Functions.pdf

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FUNCTIONS

OBJECTIVE PROBLEMS

1. Let A be a set of n distinct elements. Then the total number of distinct functions from A

to A is

a) n2 b) nn c) 2n d) n!

2. Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A into B is

a) 144 b) 12 c) 24 c) 64

3. Let n(A) = 4 and n(B) = K. The number of all possible injections from A to B is 120 then k =

a) 9 b) 24 c) 5 d) 6

4. Let n(A) = 4 and n(b) = 5. The number of all possible many-one functions from A to B is

a) 625 b) 20 c) 120 d) 505

5. Set A contains 3 elements and set B contains 2 elements. The number of onto functions from A onto B is

a) 3 b) 6 c) 8 d) 9

6. Let A = {1, 2, 3, .., n} and B = {a, b, c}, the number of functions from A to B that are onto is

a) 3n 2n b) 3n 2n = 1 c) 3(2n 1) d) 3n 3(2n 1)

7. Set A has 3 elements, Set B has 4 elements. The number of surjections that can be defined from A to B is

a) 144 b) 12 c) 0 d) 64



8. Let A, B are two sets each with 10 elements, then the number of all possible bijections from A to B is

a) 20 b) 10! C) 100 d) 1000

9. The number of one-one onto functions that can be defined from (1, 2, 3, 4) onto set B is 24 then n(B) =

a) 4 b) 2 c) 3 d) 6

10. A = {1, 2, 3, 4), B = {a, b, c, d, e}, then the number of all possible constant functions from A to B is

a) 9 b) 4 c) 5 d) 16

11. If ( ) cos(log )f x x= , then 1( ) ( ) [ ( / ) ( )]2

f x f y f x y f xy + =

(a) 1 (b) 21

(c) 2 (d)None of these

12. If ( ) sin logf x x= , then the value of ( ) 2 ( ).cos logxf xy f f x yy

+

is equal to

(a)1 (b) 0 (c)1 (d) sin log .cos logx y

13. If ( ) cos(log )f x x= , then 2 2

2 22

1( ) ( )2 2

x xf x f y f fy

+

has the value

(a)2 (b) 1 (c)1/2 (d)None of these

14. If ( ) cos(log )f x x= , then the value of 1( ). (4) (4 )2 4

xf x f f f x +

(a)1 (b) 1 (c)0 (d) 1

15. Given the function ( ) , ( 2)2

x xa af x a+

= > . Then ( ) ( )f x y f x y+ + =

(a) )().(2 yfxf (b) )().( yfxf

(c) )()(

yfxf

(d)None of these

16. Let :f R R be defined by ( ) 2 | |f x x x= + , then (2 ) ( ) ( )f x f x f x+ = (a) x2 (b) ||2 x

(c) x2 (d) ||2 x

17. If ( ) 10 , ( 10,10)10

f x xe xx

+=

and 2200( )

100xf x kfx

= +

, then k =

(a) 0.5 (b) 0.6 (c) 0.7 (d)0.8



18. If 1( ) log1

xf xx

+ =

, then 2

21

xfx

+

is equal to

(a) 2)]([ xf (b) 3)]([ xf

(c) )(2 xf (d) )(3 xf 19. If 2 2( ) cos[ ] cos[ ]f x x xpi pi= + , then

(a) 24

=

pif (b) 2)( =pif

(c) 1)( =pif (d) 12

=

pif

20. If ( ) ax by f xcx a

+= =

, then x is equal to

(a) )(/1 xf (b) )(/1 yf

(c) )(xyf (d) )(yf 21. The function equivalent to 2log x is

(a) xlog2 (b) ||log2 x

(c) |log| 2x (d) 2)(log x 22. The graph of the function ( )y f x= is symmetrical about the line 2x = , then (a) )()( xfxf = (b) )2()2( xfxf =+

(c) )()( xfxf = (d) )2()2( =+ xfxf

23. If 1 1( )2 2 4 2 2 4

f xx x x x

= ++

for 2x > , then (11)f =

(a) 7/6 (b) 5/6 (c) 6/7 (d)5/7 24. If :f IR IR is defined by ( ) 3 4f x x= , then 1 :f IR IR is

