today’s mathiit’ians..... tomorrow’s iiti’ians

8/14/2019 Todays Mathiitians..... Tomorrows IITiians.....

1/18

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Todays Mathiit ians..... Tomorrows IITi ians.....

F u n c t i o n s M a t e r i a l

UNIT - 1

CONTENTS

* Synopsis

Questions

* Level - 1

* Level - 2

* Level - 3

Answers

* Level - 1

* Level - 2

* Level - 3

e-Learn ing Resources

w w w . m a t h i i t . i n


2/18

DEFINITIONS & RESULTS1. G ENERAL DEFINITION :

If to every value (considered as real unless other-wise

stated) of a variable x, which belongs to some collection(Set) A, there corresponds one and only one finite valueof the quantity y, then y is said to be a function (Singlevalued) of x or a dependent variable defined on the set A; x is the argument or independent variable .If to every value of x belonging to some set A therecorresponds one or several values of the variable y, theny is called a multiple valued function of x defined onA.Conventionally the word "FUNCTION is used onlyas the meaning of a single valued function, if nototherwise stated .

Pictorially : xinput

f f x youtput( ) = , y is called

the image of x & x is the pre-image of y under f .

Every function from A B satisfies the followingconditions .(a) f A x B(b) a A (a, f(a)) fand

(c) (a, b) f & (a, c) f b = c2. D OMAIN , C O -DOMAIN & R ANGE O F A F UNCTION :

Let f : A B, then the set A is known as the domainof f & the set B is known as co-domain of f . The set of allf images of elements of A is known as the range of f .Thus : Domain of f = {a | a A, (a, f(a)) f}

Range of f = {f(a) | a A, f(a) B}It should be noted that range is a subset of co-domain .Sometimes if only f (x) is given then domain is set of those values of ' x ' for which f (x) exists or is defined .

To find the range of a function, there isn't any particularapproach, but student will find one of these approachesuseful .

(i) When a function is given in the form y = f (x), expressif possible ' x ' as a function of ' y ' i.e. x = g (y) . Find the domain of ' g ' . This willbecome range of ' f ' .(ii) If y = f (x) is a continuous or piece-wise continuousfunction , then range of ' f ' will be union of [ Min

m f (x),Max m f (x) ] in all such intervals where f (x) is continuous/ piece-wise continuous .

3. C LASSIFICATION O F F UNCTIONS :Functions can be classified into two categories :(i) One - One Function (Injective mapping) or Many -one function :A function f : A B is said to be a one-one function

or injective mapping if different elements of A havedifferent f images in B . Thus for x 1, x 2 A & f(x 1)

,f(x 2) B

, f(x 1) = f(x 2) x1 = x 2 or

x1

x2 f(x 1) f(x 2) .Diagramatically an injective mapping can be shown as

Note:(a) Any function which is entirely increasing ordecreasing in whole domain, then f(x) is one-one .(b) If any line parallel to x-axis cuts the graph of thefunction atmost at one point, then the function isone-one .Many - one function :A function f : A B is said to be a many onefunction if two or more elements of A have the same f image in B . Thus f : A B is many one if for ;x1,

x2 A , f(x 1) = f(x 2) but x 1 x2 .

Diagramatically a many one mapping can be shown as

Note: (a) Any continuous function which has atleast one localmaximum or local minimum, then f(x) is many-one . Inother words, if there is even a single line parallel to x-axiscuts the graph of the function atleast at two points, thenf is many-one .

(b) If a function is one-one, it cannot be many-one andvice versa .

A B

(or)A B

x1x2

A B

(or)

x1x2

A B

SYNOPSIS

FUNCTION

Mathiit e- Le arn in g Res our ce [email protected] www.mathiit.in


3/18

(c) All functions can be categorized as one-one or many-one .(ii) Onto function (Surjective mapping) or into function:If the function f : A B is such that each element in B(co-domain) must have atleast one pre-image in A, thenwe say that f is a function of A 'onto' B . Thus f : A Bis surjective iff b B, some a A such thatf (a) = b .

Diagramatically surjective mapping can be shown as

Note that : if range = co-domain, then f(x) is onto .

Into function :If f : A B is such that there exists atleast one elementin co-domain which is not the image of any element indomain, then f(x) is into .

