IIT MATHEMATICS

_By Er. Pankaj Mangla sir

FUNCTIONN
Exercise Sheet
CONTENTS
KEY CONCEPTS — Page - 2 - 9

PROFICIENCY TEST — Page - 10 - 12

EXERCISE-I — Page - 13 - 15

EXERCISE-II — Page - 16 - 18

EXERCISE-III — Page - 19 - 20

EXERCISE-IV — Page - 21 - 22

EXERCISE-V — Page - 23 - 25

ANSWER KEY — Page - 26 - 28

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 KEY CONCEPTS (FUNCTIONS)
THINGS TO REMEMBER:
1. GENERAL DEFINITION:
If to every value (Considered as real unless otherwise stated) of a variable x, which belongs to some
collection (Set) E, there corresponds one and only one finite value of the quantity y, then y is said to be a
function (Single valued) of x or a dependent variable defined on the set E ; x is the argument or independent
variable .
If to every value of x belonging to some set E there corresponds one or several values of the variable y, then
y is called a multiple valued function of x defined on E.Conventionally the word "FUNCTION” is used only as
the meaning of a single valued function, if not otherwise stated.

x f (x )  y
Pictorially :     , y is called the image of x & x is the pre-image of y under f.
input output

Every function from A  B satisfies the following conditions .

(i) f  Ax B (ii)  a  A  (a, f(a))  f and
(iii) (a, b)  f & (a, c)  f  b = c

2. DOMAIN, CODOMAIN & RANGE OF A FUNCTION :

Let f : A  B, then the set A is known as the domain of f & the set B is known as co-domain of f . The
set of all f images of elements of A is known as the range of f . T hus :
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 . If only the rule of function is given then the domain of the
function is the set of those real numbers, where function is defined. For a continuous function, the interval from
minimum to maximum value of a function gives the range.

3. IMPORTANT TYPES OF FUNCTIONS :

(i) POLYNOMIAL FUNCTION :
If a function f is defined by f (x) = a0 xn + a1 xn1 + a2 xn2 + ... + an1 x + an where n is a non negative integer and
a0, a1, a2, ..., an are real numbers and a0  0, then f is called a polynomial function of degree n .
NOTE : (a) A polynomial of degree one with no constant term is called an odd linear
function . i.e. f(x) = ax , a  0

(b) There are two polynomial functions , satisfying the relation ;

f(x).f(1/x) = f(x) + f(1/x). They are :
(i) f(x) = xn + 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 satisfies an algebraic equation of the form
P 0 (x) y n + P 1 (x) y n 1 + ....... + P n 1 (x) y + P n (x) = 0 W here n is a positive integer and
P0 (x), P1 (x) ........... are Polynomials in x.
e.g. y = x is an algebraic function, since it satisfies the equation y²  x² = 0.
Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called
T RANSCEDENTAL FUNCTION .

(iii) FRACTIONAL RATIONAL FUNCTION :

g(x )
A rational function is a function of the form. y = f (x) = , where
h (x )
g (x) & h (x) are polynomials & h (x)  0.
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(iv) EXPONENTIAL FUNCTION :
A function f(x) = ax = ex ln a (a > 0 , a  1, x  R) is called an exponential function. The inverse of the
exponential function is called the logarithmic function . i.e. g(x) = loga x .
Note that f(x) & g(x) are inverse of each other & their graphs are as shown .
 + + 
1
x ,
a>
a (0, 1)
) =  f(x) = ax , 0 < a < 1
f(x (0, 1)
)45º )45º
 
(1, 0) (1, 0)
 x

x
x

l og a

=
=

)=

y
y

g( x g(x) = loga x

(v) ABSOLUTE VALUE FUNCTION :