(a) x34 (b) 3

4+x

(c)43

1x

(d)4

3+x

25. Which of the following function is invertible?

(a) ( ) 2xf x = (b) 3( )f x x x=

(c) 2)( xxf = (d)None of these

26. The inverse of the function 10 1010 10

x x

x x

+ is

(a) 101 1log2 1x

x

+

(b) 101 1log2 1

x

x

+

(c) 101 2log4 2x

x

(d)None of these

27. If x

xxf

+=

1)( , then )(1 xf is equal to

(a) x

x)1( + (b) )1(

1x+

(c) )1()1(

x

x

+ (d) )1( x

x

28. If 2

( )1

xf xx

=

+, then =))(( xfofof

(a) 21

3

x

x

+ (b)

231 x

x

+ (c)

21

3

x

x

+ (d)None of these



29. If ( ) logaf x x= and xaxF =)( , then )]([ xfF is (a) )]([ xFf (b) )]2([ xFf (c) |)2(| xfF (d) )][(xF

30. Let f and g be functions defined by ( ) ,1

xf xx

=

+( )

1xg x

x=

, then ))(( xfog is

(a) x

1 (b)

11x

(c) 1x (d)x

31. If ( ) , 11

xf x xx

=

+. Then, for what value of is xxff =))((

(a) 2 (b) 2 (c) 1 (d)1

32. If 1/( ) ( ) ,n nf x a x= where 0>a and n is a positive integer, then =)]([ xff (a) 3x (b) 2x

(c) x (d)None of these

33. If 1 21 21 2

( ) ( )1x xf x f x f

x x

=

for 1 2, [ 1,1]x x , then )(xf is

(a) )1()1(log

x

x

+

(b) )1()1(

tan 1x

x

+

(c) )1()1(log

x

x

+ (d) all the above

34. If , when is rational( )0, when is irrationalx xf x

x

=

; 0, when is rational( )

, when is irrationalx

g xx x

=

then ( )f g is

(a) One-one onto (b)One-one not onto (c) Not one-one but onto (d)Not one-one not onto 35. If 21xe y y= + + , then y =

(a) 2

x xe e+ (b)

2

x xe e

(c) x xe e+ (d) x xe e

36. If ||||)(

x

xxxf = , then = )1(f

(a) 1 (b) 2 (c) 0 (d) +2 37. If ( , )f x ay x ay axy+ = , then ),( yxf is equal to

(a) xy (b) 222 yax

(c)4

22 yx (d) 2

22

a

yx



38. Let

1 1, 0

2 2( )1 1

, 13 2

if xf x

if x

= <

, then f is

(a) A rational function (b)A trigonometric function (c) A step function (d)An exponential function

39. Let : (2,3) (0,1)f be defined by ][)( xxxf = then )(1 xf equals (a) 2x (b) 1+x

(c) 1x (d) 2+x

40. The domain of

3logsin 3

1 x is

(a) [1, 9] (b) [1, 9] (c) [9, 1] (d) [9, 1] 41. The domain of the function 1 2( ) sin [log ( / 2)]f x x= is

(a) [1, 4] (b) [4, 1] (c) [1, 4] (d) None of these

42. The domain of the function ( ) log( 4 6 )f x x x= + is (a) ),4[ (b) ]6,(

(c) ]6,4[ (d) None of these

43. Domain of the function 1/22

105( ) log

4x xf x =

is

(a)



47. Domain of the function 2

23 2( )

6x xf xx x

+=

+ is

(a) }3,:{ xRxx (b) }2,:{ xRxx

(c) }:{ Rxx (d) }3,2,:{ xxRxx

48. The domain of the function 1| |y x x= is

(a) )0,( (b) ]0,(

(c)

)1,(

(d)

),(

49. Function x1sin is defined in the interval (a) (1, 1) (b) [0, 1] (c) [1, 0] (d) (1, 2)

50. The domain of the function 1sin (3 )( )

ln(| | 2)xf x

x

=

is

(a) [2, 4] (b) (2, 3) (3, 4] (c) [2, ) (d)

),2[)3,(

51. The domain of 22log ( 3)( )

3 2xf x

x x

+=

+ + is

(a) }2,1{ R (b) ),2( +

(c)

}3,2,1{ R (d)

}2,1{),3( +

52. The domain of the function 23( ) log ( 1)xf x x+= is (a) ),1()1,3(

(b)

),1[)1,3[

(c) ),1()1,2()2,3( (d)

),1[)1,2()2,3[

53. Domain of definition of the function )(log4

3)( 3102 xxx

xf +

= , is

(a) (1, 2) (b) )2,1()0,1(

(c) ),2()2,1( (d)