Diagramatically into function can be shown as

Note that : if a function is onto, it cannot be into and viceversaThus a function can be one of these four types :

(a) one-one onto (injective & surjective)

(b) one-one into (injective but not surjective)

(or)

A B

A B

(or)

A B

A B

(c) many-one onto (surjective but not injective)

(d) many-one into (neither surjective nor injective)

Note : (a) If f is both injective & surjective, then it is calleda Bijective mapping. The bijective functions are alsonamed as invertible, non singular or biuniform functions.(b) If a set A contains n distinct elements then thenumber of different functions defined from A A is n n& out of it n ! are one one .

4. A LGEBRAIC O PERATIONS O N F UNCTIONS :If f & g are real valued functions of x with domain set A, Brespectively, then both f & g are defined in A B . Nowwe define f + g , f - g , (f . g) & (f/g) as follows :(i) (f g) (x) = f(x) g(x) (ii) (f . g) (x) = f(x) . g(x)

(iii) f g

(x) =

f xg x

( )( )

domain is {x | x A B s . t g(x) 0}

5. Important Types Of Functions :(i) Polynomial Function :

If a function f is defined by f (x) = a 0 xn + a 1 x

n-1 + a 2 xn-2 + ...

+ a n-1 x + a n where n is a non negative integer and a 0, a1, a 2,..., a n are real numbers and a 0 0, then f is called apolynomial function of degree n .

NOTE : (a) A polynomial of degree one with no constant termis called an odd linear function . i.e. f(x) = ax , a 0

(b) There are two polynomial functions

, satisfying therelation ; f(x).f(1/x) = f(x) + f(1/x) . They are :(i) f(x) = x n + 1 &(ii) f(x) = 1 - xn , where n is a positive integer.

(ii) Algebraic Function :y is an algebraic function of x, if it is a function that satisfiesan algebraic equation of the form , P 0 (x) y

n + P 1 (x) yn-1 +

....... + P n-1 (x) y + P n (x) = 0 Where n is a positive integerand P 0 (x), P 1 (x) ........... are Polynomials in x.e.g. y = | x | is an algebraic function, since it satisfies theequation y - x = 0.Note that all polynomial functions are Algebraic but notthe converse. A function that is not algebraic is calledT RANSCEDENTAL F UNCTION .

(iii) Fractional Rational Function :A rational function is a function of the form.

y = f (x) =g xh x

( )( )

, where g (x) & h (x) are polynomials & h (x) 0.

(iv) Exponential Function :A function f(x) = a x = e x ln a (a > 0 , a 1, x R) is calledan exponential function . The inverse of theexponential function is called the logarithmic function .i.e. g(x) = log a x .

Note that f(x) & g(x) are inverse of each other & theirgraphs are as shown .



4/18

v) A BSOLUTE VALUE F UNCTION :

A function y = f (x) = | x | is called the absolute valuefunction or Modulus function. It is defined as :

y = | x | =x if x

x if x

=

0

y = -1 if x < 0

y = Sgn x> x

(viii) Fractional Part Function :It is defined as : g (x) = {x} = x - [x] .e.g. the fractional part of the no. 2.1 is 2.1- 2 = 0.1 and thefractional part of - 3.7 is 0.3 . The period of this functionis 1 and graph of this function is as shown .

(ix) Identity function :The function f : A A defined by f(x) = x x A iscalled the identity of A and is denoted by I A . It is easy toobserve that identity function is a bijection .

(x) Constant function :A function f : A B is said to be a constant function if every element of A has the same f image in B . Thusf : A B ; f(x) = c , x A , c B is a constantfunction. Note that the range of a constant function is asingleton and a constant function may be one-one ormany -one, onto .

6. Homogeneous Functions :An integral function is said to be homogeneous withrespect to any set of variables when each of its termsis of the same degree with respect to those variables.

For example 5 x2 + 3 y2 - xy is homogeneous in x & y .7. Bounded Function :

A function is said to be bounded if | f(x) | < M , where Mis a finite quantity .

8. Implicit & Explicit Function :A function defined by an equation not solved for thedependent variable is called an I MPLICIT F UNCTION . Foreg. the equation x 3 + y 3 = 1 defines y as an implicitfunction. If y has been expressed in terms of x alone thenit is called an E XPLICIT F UNCTION .