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

 x if x  0
= x 
  x if x  0
(vi) SIGNUM FUNCTION :
A function y= f (x) = Sgn (x) is defined as follows :
y
 1 for x  0 y = 1 if x > 0
y = f (x) =  0 for x  0
 > x
  1 for x  0 O y = Sgn x
It is also written as Sgn x = |x|/ x ; y = 1 if x < 0
x  0 ; f (0) = 0
(vii) GREATEST INTEGER OR STEP UP FUNCTION :
The function y = f (x) = [x] is called the greatest integer function where [x] denotes the greatest integer
less than or equal to x . Note that for :
1  x < 0 ; [x] =  1 0x< 1 ; [x] = 0
1x< 2 ; [x] = 1 2x < 3 ; [x] = 2
and so on .
Properties of greatest integer function :
y
graph of y = [x] 3
(a) [x]  x < [x] + 1 and
2  º
x  1 < [x]  x , 0  x  [x] < 1
(b) [x + m] = [x] + m if m is an integer . 1  º
(c) [x] + [y]  [x + y]  [x] + [y] + 1  º 
(d) [x] + [ x] = 0 if x is an integer 3 2 1 1 2 x
3
 º 1
=  1 otherwise .
 º 2
(viii) FRACTIONAL PART FUNCTION : 3
It is defined as :
g (x) = {x} = x  [x] .
y graph of y = {x}
e.g. the fractional part of the no. 2.1 is
2.1 2 = 0.1 and the fractional part of  3.7 is 0.3. The
1 º   º
º º


period of this function is 1 and graph of this function is as

shown . 


     x
1 1 2

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4. DOMAINS AND RANGES OF COMMON FUNCTION :

Function Domain Range

(y = f (x) ) (i.e. values taken by x) (i.e. values taken by f (x) )

A. Algebraic Functions

(i) xn , (n  N) R = (set of real numbers) R, if n is odd

R  {0} , if n is even
+

1
(ii) 
x n , (n N) R – {0} R – {0} , if n is odd

R+ , if n is even

(iii) x1 / n , (n  N) R, if n is odd R, if n is odd

R+  {0} , if n is even R+  {0} , if n is even

1
(iv) 1 / n , (n  N) R – {0} , if n is odd R – {0} , if n is odd
x
R+ , if n is even R+ , if n is even

B. Trigonometric Functions

(i) sin x R [–1, + 1]

(ii) cos x R [–1, + 1]


(iii) tan x R – (2k + 1) , kI R
2

(iv) sec x R – (2k + 1) , kI (–  , – 1 ]  [ 1 ,  )
2
(v) cosec x R – k , k  I (–  , – 1 ]  [ 1 ,  )
(vi) cot x R – k , k  I R
C. Inverse Circular Functions (Refer after Inverse is taught )

  
(i) sin–1 x [–1, + 1]  2 , 2 
 
(ii) cos–1 x [–1, + 1] [ 0, ]

  
(iii) tan–1 x R  , 
 2 2

  
(iv) cosec –1x (–  , – 1 ]  [ 1 ,  )  2 , 2  – { 0 }
 

 
(v) sec–1 x (–  , – 1 ]  [ 1 ,  ) [ 0, ] –  
2 
(vi) cot –1 x R ( 0, )

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 Function Domain Range
(y = f (x) ) (i.e. values taken by x) (i.e. values taken by f (x) )

D. Exponential Functions

(i) ex R R+
(ii) e1/x R–{0} R+ – { 1 }
(iii) ax , a > 0 R R+
(iv) a1/x , a > 0 R –{0} R+ – { 1 }

E. Logarithmic Functions

(i) logax , (a > 0 ) (a  1) R+ R

1
(ii) logxa = R+ – { 1 } R–{0}
log a x
(a > 0 ) (a  1)

F. Integral Part Functions Functions

(i) [x] R I

1 1 
(ii) R – [0, 1 )  , n  I  {0} 
[x ] n 

G. Fractional Part Functions

(i) {x} R [0, 1)

