that

&ffof .

.61' !f '$'

')(

+

= xx

xxf $ then show that ( )

xxff

')( = provided that '$ x .

.62' !f QQf : is given by $)( &xxf = then find

(i) )7('f (ii) )6(

' f (iii) )('f

.64' !f the function RRf : be defined by 76)( & ++= xxxf $ find )0('f and )7('f .

.65' !f the function CCf : be defined by ')( & = xxf $ find )0()$6( '' ff .

.66' Let RRf : be defined as ')( & += xxf . %ind :

(i) )6(' f (ii) )&1('f (iii) 94/$'8'f

.67' !f90$1$5$&89$5$4$&$'8 == BA

andBAf :

is given byxxf &)( =

$ then writef and

'f as a set of ordered pairs.

.78' Let 94$&$'8= . *etermine whether the function f : defined as below have

inverse. %ind'f $ if it eists.

(i) )94$4()$&$&()$'$'8(=f (ii) )9'$4()$'$&()$&$'8(=f

(iii) )9'$&()$&$4()$4$'8(=f

.7.' ;onsider 9$$894$&$'8: cbaf given by bfaf == )&($)'( and cf =)4( . %ind the

inverse

''

)(

f of

'

f . how that ff = ''

)( .

.7' !f RRf : is defined by /&)( += xxf . rove that f is a bi#ection. Also$ find the

inverse of f .

.70' !f RRf : is a bi#ection given by 4)( 4 += xxf find )(' xf .

.71' Let RRf : be defined by /4)( = xxf . how that f is invertible and hence find

'f .

.72' how that 9898: RRf given byx

xf 4)( = is invertible and it is inverse of itself.

.74' Let 9898: NNf be defined by

+

=ddoisif'

evenisif$')(

nn

nnxf

how that f is invertible and'= ff .

.75' rove that the functions RRf : defined as 4&)( = xxf is invertible. Also$ find 'f .

.76' how that the function RRf : is given by ')( & += xxf is not invertible.

.77' how that 9'89'8: RRf given by'

)(+

=x

xxf is invertible. Also$ find 'f .

88' !f the function )$',)$',: f defined by)'(

&)( = xx

xf is invertible$ find )('

xf .

8.' %ind the value of parameter for which the function $')( += xxf is the inverse of

itself.

45


35/55


8' Let YNf : be a function defined as 45)( += xxf $ where

9somefor45:8: NxxyNyY += . how that f is invertible. %ind its inverse.

80' Let NNnnY = 9:8 & ;onsider YNf : given by &)( nnf = . how that f is

invertible. %ind the inverse of f .

81' Let RNf : be a function defined as '6'&5)( & ++= xxxf . how that Nf :

+ange )(f is invertible. %ind the inverse of f .

82' how that Rf '$',: $ given by&

)(+

=x

xxf is one-one. %ind the inverse of the

function )(+ange'$',: ff .

84' Let 9$:&8 (baba += . Then$ prove that an operation P on defined by

)(&)()&(P)&( &'&'&&'' bbaababa +++=++ for all (baba &&'' $$ is a binary

operation on .

85' Let 95$4$&$'8= and P be an operation on defined by rba =P $ where r is the

least non-negative remainder when product is divided by 6. rove that P is a binary operation

on .

86' Let 95$4$&$'$8= and P be an operation on defined by rba =P $ where r is the

least non-negative remainder when ba+ is divided by 6. rove that P is a binary operation

on .

87' how that the operation and on + defined asba "


36/55


.2' *iscuss the commutativity and associativity of the binary operation P on + defined by

5P

abba = for all Rba $ .

.4' *iscuss the commutativity and associativity of binary operation P defined on Q by the rule

abbaba +=P for all Qba $ .

.5' Let A be a non-empty set and be the set of all functions from A to itself. rove that the

composition of functions o is a non-commutative binary operation on . Also$ prove that o

is an associative binary operation on .

.6' Let NNA = and P be a binary operation on A defined by )$()$(P)$( bdacdcba =

for all Ndcba $$$ . how that P is commutative and associative binary operation on

A.

.7' Let A be a set having more than one element. Let P be a binary operation on A defined by

aba =P for all Aba $ . !s P commutative or associative on AH

8' Let P be a binary operation on ?$ the set of natural numbers$ defined by baba =P for all

Nba $ . !s P associative or commutative on ?H

.' Let P be a binary operation on ? given by Nbaba*C+ba = $)$$(P

(i) %ind : 6P/$&5P'0$5P'&

(ii) ;hec the commutativity and associativity of P on ?.