),2()2,1()0,1(

54. Domain of the function { }2log (5 ) / 6x x is (a) (2, 3) (b) [2, 3] (c) [1, 2] (d) [1, 3]

55. The domain of the function 2log( 6 6)x x + is

(a) ),( (b) ),33()33,( +

(c) ),5[]1,( (d) ),0[



56. The domain of the function 2( ) exp( 5 3 2 )f x x x= is

(a)

23

,1 (b)

,

23

(c) ]1,[ (d)

23

,1

57. The domain of the function 1

2

sin ( 3)( )9

xf xx

=

is

(a) [1, 2) (b) [2, 3) (c) [1, 2] (d) [2, 3] 58. The domain of the function 1 1( ) sin {(1 ) }xf x e = + is

(a)

31

,

41

(b) [1, 0] (c) [0, 1] (d) [1, 1]


12x

x

is

(a) (3, 1) (b) [3, 1] (c) (3, 2] (d) [3, 1)


3( )( 1) 4

xf xx x

=

is

(a) (1, 2) (b) ),2()2,( (c) ),1()2,( (d) }2,1{),(

61. The domain of the function 1( ) log | sin |f x x= is

(a) },2{ InnR pi (b) },{ InnR pi (c) },{ pipiR (d) ),(

62. The function ,][

sec)(1

xx

xxf

=

where [.] denotes the greatest integer less than or equal to x is

defined for all x belonging to

(a) R (b) )}|()1,1{( ZnnR (c) )1,0(+R

(d)

}|{ NnnR +

63. Domain of |log|log)( xxf = is

(a) ),0( (b) ),1(

(c) ),1()1,0( (d) )1,( 64. Domain of function xxf 5sin)( 1= is

(a)

51

,

51

(b)

51

,

51

(c) R (d)

51

,0

65. The domain of the function ( ) log( 4 6 )f x x x= + is (a) ),4[ (b) ]6,(

(c) ]6,4[ (d)None of these



66. Domain of the function )231(sin)( 21 xxxf ++= is

(a) ),( (b) )1,1(

(c)

0,

23

(d) ),2(21

,

67. The range of the function 2( ) | 2 |xf xx

+=

+ is

(a) {0, 1} (b) {1, 1} (c) R (d) }2{R 68. Range of the function

x3sin21

is

(a) [1, 3] (b)

1,31

(c) (1, 3) (d)

1,31

69. If x is real, then value of the expression 32914

2

2

++

++

xx

xx lies between

(a) 5 and 4 (b) 5 and 4 (c) 5 and 4 (d) None of these 70. If the function :[1, ) [1, )f is defined by ( 1)( ) 2 ,x xf x = then 1f (x) is

(a) )1(

21

xx

(b) )log411(21

2 x++

(c) )log411(21

2 x+ (d) Not defined

71. The range of function The range of function 76)( 2 += xxxf is

(a) ),( (b) ),2[

(c) )3,2( (d) )2,(

72. Range of the function 2

22( ) ;1

x xf x x Rx x

+ +=

+ + is

(a) ),1( (b) ]7/11,1(

(c) ]3/7,1( (d) ]5/7,1( 73. The function RRf : is defined by xxxf 42 sincos)( += for Rx , then =)(Rf

(a)

1,43

(b)

1,43

(c)

1,43

(d)

1,43

74. If RRf : , then the range of the function 1

)( 22

+=

x

xxf is

(a) R (b) +R

(c) R (d) RR 75. Range of the function ( ) 9 7sinf x x= is (a) (2, 16) (b) [2, 16] (c) [1, 1] (d) (2, 16] 76. The range of



77. If dcbxaxf ++= )cos()( , then range of )(xf is

(a) ]2,[ adad ++ (b) ],[ dada +

(c) ],[ daad + (d) ],[ adad +

78. Range of the function 1

)(2

2

+=

x

xxf is

(a) (1, 0) (b) (1, 1) (c) [0, 1) (d) (1, 1)

79. The domain of )(logsin 31 x is

(a) [1, 1] (b) [0, 1] (c) [0, ] (d) R 80. For

3pi > , the value of 22 cossec)( +=f always lies in the interval

(a) (0, 2) (b) [0, 1] (c) (1, 2) (d) ),2[ 81. The Domain of function ])[(log)( xxxf e = is