1
(ii) R–I (1, )
{x}

H. Modulus Functions
(i) |x| R R+  { 0 }

1
(ii) R–{0} R+
|x|

I. Signum Function

|x|
sgn (x) = ,x0 R {–1, 0 , 1}
x
=0,x=0

J. Constant Function

say f (x) = c R {c}

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5. 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 common domain. eg.
1 x
f(x) = & g(x) = are identical functions .
x x2
6. CLASSIFICATION OF FUNCTIONS :
One  One Function (Injective mapping) :
A function f : A  B is said to be a oneone function or injective mapping if different elements of A have
different f images in B . Thus for x1, x2  A & f(x1) ,
f(x2)  B , f(x1) = f(x2)  x1 = x2 or x1  x2  f(x1)  f(x2) .
Diagramatically an injective mapping can be shown as

OR

Note : (i) Any function which is entirely increasing or decreasing in whole domain, then
f(x) is oneone .
(ii) If any line parallel to xaxis cuts the graph of the function atmost at one point,
then the function is oneone .
Many–one function :
A function f : A  B is said to be a many one function 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(x1) = f(x2) but x1  x2 .
Diagramatically a many one mapping can be shown as

OR

Note : (i) Any continuous function which has atleast one local maximum or local minimum, then f (x) is
manyone . In other words, if a line parallel to xaxis cuts the graph of the function atleast at
two points, then f is manyone .
(ii) If a function is oneone, it cannot be manyone and vice versa .
Onto function (Surjective mapping) :
If the function f : A  B is such that each element in B (codomain) is the f image of atleast one element
in A, then we say that f is a function of A 'onto' B . Thus f : A  B is surjective iff  b  B,  some a 
A such that f (a) = b .
Diagramatically surjective mapping can be shown as

OR

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

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 Into function :
If f : A  B is such that there exists atleast one element in codomain which is not the image of any
element in domain, then f(x) is into .
Diagramatically into function can be shown as

OR

Note that : If a function is onto, it cannot be into and vice versa . A polynomial of degree even will always be
into.
Thus a function can be one of these four types :

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

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

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

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

Note : (i) If f is both injective & surjective, then it is called a Bijective mapping.
The bijective functions are also named as invertible, non singular or biuniform functions.
(ii) If a set A contains n distinct elements then the number of different functions defined from A  A
is nn & out of it n ! are one one.
Identity function :
The function f : A  A defined by f(x) = x  x  A is called the identity of A and is denoted by IA.
It is easy to observe that identity function is a bijection .
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 . Thus
f : A  B ; f(x) = c ,  x  A , c  B is a constant function. Note that the range of a constant function
is a singleton and a constant function may be one-one or many-one, onto or into .

7. ALGEBRAIC OPERATIONS ON FUNCTIONS :

If f & g are real valued functions of x with domain set A, B respectively, then both f & g are defined in
A  B. Now we 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)

f f (x )
(iii)   (x) = domain is {x  x  A  B s . t g(x)  0} .
 g g (x)
8. COMPOSITE OF UNIFORMLY & NON-UNIFORMLY DEFINED FUNCTIONS :
Let f : A  B & g : B  C be two functions. Then the function gof : A  C defined by
(gof) (x) = g (f(x))  x  A is called the composite of the two functions f & g .
x f (x)
Diagramatically     g (f(x)) .
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 Thus the image of every x  A under the function gof is the gimage of the fimage of x .
Note that gof is defined only if  x  A, f(x) is an element of the domain of g so that we can take its g-image.
Hence for the product gof of two functions f & g, the range of f must be a subset of the domain of 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 are defined, 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 is also a bijection.
9. HOMOGENEOUS FUNCTIONS :
A function is said to be homogeneous with respect to any set of variables when each of its terms
is of the same degree with respect to those variables .
For example 5 x2 + 3 y2  xy is homogeneous in x & y . Symbolically if ,
f (tx , ty) = tn . f (x , y) then f (x , y) is homogeneous function of degree n .
10. BOUNDED FUNCTION :
A function is said to be bounded if f(x)  M , where M is a finite quantity .
11. IMPLICIT & EXPLICIT FUNCTION :
A function def ined by an equation not solved for the dependent variable is called an
IMPLICIT FUNCTION . For eg. the equation x3 + y3 = 1 defines y as an implicit function. If y has been expressed
in terms of x alone then it is called an EXPLICIT FUNCTION.