' ;onsider the binary operations RRR P:P and RRRo : defined as

baba =P and aaob= for all Rba $ . how that P is commutative but not

associative$ o is associative but not commutative. %urther show that P is distributive over o

. *oes o distribute over PH ustify your answer.

0' !f P is defined on the set + of real numbers by/

4P

abba = $ find the identity elements in +

for the binary operation P.

1' %ind the identity element in the set+Q of all positive rational numbers for the operation P

defined by&

P ab

ba = for all +Qba$ .

2' !f P is defined on the set + of all real numbers by &&P baba += $ find the identity element

in R with respect to P.

4' Let be a non-empty set and )(P be the power set of set . %ind the identity element for

the union )( as a binary operation on )(P .

5' !n above eample find the identity element for intersection )( as a binary operation on

)(P .

41


37/55


6' Sn G$ the set of all rational numbers$ a binary operation P is defined by6

P ab

ba = for all

Qba $ . %ind the identity element for P in G. Also$ prove that every non->ero element of G

is invertible.

7' Let P be a binary operation on set 9'8Q defined by 9'8$MP += Qbaabbaba .

%ind the identity element with respect to P on G. Also$ prove that every element of 9'8Q

is invertible.

08' Sn the set 9'8R a binary operation P is defined by abbaba ++=P for all

9'8$ Rba . rove that P is commutative as well as associative on 9'8R . %ind the

identity element and prove that every element n 9'8R is invertible.

0.' Let P be a binary operation on Q (set of all non->ero rational numbers) defined by

$

5P Qba

abba = Then$ find the

(i) identity element in Q (ii) inverse of an element in Q

0' Let I be a non-empty set and let P be a binary operation on )(XP (the power set of set I)

defined by BABA =P for all )($ XPBA . rove that P is both commutative and

associative on )(XP . %ind the identity element with respect to P on )(XP . Also show

that )(XP is the only invertible element of )(XP .

00' Let I be a non-empty set and let P be a binary operation on )(XP (the power set of I)

defined by BABA =P for )($ XPBA

(i) %ind the identity element with respect to P in )(XP .

(ii) how that I is the only invertible element of )(XP .

01' Let I be a non-empty set and let P be a binary operation on )(XP (the powers set of set I)

defined by )()(P ABBABA = for all )($ XPBA . how that :

(i) is the identity element for P on )(XP .

(ii) A is invertible for all )(XPA and the inverse of A is A itself.02' Let QQA = and let P be a binary operation on A defined by

)$()$(P)$( adbacdcba += for Adcba )$()$$( . Then$ with respect to P on A

(i) %ind the identity element in A

(ii) %ind the invertible elements of A

04' Let 9898 = NNA and let P be a binary operation on A defined by

)$()$(P)$ dbcadcba ++= for all Adcba )$()$$( . how that

(i) P is commutative on A

(ii) P is associative on A.

Also find the identity element$ if any$ in A

4/


38/55


05' Let NNA = $ and let P be a binary operation on A defined by

)$()$(P)$( bdbcaddcba += for all NNdcba )$()$$( $ how that

(i) P is commutative on A.

(ii) P is associative on A.

(iii) A has no identity element.

06' Let P be a binary operation on ? given by )$(...P ba)Cba = for all Nba $

(i) %ind '1P&$/P6 (ii) !s P commutativeH

(iii) !s P associative (iv) %ind the identity element in ?

;v< Fhich elements of ? are invertibleH %ind them.

07' *efine a binary operation P on the set 96$5$4$'$8=A given by )1(modP abba = .

how that ' is the identity for P$ ' and 6 are the only invertible elements with '' ' = and

66 ' = .

18' Sn the set

== Rx

xx

xxxA) :)( of && matrices$ find the identity element for

the multiplication of matrices as a binary operation. Also$ find the inverse of an element of