(a) R (b) R-Z (c) ),0( + (d) Z 82. If 1)( 2 += xxf , then )17(1f and )3(1 f will be

(a) 4, 1 (b) 4, 0 (c) 3, 2 (d) None of these

83. If |cos|)( xxf =

and ][)( xxg = , then )(xgof is equal to

(a) |][cos| x (b) |cos| x

(c) |]cos[| x (d) |][cos| x

84. Let ][1)( xxxg += and

>

=

=



88. If RRf : , then ||)( xxf = is

(a) One-one but not onto (b) Onto but not one-one (c) One-one and onto (d) None of these

89. Which one of the following is a bijective function on the set of real numbers

(a) 52 x (b) || x

(c) 2x (d) 12 +x

90. If Ryx ),( and 0, yx ; yxyxf ),( , then this function is a/an

(a) Surjection (b) Bijection (c) One-one (d) None of these 91. The function ( )2( ) sin log( 1)f x x x= + + is (a) Even function (b) Odd function (c) Neither even nor odd (d) Periodic function 92. If xxf 2sin)( = and the composite function |sin|)}({ xxfg = , then the function )(xg is equal to

(a) 1x (b) x (c) 1+x (d) x

93. If 246 432)( xxxxf ++= then )(' xf is

(a) Even function (b) An odd function (c) Neither even nor odd (d) None of these

94. The function )1log()( 2 ++= xxxf , is

(a) An even function (b) An odd function (c) A Periodic function (d) Neither an even nor odd function 95. The period of ( ) [ ]f x x x= , if it is periodic, is

(a) )(xf is not periodic (b) 21

(c) 1 (d)2

96. A function f from the set of natural numbers to integers defined by

=

even is when,2

oddiswhen,2

1

)(n

n

nn

nf , is

(a) One-one but not onto (b) Onto but not one-one (c) One-one and onto both (d) Neither one-one nor onto



97. Let NNf : defined by 1)( 2 ++= xxxf , Nx , then f is (a) One-one onto (b) Many one onto

(c) One-one but not onto (d) None of these 98. The function RRf : defined by )1()( = xxf )3)(2( xx is

(a) One-one but not onto (b) Onto but not one-one (c) Both one-one and onto (d) Neither one-one nor onto

99. If :[0, ) [0, )f and ( ) ,1

xf xx

=

+then f is

(a) One-one and onto (b) One-one but not onto (c) Onto but not one-one (d) Neither one-one nor onto

100. Let the function RRf : be defined by ( ) 2 sin ,f x x x x R= + . Then f is (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

101. 2)( xxxf += is a function from R R , then )(xf is

(a) Injective (b) Surjective (c) Bijective (d) None of these

102. Range of 2

234 71( )2 7

x xf xx x

+ =

+ is

(a) [5, 9] (b) ),9[]5,(

(c) (5, 9) (d) None of these 103. Which of the following functions is inverse of itself

(a) x

xxf

+

=

11)( (b) xxf log5)( =

(c) )1(2)( = xxxf (d) None of these

104. If 53)( = xxf , then )(1 xf

(a) Is given by 53

1x

(b) Is given by 3

5+x

(c) Does not exist because f is not one-one (d) Does not exist because f is not onto 105. If SRf : defined by ( ) sin 3 cos 1f x x x= + is onto, then the interval of S is (a) [1, 3] (b) [1, 1] (c) [0, 1] (d) [0, 1] 106. Let RRf : be a function defined by

nx

mxxf

=)( , where nm . Then

(a) f is one-one onto (b) f is one-one into (c) f is many one onto (d) f is many one into



107. Let 44)( 2

2

+

=

x

xxf for 2|| >x , then the function : ( , 2] [2, ) ( 1,1)f is

(a) One-one into (b) One-one onto (c) Many one into (d) Many one onto

108. Let X and Y be subsets of R, the set of all real numbers. The function YXf : defined by 2)( xxf = for Xx is one-one but not onto if (Here +R is the set of all positive real numbers)

(a) +== RYX (b)

+== RYRX ,

(c) RYRX == + , (d)

RYX ==

109. Function xxxfRRf += 2)(,: is

(a) One-one onto (b) One-one into (c) Many-one onto (d) Many-one into



FUNCTIONS

HINTS AND SOLUTIONS

1. (b).