12. INVERSE OF A FUNCTION :

Let f : A  B be a oneone & onto function, then their 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 = IB and
gof = IA , where IA & IB are identity functions on the sets A & B respectively.
Note that the graphs of f & g are the mirror images of each other in the line y
2
= x . As shown in the figure given below a point (x ',y ' ) corresponding to y = x (x >0) changes to (y
',x ' ) corresponding to y   x , the changed form of x = y .

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

(iv) If f & g are two bijections f : A  B , g : B  C then the inverse of gof exists and
(gof)1 = f 1 o g1 .
13. ODD & EVEN FUNCTIONS :
If f (x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function.
e.g. f (x) = cos x ; g (x) = x² + 3 .
If f (x) = f (x) for all x in the domain of ‘f’ then f is said to be 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) = 0 => f (x) is odd .
(b) A function may neither be odd nor even .
(c) Inverse of an even function is not defined .

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 (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 an even & an odd function.
f ( x )  f (  x ) f ( x)  f (  x)
e.g. f ( x)  
2 2

(f) The only function which is defined on the entire number line & 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 function f.g will be even but if any one of
them is odd then f.g will be odd .
14. PERIODIC FUNCTION :
A function f(x) is called periodic if there exists a positive number T (T > 0) called the period of the function
such that 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 no fundamental period.
(d) If f (x) has a period T & g (x) also has a period T then it does not mean that
f (x) + g (x) must have a period T . e.g. f (x) = sinx+ cosx.

1
(e) If f(x) has a period p, then and f (x) also has a period p.
f (x)
(f) if f(x) has a period T then f(ax + b) has a period T/a (a > 0).

15. 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) = xn, n  R
(iii) f(x + y) = f(x) . f(y)  f(x) = akx.
(iv) f(x + y) = f(x) + f(y)  f(x) = kx, where k is a constant.

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 PROFICIENCY TEST-01
1. Which of the following is a function ?
(A) {(2, 1), (2, 2), (2, 3), (2, 4)} (B) {(1, 4), (2, 5), (1, 6), (3, 9)}
(C) {(1, 2), (3, 3), (2, 3), (1, 4)} (D) {(1, 2), (2, 2), (3, 2), (4, 2)}
2. Find the domains of definitions of the following functions :
(Read the symbols [*] and {*} as greatest integer and fractional part functions respectively.)

(i) f (x) = cos2x  16  x 2 (ii) f (x) = log7 log5 log3 log2 (2x3 + 5x2  14x)

1  5x
(iii) f (x) = ln  x 2  5x  24  x  2  (iv) f (x) =
  7 x 7
2
 2 log10 x  1 
(v) y = log10 sin (x  3)  16  x (vi) f (x) = log100x  
 x 
1 x
(vii) f (x) =  ln x(x 2  1) (viii) f (x) = log 1 2
2 2 x 1
4x  1
1
(ix) f (x) = x2 | x |  (x) f (x) = ( x 2  3x  10) . ln 2 ( x  3)
9  x2
cos x  (1 2)
(xi) f(x) = logx (cos 2x) (xii) f (x) =
6  35x  6 x 2
[x]
(xiii) f (x) = 
log1 / 3 log 4  [x] 2
5  (xiv) f (x) =
2x  [ x ]
(xv) f (x) = logx sin x