39/55


16' ;onsider the infimum binary operation on the set 96$5$4$&$'8= defined by

= ba


40/55


(c) refleive$ not symmetric and transitive

(d) not refleive$ symmetric and transitive

6. Let ( ) ( ) ( ) ( ) ( )8 4$4 $ 1$1 $ 7$7 $ '&$'& $ 1$'& $R= ( ) ( ) ( )4$7 $ 4$'& $ 4$1 9be a relation on the set

84$1$7$'&9.A= The relation is ,A!@@@ &6(a) refleive and symmetric only

(b) an equivalence relation

(c) refleive only

(d) refleive and transitive only

1. Let ( ): '$' $f B be a function defined by ( ) '&

&tan $

'

xf x

x

= thenf is both one-one and

onto when = is in the interval ,A!@@@ &6

(a) $& &

(b) $& &

(c) $

&

(d) $&

/. A real valued functionf-x.satisfies the %unctional equation

( ) ( ) ( ) ( ) ( )f x y f x f y f a x f a y = +

Fhere a is given constant and ( ) '$f = ( )&f a x is equal to ,A!@@@ &6

(a) ( )f x (b) ( ) ( )f a f a x+ (c) ( )f x (d) ( )f x

0. Let ( ) ( ) ( ) ( ) ( )8 '$4 $ 5$& $ &$5 $ &$4 $ 4$' 9R= be a +elation on the set 8'$&$4$59.A= The relation +is ,A!@@@ &5

(a) a function (b) transitive (c) not symmetric (d) refleive

7. The graph of the function ( )y f x= is ymmetrical about the line &$x= then ,A!@@@ &5

(a) ( ) ( )& &f x f x+ = (b) ( ) ( )& &f x f x+ =

(c) ( ) ( )f x f x= (d) ( ) ( )f x f x=

'. The domain of the function$ ( ) ( )'

&

sin 4

7

xf x

x

=

is ,A!@@@ &5

(a) ,&$ 4 (b) ,&$ 4) (c) ,'$ & (d) ,'$ &)

''. A function f from the set of natural numbers to !ntegers defined by

( )

'$

&

$&

n

f nn

=

,A!@@@ &4

(a) ne-one but not onto

(b) Snto but not one-one

5

Fhen n is even

Fhen n is odd


41/55


(c) Sne-one and onto both

(d) ?either one-one nor onto

'&. !f :f R R satisfies ( ) ( ) ( )$f x y f x f y+ = + %or all $x y R and ( )' /$f = then ( )'

n

f rr= is

,A!@@@ &4

(a)/

&

n (b)

( )/ '&

n+ (b) ( )/ 'n n+ (d) ( )/ '

&

n n+

5'


42/55


'4. *omain of definition of the function ( ) ( )4'&4

log $5

f x x xx

= +

is ,A!@@@ &4

(a) ('$ &) (b) ( ) ( )'$ '$&

(c) ( ) ( )'$& &$ (d) ( ) ( ) ( )'$ '$& &$ '5. The function ( ) &log( ')$f x x x= + + is ,A!@@@ &4

(a) an even function (b) an odd function

(c) a periodic function (d) neither an even nor an odd function

'6. The period of &sin is ,A!@@@ &&

(a) & (b) (c) & (d)&

'1. The domain of'

4sin log

4x

is ,A!@@@ &&

(a) ,'$ 7 (b) ,V'$ 7 (c) ,V7$ ' (d) ,V7$V'

'/. The period of the function ( ) 5 5sin cosf x x x= + is ,A!@@@ &&

(a) (b)&

(c) & (d) ?one of these

'0. The domain of definition of the function ( ) &

'

6log

5

x xf x

=

is ,A!@@@ &&

(a) ,'$ 5 (b) ,'$ (c) ,$ 6 (d) ,6$

'7. Let ( ) ' .f x x= Then: ,!!T '704M '


43/55


(b) ( ) 'sinf xx

= for ( )$ x f =

(c) ( ) cosf x x x=

(d) none of these

&4. %or realx$ the function( ) ( )x a x b

x c

will assume all real values provided: ,!!T '705M 4 > (b) a b c< < (c) a c b> < (d) a c b

&5. !f ( )( ) sing f x x= and ( )( ) ( )&

sin $f g x x= then: ,!!T '770M & (d) logex x>

4. Let ( ): $f R and ( ) ( )

.x

+ x f & d&= !f ( ) ( )& & ' $+ x x x= + then ( )5f equals: ,!!T &

(a) 635 (b) / (c) 5 (d) &

4'. Let ( ) [ ]'g x x x= + and ( )'$

$ $

'$

x

f x x

x

then for all ( )$x f g x is equal to: ,!!T &'

54


44/55


(a) x (b) ' (c) ( )f x (d) ( )g x

4&. !f [ ) [ ): '$ &$f is given by ( ) ' $f x xx

= + then ( )'f x equals: ,!!T &'

(a)&

5&

x x+ (b) &'x

x+ (c)