( ) ( )( ) ( ). n A

n A n B n

and no functions n B= =

=

2. (c) Synopsis 3. (c) Synopsis 4. (d) Synopsis 5. (b) Synopsis 6. (d) 7. (c) 8. (b) 9. (a) 10. (c) Synopsis 11. (d) )(logcos)()(logcos)( yyfxxf ==

+

)(

21)().( xyf

yxfyfxf

+

= )(logcoslogcos

21)(logcos)(logcos xy

yxyx

)](logcos)(logcos2[21)(logcos)(logcos yxyx = = 0.

12. (b) )log(logsinlogsin)( yxxyxyf +== .....(i)

)log(logsin)/log(sin)/( yxyxyxf == .....(ii)

yxyxfxyf logcoslogsin2)/()( =+

0logcoslogsin2logcoslogsin2 = yxyx .

13. (d) 14. (c) 15. (a) )()( yxfyxf ++

12

x y x y x y x ya a a a+ + = + + +

1 ( ) ( )2

x y y x y ya a a a a a = + + +

1 ( ) ( ) 2 ( ) ( )2

x x y ya a a a f x f y = + + = .

16. (b) (2 ) 2(2 ) | 2 | 4 2 | |f x x x x x= + = + ,

2 1y x= + ,

||2)( xxxf += )()()2( xfxfxf +

4 2 | | | | 2 2 | |x x x x x x= + + ||2 x= .



17. (a) ( ) 1010

f x xex

+=

+=

x

xxf

1010log)(

2

2

2

2 )10(10)10(10log

10020010

10020010

log100

200

+=

+

++

=

+ x

x

x

xx

x

x

xf

)(21010log2 xf

x

x=

+=

.5.021

100200

21)( 2 ==

+= k

x

xfxf

18. (c)

2( ) log( 1)f x x x= + +

+

++=

+

++

=

+

xx

xx

x

x

x

x

x

xf2121log

121

121

log1

22

2

2

2

2

)(211log2

11log

2

xfx

x

x

x=

+=

+= .

19. (d)

xxxf ][cos][cos)( 22 pipi +=

)10cos()9cos()( xxxf += )10cos()9cos( xx +=

=

2cos

219

cos2 xx

=

4cos

419

cos22

pipipif ; 12

1212

2=

=

pif .

20. (d) acx

baxy

+=

aybacyx += )( )(yfacybay

x =

+= .

21. (b) domain of 2log x is R-{0} and the domain of ||log x is also R-{0}.

2log x and ||log2 x are identical functions.

22. (b) )0()0()()( xfxfxfxf =+= is symmetrical about 0=x . )2()2( xfxf =+ is symmetrical about 2=x .

23. (c) 422

1

422

1)(

++

=

xxxx

xf

18211

1

18211

1)11(

++

=f

76

723

723

231

231

=

++

=

++

= .

24. (b) 43)( = xxf . let xyfxfy == )()(1

4343 +== xyxy



34)(

34 1 +

=+

= x

xfxy .

25. (a) A function is invertible if it is bijection.

26. (a)

+=

+

=

yy

xyxx

xx

11log

21

10101010

10

Let )(xfy = )(1 yfx =

+=

yyyf

11log

21)( 101

+=

x

xxf

11log

21)( 101

27. (d) x

xxf

+=

1)( . Let )()( 1 yfxxfy ==

xyxyx

xy =++

=

1

yy

x

=

1

y

yyf

=

1)(1

x

xxf

=

1)(1 .

28. (b)

+==

21)())(()()()(

x

xfofxffofxfofof

++

+=

++

+=

22

2

2

2

2

211

1

11

1

xx

xxf

x

x

x

x

f

.

3121

1

2121 2

2

2

2

2x

x

x

x

x

x

x

xf+

=

++

+=

+=

29. (a) xaxFxfF xa a === log)(log)]([ xaxaafxFf axax ==== loglog)()]([ .

30. (d) xxx

x

x

x

x

x

x

xfxgfxfog =+

=

+

=

==

111

11

))(())(( .

31. (d)

1.

11

11)()())((

2

++=

+

+

+=

+=

xx

x

x

x

x

x

xfxf

xff

1)1(

.

2

++=

x

xx

or 0)1)1(( 2 =++ xx

Or 0)1()1( 22 =++ xx .

01,01 2 ==+ , 1= .

32. (c) { } xxaaxfaxff nnnn === /1/1 )]([])([)]([ .