9. Odd & Even Functions :If f (-x) = f (x) for all x in the domain of f then f is said tobe an even function.e.g. f (x) = cos x ; g (x) = x + 3 .If f (-x) = -f (x) for all x in the domain of f then f is said tobe an odd function.e.g. f (x) = sin x ; g (x) = x 3 + x . NOTE :

(a) f (x) - f (-x) = 0 f (x) is even & f (x) + f (-x) = 0f (x) is odd .(b) A function may neither be odd nor even .(c) Inverse of an even function is not defined .(d) Every even function is symmetric about the y-axis &every odd function is symmetric about the origin .(e) Every function can be expressed as the sum of aneven & an odd function.

y =

x

45 )(0 ,1 )

(1 ,0 )

g(x) = log a x

f(x) = a x , 0 < a < 1

+

y =

x

45 )(1 ,0 )

(0 ,1 )

+

f ( x ) =

a x , a

> 1

g ( x ) =

l o g a

x

-1 1 2

y

1

- - - - -

-

-

-

-

-

-

graph of y = {x}

x

-3 -2 -1 1 2

x

y

3

2

1

-1

-2

-3

graph of y = [x]



5/18

e.g. f xf x f x f x f x

( )( ) ( ) ( ) ( )= + +

2 2|________| |________|

| | EVEN ODD

(f) The only function which is defined on the entire numberline & is even and odd at the same time is f(x) = 0 .

(g) If f and g both are even or both are odd then the functionf.g will be even but if any one of them is odd then f.gwill be odd .

10. Periodic Function :

A function f(x) is called periodic if there exists a + venumber T (T > 0) called the period of the function suchthat f (x + T) = f(x), for all values of x within the domain of x .

e.g. The function sin x & cos x both are periodic over 2 & tan x is periodic over .

NOTE :

(a) f (T) = f (0) = f (-T) , where T is the period .

(b) Inverse of a periodic function does not exist .

(c) Every constant function is always periodic, with nofundamental period .

(d) If f (x) has a period T & g (x) also has a period T thenit does not mean that

f (x) + g (x) must have a period T .

e.g. f (x) = | sinx | + | cosx | .

(e) If f(x) has a period p, then1

f x( )and f x( ) also has a

period p .

(f) If f(x) has a period T then f (ax + b) has a period T a| |

11. Composite Functions :

Let f : A B & g : B C be two functions . Then thefunction gof : A C defined by (gof) (x) = g (f(x)) x

A is called the composite of the two functions f & g .

Diagramatically x f f x( ) g g (f(x)) .Thus the image of every x A under the function gof isthe g-image of the f-image of x.

Note that gof is defined only if x A, f(x) is anelement of the domain of g so that we can take its g-

image . Hence for the product gof of two functions f & g,

Properties Of Composite Functions :

(i) The composite of functions is not commutative

i.e. gof fog .(ii) The composite of functions is associative i.e. if f, g, h

are three functions such that fo (goh) & (fog) oh aredefined, then fo (goh) = (fog) oh .

(iii) The composite of two bijections is a bijection i.e. if f &g are two bijections such that gof is defined, then gof isalso a bijection .

12. Inverse Of A Function :

Let f : A B be a one-one & onto function, thentheir exists a unique function

g : B A such that f(x) = y g(y) = x, x A &y B . Then g is said to be inverse of f . Thus

g = f -1 : B A = {(f(x), x) | (x, f(x)) f} .

Properties Of Inverse Function :

(i) The inverse of a bijection is unique .

(ii) If f : A B is a bijection & g : B A is the inverse of f, then fog = I B & gof = I A , where I A & I B are identityfunctions on the sets A & B respectively. Note that thegraphs of f & g are the mirror images of each otherin the line y = x .

(iii) The inverse of a bijection is also a bijection .

(iv) If f & g are two bijections f : A B , g : B C thenthe inverse of gof exists and (gof) -1 = f -1 o g -1 .

(v) Inverse of an even function is not defined .

13. Equal or Identical Function :

Two functions f & g are said to be equal if :

(i) The domain of f = the domain of g .

