functions.pdf
TRANSCRIPT
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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
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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
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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
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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
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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)
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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[
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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]
59. Domain of the function 29
12x
x
is
(a) (3, 1) (b) [3, 1] (c) (3, 2] (d) [3, 1)
60. Domain of the function 2
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
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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
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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
>
=
=
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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
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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
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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
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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= .
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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
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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 )]([])([)]([ .
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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)
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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,( .
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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
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57. (b) 3339 2
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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
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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,
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( ) ( ) 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.
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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.
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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.