33. (d) When 11 =x and 12 =x , then

0)1()1()1(1111)1()1( ==

+

= fffff

Which is satisfied when

+

=

x

xxf

11

tan)( 1

When 021 == xx , then

0)0()0(0100)0()0( ==

= fffff

When 11 =x and 02 =x then

0)0()1(0101)0()1( ==

= fffff

Which is satisfied when

+

=

x

xxf

11log)( and

+=

x

xxf

11log)( .

34. (a)

= QxxQxx

xgf,

,))((

35. (b)

21 yye x ++=

21 yye x +=

)1()( 22 yye x +=

xxxx yeeyyeye 2112 2222 =+=+

xxx

x

eeye

ey == 2122

2

xx eey

= .

36. (b)

21

11|1|

|1|1)1( ==

=f .

37. (c) axyayxayxf =+ ),( ..(i) Let uayx =+ and vayx =

Then 2

vux

+=

and a

vuy2

=

Substituting the value of x and y in (i), we obtain

4),(

22 vuvuf =

4),(

22 yxyxf = .

38. (c)



39. (d) ][)( xxxf =, here [x]=2

2)( == xyxf 2)()(2 11 +==+= xxfyfyx .

40. (a)

=

3logsin 3

1 xy 1

3log1 3

x

333

1 x 91 x ]9,1[x .

41. (a) )]2/([logsin)( 21 xxf = , Domain of x1sin is ]1,1[x

1)2/(log1 2 x 2221 x

41 x

]4,1[x .

42. (c)

)64log()( xxxf +=

04 x and 06 x 4x and 6x

Domain of )(xf = ]6,4[ .

43. (b) We have 2/1

2

10 45log)(

=

xxxf ..(i)

From (i), clearly )(xf is defined for those values of x for which 04

5log2

10

xx

14

5104

5 202

xxxx

0452 + xx 0)4)(1( xx

44. (b) 22 2 0x x 1 3 1 3.x + 45. (d) 101 + xx ; 0,101 xxx

Domain is }0{]1,1[ .

46. (d)

1011 >> xx

. Also, 0x .

Required interval ),1()0,( =

47. (d)

( 2) ( 3) 0x x + Domain is { }3,2,: xxRxx . 48. (a) 0|| >xx

xx >|| but xx =|| for x positive and xx >|| for x negative. Domain is )0,( .



49. (b) Let yxxy sinsin 1 == 10,sin 2 = xyx

50. (b) [ ]2||log)3(sin)(

1

=

x

xxf

Let )3(sin)( 1 xxg = 131 x

Domain of )(xg is [2, 4] And let [ ]2||log)( = xxh 02|| >x 2|| >x 2x ),2()2,(

We know that

=))(/( xgf { }0)(:)()(

21 = xgRxDDxxgxf

Domain of }3{]4,2()( =xf ]4,3()3,2( = .

51. (d) Here 03 >+x and 0232 ++ xx 3>x and ,0)2)(1( ++ xx i.e. 2,1 x

Domain }2,1{),3( =

52. (c)

012 >x

,12 >x 1or 1 >< xx and 03 >+ x

3>x and 2x

),1()1,2()2,3( =fD .

53. (d) )(log4

3)( 3102 xxxxf +

= . 04 2 x 4x

0)1(0 23 >> xxxx 1,0 >> xx

54. (b) 16

506

5log22

xxxx or 0652 + xx or 0)3()2( xx . Hence .32 x

55. (c) 0)66log( 2 + xx 1662 + xx 0)1)(5( xx

1x or 5x . Domain is ),5[]1,( .

56. (d) 0235 2 xx 023)1(

xx

Domain is [1,3 / 2] .

+ +

1 0 1



57. (b) 3339 2



023 x

0,

23

x

Domain of function =

0,23

.

67. (b) |2|2)(

+

+=

x

xxf

22

,1,1)(

>

yyy

0)1293(4)1449(4 22 >++ yyyy

0483612561964 22 >++ yyyy

016088 2



72. (c)

43

2111)( 2

+

+

+=

x

xf Range ]3/7,1(= .

73. (c) xxxfy 42 sincos)( +== )cos1(sincos)( 222 xxxxfy +==

xxxxy 2222 cossinsincos +=

xxy 22 cossin1 = xy 2sin.411 2=

1)(43 xf , )12sin0( 2 x

]1,4/3[)( Rf .

74. (b)

+R {as y is always positive }Rx .