(ii) The range of f = the range of g and

(iii) f(x) = g(x) , for every x belonging to their commondomain . eg.

f(x) =1x

& g(x) =x

x 2are identical functions.

14. General :

If x, y are independent variables, then :

(i) f(xy) = f(x) + f(y) f(x) = k ln x or f(x) = 0 .

(ii) f(xy) = f(x) . f(y) f(x) = x n , n R

(iii) f(x + y) = f(x) . f(y) f(x) = a kx .

(iv) f(x + y) = f(x) = f(y) f(x) = kx, where k is a constant.

the range of f must be a subset of the domain of g .



6/18

1. The period of the function, f(x) = [sin 3x] + |cos 6x| is : ( [.] denotes the greatest integer less than or equal to x)

(a) (b) 32

(c) 2 (d) 2

2. The function f(x) =[ ]

sec

1 x

x x, where [ x] denotes the greatest integer less than or equal to x is defined for all

x belonging to :(a) R (b) R - {(-1, 1) {n : n I)} (c) R - (0, 1) (d) R - {n : n N}

3. The function f(x) = )x(log 2x is defined for x belonging to :

(a) (- , 0) (b) (1, ) (c) (0, ) (d) none of these

4. If f(x) = ,145

gand3

xcos.xcos3

xsinxsin 22 =

++

++ then (gof)(x) =

(a) 1 (b) -1 (c) x (d) none of these

5. Let [x] denote the greatest integer x . The domain of definition of function 2]x[x4

)x(f 2

+= is

(a) ]2,1[)2,( (b) [0, 2] (c) [-1, 2] (d) (0, 2)

6. The domain of definition of the function

=

4xx5

log)x(f 2

10 is

(a) [1, 4] (b) (1, 4) (c) (0, 5) (d) [0, 5]

7. The range of the functionx3cos2

1)x(f

=is

(a) 0,31

(b) R (c) 1,31

(d) none of these

8. The domain of definition of the functionx|x|

1)x(f

= is

(a) R (b) (0, ) (c) (- ,0) (d) none of these

w w w . m a t h i i t . i nLEVEL - 1 (Object ive)



7/18

9. The function )1xxlog()x(f 2 ++= is(a) an even function (b) an odd function (c) periodic function (d) none of these

10. The function ))1xxlog(cos)x(f 2 ++= is(a) even (b) odd (c) constant (d) none of these

11. Thr period of the function f(x) = sin4

x+ cos4

x is(a) (b) /2 (c) 2 (d) none of these

12. Which of the following functions is an odd function

(a) f(x) = constant (b) f(x) = sinx + cosx (c) ))1xxlog(sin)x(f 2 ++= (d) f(x) = 1 + x + 2x 313. The domain of definition of the functionf(x) = 7-xPx-3 is

(a) [3, 7] (b) {3,4,5,6,7} (c) {3,4,5} (d) none of these

14. The functionxtanxx

xcosxsin)x(f

2

44

++= is

(a) even (b) odd (c) periodic with period (d) periodic with period 2 15. Range of f(x) = |x| + |x+1| is

(a) [0, ) (b) (0, ) (c) (1, ) (d) [1, )16. Range of tan -1x - cot -1x is

(a) (0, ) (b)2

,23

(c)

1,32

(d) 1,32

17. The range of x|x| is(a) (0, ) (b) [0, ) (c) (- ,0) (d) (- ,0]

18. If

+=

110110

x)x(f x2x2

2 then f is

(a) an even function (b) an odd function (c) neither even nor odd (d) cannot be determined

19. The domain of the function f(x) =)x1(log

1

10 + )2x( + is

(a) [-3, -2] excluding (-2.5) (b) [0, 1] excluding 0.5

(c) [-2, 1], excluding 0 (d) None of these

20. The period of xcos]x[xxcos24

e ++ is ([.] denotes the greatest integer function)