3. Find the domain & range of the following functions.

(Read the symbols [*] and {*} as greatest integer and fractional part functions respectively.)
2x
(i) f(x) = log 5  2 (sin x  cos x)  3  (ii) f(x) =
1 x2
x 2  3x  2 x
(iii) f(x) = (iv) f (x) =
x2  x  6 1 | x |
(v) f(x) = 2 + x – [x – 3] (vi) f(x) = sin–1 x2  x  1
(vii) f(x) = log3(5 + 4x – x2)
4. The range of the function f(x) = |x – 1| + |x – 2|, – 1  x 3 is
(A) [1, 3] (B) [1, 5] (C) [3, 5] (D) None of these

5. The range of the function f(x) = 2 |sin x| – 3 |cos x| is :

(A) [–2, 13 ] (B) [–2, 3] (C) [3, 13 ] (D) [–3, 2]

6. (i) The function f(x) is defined on the interval [0,1]. Find the domain of definition of the functions.
(a) f (sin x) (b) f (2x+3)
(ii) Given that y = f (x) is a function whose domain is [4, 7] and range is [–1, 9]. Find the range and domain of
1
(a) g (x) = f (x) (b) h (x) = f (x – 7)
3

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 PROFICIENCY TEST-02
1. Classify the following functions f(x) definzed in R  R as injective, surjective, both or none .
x 2  4x  30
(a) f(x) = (b) f(x) = x3  6 x2 + 11x  6 (c) f(x) = (x2 + x + 5) (x2 + x  3)
x 2  8x  18

2 2
x2 e x – e–x
(d) f(x) = (e) f(x) = x + |x| (f) f(x) = ex – e–x (g) f(x) = 2 2
1  x2 e x  e–x

 1  1
2. If f(x) = |x| and g(x) = [x], then value of fog  –   gof  –  is, ([x] denotes greatest integer function)
 4  4

(A) 0 (B) 1 (C) – 1 (D) 1/4

3. If f : R  R, f(x) = x3 + 3, and g : R  R, g(x) = 2x + 1, then f –1 og–1(23) equals :

(A) 2 (B) 3 (C) (14)1/3 (D) (15)1/3

4. Which of the following functions has its inverse :

(A) f : R  R, f(x) = ax (B) f : R R, f(x) = |x| + |x – 1|
(C) f : R  R+, f(x) = |x| (D) f : [, 2]  [– 1, 1], f(x) = cos x

 2, when x  Q
5. If function f(x) =  , (fof) 4 the value will be :
0, when x  Q

(A) 0 (B) 2 (C) 2 (D) None of these

 1 x   3x  x3 
6. If f(x) = log  1 – x  and g(x) =  
   1  3 x 2  , then f[g(x)] is equal to :
 
(A) – f(x) (B) 3f(x) (C) [f(x)]3 (D) None of these

7. If f : R  R, g : R  R and f(x) = 3x + 4 and (gof) (x) = 2x – 1, then the value of g(x) is :

1
(A) 2x – 1 (B) 2x – 11 (C) (2x – 11) (D) None of these
3

8. If f : R  R, f(x) = x2 + 2x – 3 and g : R  R, g(x) = 3x – 4, then the value of fog (x) is :

(A) 3x2 + 6x – 13 (B) 9x2 – 18x + 5 (C) (3x – 4)2 + 2x – 3 (D) None of these

   
9. If f(x) = sin²x + sin²  x    cos x cos  x   and g(x) is a one-one function defined in RR, then (gof) (x)
 3 3
is
(A) One-one (B) Onto
(C) Constant function (D) Periodic with fundamental period 

10. Compute the inverse of the functions:

x
10 x  10  x
f (x) = ln  x  x 2  1
x 1
(a) (b) f (x) = 2 (c) y=
10 x  10  x

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 PROFICIENCY TEST-03
1. Find whether the following functions are even or odd or none
x (a x  1)
(a) f(x) = log x  1  x 2  (b) f(x) = (c) f(x) = sin x + cos x
  a x 1 2