&

5&

x x (d) &' 5x+

44. The domain of definition of ( ) ( )&

&

log 4

4 &

xf x

x x

+=

+ +is: ,!!T &'

(a) { }3 '$ &R (b) ( )&$ (c) { }3 '$ &$ 4R (d) ( ) { }4$ 3 '$ &

45. Let ( ) ( )& &' & 'f x b x bx= + + + and let m-b.be the minimum value of ( ) .f x As b varies$ therange of m-b.is: ,!!T &'

(a) ,$ ' (b)'$&

(c)

' $'&

(d) ( ]$'

46. Let { }'$&$4$5/= and { }'$ & .+= Then the number of onto function from @ to % is : ,!!T &'

(a) '5 (b) '1 (c) '& (c) 0

41. Let ( ) $ '.'

xf x x

x

=

+Then$ for what value of is ( ) H :f f x x= ,!!T &'

(a) & (b) & (c) ' (d) '

4/. uppose ( ) ( )&'f x x= + for '.x !f ( )g x is the function whose graph is reflection of the

graph of ( )f x with respect to the liney = x$ then ( )g x equals: ,!!T &&

(a) '$ x x (b) ( )&'

$ ''

xx

> +

(c) '$ 'x x+ (d) '$ x x

40. Let function :f R R be defined by ( ) & sinf x x x= + for .x R Thenf is : ,!!T &&

(a) one-to -one and onto

(b) one-to-one but not onto(c) onto but not one-to-one

(d) neither one-to-one nor onto

47. !f [ ) [ ): $ $f and ( ) $'

xf x

x=

+then f is: ,!!T &4

(a) one-one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) neither one-one nor onto

5. +ange of the function ( ) &

&

&M

'

x xf x x R

x x

+ +=

+ +is: ,!!T &4

55


45/55


(a) ( )'$ (b) ( )'$''3 / (c) ( )'$/ 3 4 (d) ( )'$/ 3 6

5'. *omain of definition of the function ( ) ( )'sin &1

f x x = + for real valuedx$ is ,!!T &4

(a) ' '$5 &

(b) ' '$

& &

(c) ' '$& 7

(d) ' '$5 5

5&. !f ( ) ( ) &sin cos $ '$f x x x g x x= + = then ( )( )g f x is invertible in the domain: ,!!T &5

(a) $&

(b) $5 5

(c) $

& &

(d) [ ]$

54. ( )$

xf x

=

( )$

$g x

x

=

Then f g is: ,!!T&6

(a) one-one and into (b) neither one-one nor onto

(c) many one and onto (d) one-one and onto

55. !f I and J are two non-empty sets where :f X Y is function is defined such that

( ) ( )8 : 9f c f x x C= for C X and ( ) ( ){ }'

:f D x f x D = for $D Y for any

A Y and B Y then : ,!!T&6

(a) ( )( )'f f A A = (b) ( )( )'f f A A = only if ( )f X Y=

(c) ( )( )'f f B B = only if ( )B f x (d) ( )( )'f f B B =

56

ifxis rational

ifxis irrational

ifxis irrational

ifxis rational


46/55


ANS>ERS

TYPE 1 (NCERT QUESTIONS)

E,ERCISE .'.

Q' .' ;< ?either refleive nor symmetric nor transitive.

;< ?either refleive nor symmetric nor transitive.

(iii) +efleive and transitive but not symmetric.

(iv) +efleive$ transitive and symmetric.

(v) ;a< +efleive$ transitive and symmetric.

;=< +efleive$ transitive and symmetric.

;c< ?either refleive nor symmetric nor transitive.

;$< ?either refleive nor symmetric nor transitive.

;e< ?either refleive nor symmetric nor transitive.

Q' 0' ?either refleive nor symmetric nor transitive.

Q' 2' ?either refleive nor symmetric nor transitive.

Q' 7' ;< 8'$ 6$ 79$ ;< 8'9 Q' .' '% is related to 4%

Q' .0' The set of all triangles Q' .1' The set of all lines

R+= ccxy $&

Q' .2' = Q' .4' ;

E,ERCISE .'