(a) 2 (b) 1 (c) 0 (d) -1

21. The domain of the function f(x) =

+++

x2x1

sin)xcos(sin)x(logsin2

12

1

(a) {x : 1 < x < 2} (b) {1} (c) Not defined for any value x (d) {-1, 1}



8/18

22. If [.] denotes the greatest integer function then the domain of the real valued function is|2xx|log 2]2 / 1x[ +

(a) ,

23

(b) ),2(2,23

(c) ),2(2,21

(d) None of these

23. The domain of the function f(x) =6 )1x(2)2x(3

2x

25284

+ is(a) (0, 1) (b) [3, ) (c) (1, 0) (d) none

24. A function whose graph is symmetrical about the y-axis is given by

(a) )1xx(log)x(f 2e ++= (b) f(x + y) = f(x) + f(y) for all x, y R(c) f(x) = cos x + sin x (d) none of these

25. If += ,

21

x,1xx)x(f 2 then value of x satisfying f(x) = f -1(x) is

(a) 1 (b) 2 (c)21

(d) none of these

26. If

=

5x1x

log)x(f 4.0 and g(x) = x 2 - 36, then D f/g is

(a) }6{~)0,( (b) }6,1{~),0( (c) }6{~),1( (d) }6{~),1[

27. The domain of the real-valued function f(x) = log e |log e x| is

(a) ),1()1,0( (b) ),0( (c) ),e( (d) ),1( 28. If f(x) and g(x) are two functions of x such that f(x) + g(x) = e x and f(x) - g(x) = e -x, then

(a) f(x) is odd, g(x) is odd (b) f(x) is even, g(x) is even

(c) f(x) is even, g(x) is odd (d) f(x) is odd, g(x) is even

29. The inverse of the function )1a,0a()1xx(logy 2a >++= is

(a) )aa(21 xx (b) not defined for all x

(c) defined for only positive x (d) none of these

30. Let f(x) = cos p x, where p = [a] = the greatest integer less than or equal to a. If the period of f(x) is , then

(a) ]5,4[a (b) ]5,4[a = (c) )5,4[a (d) none of these

31. If f(x) = ,e|xn|cos........|x2|cos|x|cos]x[x ++++

then period of f(x) is

(a) 1 (b)n1 (c)

1n1n 2

2

+ (d) n......3.2.1 1



9/18

32. Let

+

+

=

4|x|2

tan4

|x|2cos

4|x|2

sin)x(f 111 , then D f is

(a) [-6, 6] (b) ),6[ (c) [-6, 3] (d) [-3, 6]33. Identify the statement(s) which is/are incorrect ?

(a) The function f(x) = cos(cos -1 x) is neither odd nor even

(b) The fundamental period of f(x) = cos(sin x) + cos(cos x) is (c) The range of the function f(x) = cos (3 sin x) is [- 1, 1]

(d) None of these

34. Let9

xcos4)x(f 2

2 = . Then

(a) ]1,1[R,,,3

D f f = = (b) ]2,2[R,,,

3D f f =

=

(c) ]4,4[Rand,33

,D f f =

= (d) ]4,0(R,

3,D f f =

=

35. The range of the function ),1(x),xxxesin()x(f 2]x[ += where [x] denotes the greatest integer function is:

(a) (b) [0, 1] (c) [-1, 1] (d) R

36. The domain of f(x) = log 2 log3 / 4log (tan-1 x)-1 =

(a) R (b) ), / 4( (c) (0, 1) (d) None of these

37. If f(x) is a periodic function of the period then f( x + a), where a is a constant, is a periodic function of theperiod

(a) 1 (b) (c) a

(d) none of these

38. If ]1x[

1|x|

11)xx(cos)x(f 2

21

+

+= then domain of f(x) is (where [.] is the greatest integer)

(a)+

251

,2 (b) +

251

,2 (c)

2

21,2 (d) none of these

39. Let f(x) = sin x, g(x) = ln |x|. If the ranges of the composite functions fog and gof are R 1 and R 2 respectively, then

(a) )0,(R),1,1(R 21 == (b) ]1,1[R],0,(R 21 ==(c) ]0,(R),1,1(R 21 == (d) ]0,(R],1,1[R 21 ==

40. If x1x1

log)x(f += , then

(a) f(x) is even (b) f(x 1).f(x 2) = f(x 1 + x 2) (c) )xx(f )x(f )x(f

212

1 = (d) f(x) is odd



10/18

1. Find the value of x for which, 0)4x(

)2x()1x)(1x2()x(f 4

32

>=

2. Find the values of x for which 0)1x(

)4x()3x()x1()2x()x(f

232

+= .