(d) f(x) = x sin2 x  x3 (e) f(x)= sin x  cos x (f) f(x) =

1  2 x

2x

 x2n  e1/x – e –1/ x  

(g) f(x) =  2n 2n1 
 e1/x  e –1/ x   , x  0 and n  N (h) f(x) = [(x+1)²]1/3 + [(x 1)²]1/3
 (x sgn x)   

 1 1
2. Let f  x    x 2  2 (x  0), then f(x) equals :
 x x

(A) x2 – 2 (B) x2 – 1 (C) x2 (D) None of these

3. Find the period of following function:

(i) f(x)= |sin 2x| is :
(A) /4 (B) /2 (C)  (D) 2

 x   x 
(ii) f(x)= sin    cos   is
2
   2 
(A) 4 (B) 6 (C) 12 (D) 24

(iii) f(x) = log cos 2x + tan 4x is

(A) /2 (B)  (C) 2 (D) 2/5
4. In the following which function is not periodic
(A) tan 4x (B) cos 2x (C) cos x2 (D) cos2x

5. Suppose f is a real function satisfying f (x + f (x)) = 4 f (x) and f (1) = 4. Find the value of f (21).

2
6. Let 'f' be a function defined from R+  R+ . If [ f (xy)]2 = x  f(y) for all positive numbers x and y and

f (2) = 6, find the value of f (50).

7. Let f (x) be a function with two properties

(i) for any two real number x and y, f (x + y) = x + f (y) and
(ii) f (0) = 2.
Find the value of f (100).
8. The period of cos(x + 4x + 9x + .... + n2x) is /7, then n  N is equal to :
(A) 2 (B) 3 (C) 4 (D) 5

9. Write explicitly, functions of y defined by the following equations and also find the domains of definition of the
given implicit functions :
(a) 10x + 10y = 10 (b) x + y= 2y
1
10. Function f & g are defined by f(x) = sin x, xR ; g(x) = tan x , xR   K   
 2
where K I . Find (i) periods of fog & gof. (ii) range of the function fog & gof .

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 EXERCISE–I
1. Find the domains of definitions of the following functions :
(Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.)

  
  1 
(i) f (x) = log2   log1/ 2 1 
 
+ log10 log10 x  log10  4  log10 x  log10 3


  x
sin 100  

1 1 1
(ii) f (x) = + log1 – {x}(x2 – 3x + 10) + +
[x ] 2| x| sec(sin x)

1
 7 
(iii) f (x) = (5x  6  x 2 ) lnx + (7 x  5  2x 2 ) +  ln   x  
 2 

2 16 x 203x
(iv) f (x) = log  1  x  x  6  C 2 x 1  P2 x 5
x 
 x

f(x) = log 
3 .
(v) 2

log|sin x| x  8x  23 

 log 2 | sin x | 

2. Find the domain & range of the following functions.

(Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.)

(i) y = 2  x  1  x

(ii) f (x) = log(cosec x - 1) (2  [sin x]  [sin x]2)

x 4 3
(iii) f (x) =
x 5

3. Column-I Column-II

  2 
(A) f (x) = cos  sin x   cos x  (P) Domain of f (x) is (– , )
 3 3 
(B) f (x) = log2 (| sin x | + 1) (Q) Range of f (x) contains only one positive integer
(C) f (x) = cos {[x] + [–x]} (R) f (x) is many-one function
(D) f (x) = [{| ex |}] (S) f (x) is constant function
where [x] and {x} denotes greatest integer and fractional part function.

4. (a) The function f (x) defined on the real numbers has the property that f  f ( x ) · 1  f ( x )  = – f (x)
for all x in the domain of f. If the number 3 is in the domain and range of f, compute the value of f (3).