Q' .' ?o

Q' ' ;< !n#ective but not sur#ective ;< ?either in#ective nor sur#ective

;


47/55


48/55


;< +ange of 91$5$4$&$'89)$(:8 == RbabR

&. ;< 9)&6$6()$'1$5()$7$4()$5$&()$'$'(8=R $ *omain of 96$5$4$&$'8=R

and +ange of 9&6$'1$7$5$'8=R

;< 9)15$5()$&/$4()$0$&()$'$'(8=R 9 *omain of 95$4$&$'8=R and +ange of

915$&/$0$'8=R

0' ;< )($ aaRaRa for 3a 9 9)$$8(9:)$8( 33 = aaaaaaR

;< *omain of 33 == 9)$(:8 baaR

;< +ange of 33 == 9)$(:8 babR

1' ;< *omain of 96$5$4$&$'8'=R $ +ange of 96'$5'$4'$&'$'8' =R

;< *omain of 9.......$0$1$5$&8& =R $ +ange of

9..........$/'$54$&4$''8& =R4' &61

5' 9)6$'()$5$4()$4$6()$&$/()$'$7(8=R

*omain of 9'$4$6$/$78=R $ +ange of 96$5$4$&$'8=R

9)'$6()$4$5()$6$4()$/$&()$7$'(8' =R

6' 'R and 5R 7' &R and 5R .8' 4R

..' ;< 'R is refleive$ 'R is not symmetric$ 'R is not transitive

;< &R is not refleive$ &R is symmetric$ &R is not transitive

;< 4R is not refleive$ 4R is not symmetric$ 4R is not transitive

;v< 5R is refleive$ 5R is symmetric$ 5R is transitive

.2' ;< 'R is refleive$ 'R is symmetric$ 'R is not transitive

;< &R is not refleive$ &R is symmetric$ &R is not transitive

;< 4R is not refleive$ 4R is not symmetric$ 4R is not transitive

;v< 5R is not refleive$ 5R is not symmetric$ 5R is not transitive

.4' 8('$ ')$ (&$ &)$ (4$ 4)$ ('$ &)$ (&$ 4)$ (&$ ')$ (4$ &)$ ('$ 4)$ (4$ ')9

.5' 8('$ ')$ (&$ &)$ (4$ 4)$ ('$ &)$ (&$ 4)$ ('$ 4)9

8('$ ')$ (&$ &)$ (4$ 4)$ ('$ &)$ (&$ 4)$ ('$ 4)$ (&$ ')9

8('$ ')$ (&$ &)$ (4$ 4)$ ('$ &)$ (&$ 4)$ ('$ 4)$ (4$ &)9

8('$ ')$ (&$ &)$ (4$ 4)$ ('$ &)$ (&$ 4)$ ('$ 4)$ (4$ ')9

.7' + is equivalence relation

0.' 9&$'89)'$(:8', == RxAx

9&$'89)&$(:8&, == RxAx

9489)4$(:84, == RxAx

9589)5$(:85, == RxAx

0' @quivalence of class of ('$ 4) " 8('$ 4)$ (&$ 1)$ (4$ 7)$ (5$ '&)$ (6$ '6)$ (1$ '0)9$

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@quivalence of class of (5$ ') " 8(5$ ')$ (0$ &)$ ('&$ 4)$ ('1$ 5)$ (&$ 6)9

00' @quivalence of class of " 8......$ 2 0$ 2 5$ $ 5$ 0$ .......9

@quivalence of class of ' " 8......$ 2 /$ 2 4$ '$ 6$ 7$ .......9

@quivalence of class of & " 8......$ 2 1$ 2 &$ &$ 1$ '$ .......9

@quivalence of class of 4 " 8......$ 2 6$ 2 '$ 4$ /$ ''$ .......9

01' 8'$ 69$ 8&$ 4$ 19$ 8&$ 4$ 19$ 859$ 8'$ 69$ 8&$ 4$ 19

02' ;< fis a function ;< fis not a function

;< fis a function ;v< fis not a function

;v< fis a function

04' &.'

05' ;< represents a function ofx ;< does not represent a function ofx

06 ;< 4& ;< 0&.5 ;< '5 ;v< '

18' xxxx 4'1 &45 + $ '' 1.' fandgare equal functions

1' ;< 9&8)( = RfD $ 98)( = RfR ;< 968)( = RfD $

9'8)( =RfR

;< R=)(fD $

=

&&

'$

&&

')(fR ;v< 96$5$48)( =fD $

94$&$'8)( =fR

10' ;< )$&,)( =fD $

9$,)( =fR;< )$5,)( =fD $

9$,)( =fR

;< 9&4$'8)( =RfD $ )$,&60$,)( =fR;v< 4$',)( =fD $ '$,)( =fR

11' ;< 4 ;< 01 ;< ? 2.0 ;v

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Relations and Functions (40)