3. Solve 12x

x|3x| >+ ++ .

4. Find the domain of the function;

+=

|xsin|log3

)23x8x(loglog)x(f 2

2|xsin|

5. Find domain for

+=

5x1x

log)x(f 4.0 .

6. Find domain for xlog3

|x|21cosy |1x|

1

+

= .

7. Find domain for

=

]1x[cos]x[sin)x(f where [ ] denotes greatest integer function.

8. Find the range forx]x[1

]x[xy

+

= where [ ] denotes greatest integer function.

9. Find domain and range of the function y = log e(3x2 - 4x + 5).

10. If f is an even function, find the real values of x satisfying the equation

++=

2x1x

f )x(f .

11. Find whether the given function is even or odd function, where

2

1x)xtanx(sinx

)x(f

++= .where [ ] denotes

greatest integer function.

w w w . m a t h i i t . i nLEVEL - 2 (Subjective)



11/18

12. If ),2[),1[:f is given byx1

x)x(f += then find )x(f 1 . (assume bijective).

13. Let ),4 / 3[),2 / 1[:f , where f(x) = x 2 - x + 1. Find the inverse of f(x).

14. ,x]x[

1)x(f = where [ ] denotes greatest integral function less than or equals to x. Then find domain of f(x).

15. Find the domain of )13x7x(log

1)x(f

22 / 1 +

=

16. Find the domain of single valued function y = f(x) given by the equation 10 x + 10 y = 10.

17. Let

2

,0x , then find the solution of the functionxtanlog

1)x(f

xsin= .

18. Find the range of log 3(log 1/2(x2 + 4x + 4)).

19. Find the domain & range of : 22

x9

sin3)x(f = .

20. Find the inverse of following functions:

(i) ]3,3[x),3 / x(sin)x(f 1 = [assuming bijective]

(ii) ]3,1[x),1x3x(ln)x(f 2 ++= . [assuming bijective]

21.=

1x0,x

0x1,1x)x(f 2 and g(x) = sinx. Find h(x) = f(|g(x)|) + |f(g(x))|.

22. If ,x]cos[x]cos[)x(f 22 += where [x] stands for the greatest integer function, then evaluate)4 / (f and)(f ),(f ),2 / (f .

23. A cubic expression f(x) satisfies the condition

=

+

x1

f )x(f x1

f )x(f , then prove that f(x) = 1 + x 3or1 - x 3.

If f(3) = 28. Then prove that f(2) = 9.

24. Let f(x) be a polynomial function satisfying, Ry,x2)xy(f )y(f )x(f )y(f )x(f ++= . If f(2) = 5 thenprove that f(5) = 26.

25. If for non-zero x, 5x1

x1bf )x(af = +

where ba then find f(x).



12/18

1. The domain of the function 1x)x1log()x(f 2 += is

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

2. The range of the function 22

xx1

)x(f += is equal to

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

3. The curves 2x3xyand2|x|3|x|y 2323 ++=++= have the same graph for

(a) x > 0 (b) 0x (c) all x except 0 (d) all x

4. Domain of the function 73x

12

x1

)x(f xsin21 +

++= is

(a) (b) R - {0} (c) R (d) None of these

5. The domain of definition of the function )1x(loge3y 1x2 = is

(a) ),1( (b) ),1[ (c) R ~ {1} (d) ),1()1,(

6. The range of the function f(x) = cos [x], where2

x2


13/18

10. The range of the function1xx1xx

)x(f 22

+++= is

(a) R (b) ),3[ (c) 3,31

(d) none of these

11. The domain of the function 2xx22)x(f = is

(a) 3x3 (b) 31x31 + (c) 2x2 (d) none of these

12. If the domain of the function ),,(is7x6x)x(f 2 += then range of the function is

(a) ),( (b) ),2[ (c) (-2, 3) (d) )2,(

13. The domain of the function

=

2x

logsin)x(f 2

21 is

(a) 1x1 (b) 1x0 (c) 2x1 (d) none of these

14. If f(x) = cos (log x) then

+

)xy(f yx

f 21

)y(f )x(f has the value

(a) 1 (b)21

(c) -2 (d) 0

15. The inverse of the function 2ee ee)x(f xxxx

++=

is given by

(a)2 / 1

e 1x2x

log

(b)3 / 1

e x31x

log

(c)2 / 1

e x2x

log

(d)2

e 1x1x

log

+

16. The range of the function for real x of x3sin2

1y

= is

(a) 1y31

(b) 1y31

> (d) 1y31

>>

17. The domain of the function2xx2

]x[1 + , where [ . ] denotes greatest integer function.