75. (b)

.sin79)( xxfy == Range ].16,2[=

76. (a)

= xxf 2cos

4sec)( pi

1cos0 2 x at ,0cos =x 1)( =xf and at ,1cos =x 2)( =xf ; 21 x ]2,1[x .

77. (d)

dcbxaxf ++= )cos()( ..(i) For minimum 1)cos( =+ cbx

)()( addaxf =+=

For maximum 1)cos( =+ cbx

)()( addaxf +=+=

Range of ],[)( adadxf += .

78. (c) Let 12

2

+=

x

xy

1,10)1( 2 =++ yyxxy For real values of x,

We have )1,0[0)1(0)1(40 yyyyyD

11

0 22

, 2sec

4sec 2 . Required interval ),2[ =

81. (a) The domain of { }log [ ]e x x is R, because [x] 0

82. (d) Let 12 += xy 1= yx

1)(1 = yyf 1)(1 = xxf

4117)17(1 ==f and 413)3(1 ==f , which is not possible

83. (c) |].cos[|)}({)( xxfgxgof == 84. (b) Here Znxnnxg ==+= ,11)(

knkn +=++ 11 , knx += (where 10,



( ) ( ) 0f x f x + =

)(xf is odd function.

92. (b) ( )( ) | sin |gof x x= and xxf 2sin)( = 2(sin ) | sin |g x x= ; ( )g x x= . 93. (b) 6 4 2( ) 2 3 4f x x x x= + +

6 4 2( ) 2( ) 3( ) 4( ) ( )f x x x x f x = + + =

)(xf is an even function and derivative of an even function is always odd.

94. (b) )1log()( 2 ++= xxxf and )1log()( 2 ++= xxxf )(xf=

)(xf is odd function.

95. (c) 96. (c) (1) 0, (2) 1, (3) 1, (4) 2, (5) 2f f f f f= = = = =

And 3)6( =f so on.

No two elements of domain have same image. And range of the function is set of all integers. Hence f is one-one and onto function.

97. (a) Let Nyx , such that )()( yfxf = Then 11)()( 22 ++=++= yyxxyfxf

( )( 1) 0x y x y x y + + = = Or Nyx = )1(

f is one-one. Again, since for each Ny , there exist Nx

f is onto. 98. (b) )3)(2)(1()( = xxxxf

0)3()2()1( === fff )(xf is not one-one.

For each Ry , there exists Rx such that yxf =)( . Therefore f is onto. Hence RRf : is onto but not one-one.

99. (b) ),0[,0)1(1)( 2 >+=

xx

xf and range )1,0[

Function is one-one but not onto

100. (a)

0cos2)( >+= xxf . So, )(xf is strictly monotonic increasing so, )(xf is one-to-one and onto.



101. (d) ||)( 2 xxxxxf +=+= Clearly f is not one-one as 0)2()1( == ff f is not one-one

,,0)( Rxxf Range of Rf = ),0( . f is not onto.

102. (b) Let yxx

xx=

+

+

727134

2

2

0)717()17(2)1(2 =++ yxyyx

0

5,9045142 + yyyy .

103. (a) 111 1( ) ( ( ) ) ,11 11

x

x xfof x f f x f x xxx

x

+= = = =

+ ++

fIfof = is the inverse of itself.

104. (b) f is one-one and onto, so 1f exists and is 3

5)(1 += xxf .

105. (a) 22 )3(1)cos3(sin)3(1 ++ xx 2)cos3(sin2 xx

12)1cos3(sin12 +++ xx

3)1cos3(sin1 + xx i.e., range ]3,1[=

For f to be onto ]3,1[=S 106. (b) For any ,, Ryx we have

yxnymy

nx

mxyfxf =

=

= )()(

f is one-one.

Let R such that =

=nx

mxxf )(

=

1nm

x

Clearly Rx for 1= . So, f is not onto.

107. (c) Let )()( yfxf = 44

44

2

2

2

2

+

=

+

yy

x

x

441441

44 22

2

2

2

2+=+

+

=

+

yxyy

x

x

yx = , )(xf is many-one.



Now for each ),1,1(y there does not exist Xx such that yxf =)( . Hence f is into. 108. (c) 21222121 )()( xxxxxfxf === , [if ]+= RX

f is one-one. Since YRRR f == + ; f is not onto.

109. (d) 0)1()0( == ff hence )(xf is many one. But there is no pre-image of 1 . Hence )(xf is into function. So function is many-one into.

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