(a) [1, 2] (b) [0, 2] (c) [0, 1) (d) [1, 2)

18. Domain of sin -1(log 3(x/3)) is

(a) [1, 9] (b) [-1, 9] (c) [-9, 1] (d) [-9, -1]

19. The range of f(x) = 7-xPx - 3 is

(a) {1, 2, 3} (b) {1, 2, 3, 4, 5, 6} (c) {1, 2, 3, 4} (d) {1, 2, 3, 4, 5}



14/18

20. The value of Zn for which the function

=

nx

sin

nxsin)x(f has 4 as its period is

(a) 2 (b) 3 (c) 5 (d) 4

21. The domain of the function f(x) = log2(log

3(log

4x)) is

(a) x < 4 (b) x > 4 (c) 0 < x < 2 (d) 2 < x < 4

22. If f(x) is an odd periodic with period 2, then f(4) is

(a) 0 (b) 2 (c) 4 (d) -4

23. If f(x) = 1 + x , ( 0 ) is the inverse of itself then the value of is(a) -2 (b) -1 (c) 0 (d) 2

24. If g(x) be a function defined on [-1, 1] and if the area of the equilateral triangle with two of its vertices at (0, 0)

and (x, g(x))) is4

3, then the function is

(a) 2x1)x(g = (b) 2x1)x(g = (c) 2x1)x(g = (d) g(x) = 2x1+25. Which of the following functions is periodic

(a) f(x) = x - [x] (b)==

0x,0

0x,x1

sinx)x(f (c) f(x) = x cos x (d) none of these

26.2x

)3x)(1x()x(f

+= is real valued in the domain

(a) ),3[]1,( (b) ]3,2(]1,( (c) ),3[)2,1[ (d) none of these27. The domain of definition of the function y(x) given by the equation 2 x + 2 y = 2 is

(a) 1x0 < (b) 1x0 (c) 0x


15/18

31. Let>= 0, is

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

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

35. Range of the function Rx,1xx2xx

)x(f 22

++++= is

(a) ),1( (b)

711

,1 (c)

37

,1 (d) 57

,1



16/18

ANSWE R KE Y

1. b

2. b

3. b

4. a

5. a

6. a

7. c

8. c

9. b

10. a

11. b

12. c

13. c

14. b

15. d

16. b

17. b

18. b

19. d

20. b

21. b

22. b

23. b

24. d

25. a

26. c

27. a

28. c

29. a

30. c

31. a

32. a

33. a

34. c

35. c

36. c

37. a

38. a

39. d

40. d

LEVEL - 1 (Objective)



17/18

ANSWE R KE Y

1. }4{~),2()2 / 1,(x

2. ),3[]1,1(x

3. ),1()2,5(x

4.

5,

23

23

,),3()x(f

5. ),1()x(f

6. ]2,1()1,0(x

7. R ~ [1, 2)

8. Range =

21,0

9. Range is

,

311

log

10.++=2

53,

253

,2

51,

251

x

11. f(x) is an odd function (if nx ) and f(x) isan even function if ( =nx ).

12.2

4xx 2 +

13.4

3x

2

1 +

14.

15. (3, 4)

16. )1,(x

17.

2,

4

18. RRange

19.3

,3

Domain , 233

,0Range

20. (i) 3 sin x (ii)2

e53 x+

21.


18/18

1. d

2. c

3. b

4. a

5. a

6. b

7. c

8. c

9. a

10. c

11. b

12. b

13. d

14. d

15. b

16. a

17. a

18. a

19. a

20. a

21. b

22. a

23. b

24. b

25. a

26. c

27. d

28. b

29. a

30. a

31. b

32. d

33. d

34. b

35. c

LEVEL - 3 (Questions asked from previous Engineering Exams)

ANSWE R KE Y


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