(b) Let f be a function such that f (3) = 1 and f (3x) = x + f (3x – 3) for all x. Then find the value of f (300).
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  1 x 
5. A function f : R  R is such that f   = x for all x  – 1. Prove the following.
1 x 

(a) f  f ( x)  = x (b) f 1 x  = – f (x), x  0 (c) f (– x – 2) = – f (x) – 2.

 x  1,  1  x  2
6. f(x) =  . Find domain and range of f(f(x)).
4  x, 2  x  5

7. Let f (x) = ax 2  bx . Find the set of real values of 'a' for which there is at least one positive real value of 'b'
for which the domain of f and the range of f are the same set.

x if x 1
 1  x if x  0 
8. f (x) =  and g (x) =  find (fog)(x) and (gof)(x)
 x2 if x0  1  x if x 1

log10 x
9. Find the inverse of f (x) = 2  8 and hence solve the equation f (x) = f–1(x).

10. Suppose p(x) is a polynomial with integer coefficients. The remainder when p(x) is divided by x – 1 is 1 and
the remainder when p(x) is divided by x – 4 is 10. If r (x) is the remainder when p(x) is divided by (x – 1)(x –
4), find the value of r (2006).

1
e  | ln{ x }| | ln{ x }|
 {x} where ever it exists
11. (i) Prove that the function defined as , f (x) = 

 {x} otherwise , then

f (x) is odd as well as even. (where {x} denotes the fractional part function )
x x
(ii) If f(x)= x
  1 , then f(x) is
e 1 2
(A) an odd function (B) an even function (C) neither even nor odd (D) both even and odd

 1    1  x 
12. In a function 2 f(x) + xf    2f  2 sin    x     = 4 cos2 + x cos
 x    4  2 x
Prove that (i) f(2) + f(1/2) = 1 and (ii) f(2) + f(1) = 0

13. Let f (x) = x135 + x125 – x115 + x5 + 1. If f (x) is divided by x3 – x then the remainder is some function of x say
g (x). Find the value of g (10).

14. Let f (x) = (x + 1)(x + 2)(x + 3)(x + 4) + 5 where x  [–6, 6]. If the range of the function is
[a, b] where a, b  N then find the value of (a + b).

15. If a, b  R be fixed positive numbers such that f(a + x) = b + [b3 + 1 – 3b2. f(x) + 3b{f(x)}2 – {f(x)}3]1/3
for all x  R then prove that f(x) is a periodic function.

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16. The graph of the function y = f (x) is as follows.

Match the function mentioned in Column-I with the respective graph given in Column-II.
Column- I Column-II
y
1
–2 –1 1 2 x
(A) y = | f (x) | (P) O
–1
y
1

x
(B) y = f ( | x | ) (Q) –2 –1 O 1 2
–1
y
1
x
(C) y = f (– | x | ) (R) –2 –1 O 1 2
–1

y
1
1
(D) y = ( | f (x) | – f (x) ) (S) –2
x
2 –1 O 1 2
–1

17. Column I contains functions and column II contains their natural domains. Exactly one entry of column II
matches with exactly one entry of column I.
Column I Column II

1 x 1
(A) f (x) = sin   (P) (1, 3)  (3, )
 x 
 x 2  3x  2 
(B) g (x) = ln  (Q) (– , 2)
 x  1 

1  1
(C) h (x) = (R)   ,  
 x 1  2
ln  
 2 

 2 
(D) (x) = ln  x  12  2 x  (S) [–3, –1)  [1, )
 

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 EXERCISE–II
1. Let f be a oneone function with domain {x,y,z} and range {1,2,3}. It is given that exactly one of the following
statements is true and the remaining two are false.
f(x) = 1 ; f(y)  1 ; f(z)  2. Determine f1(1)

2. Let x = log49 + log928

show that [x] = 3, where [x] denotes the greatest integer less than or equal to x.

3.(a) A function f is defined for all positive integers and satisfies f(1) = 2005 and f(1)+ f(2)+ ... + f(n) = n2f(n)
for all n > 1. Find the value of f(2004).
(b) If a, b are positive real numbers such that a – b = 2, then find the smallest value of the constant L for which

x 2  ax  x 2  bx < L for all x > 0.

(c) Let f (x) = x2 + kx ; k is a real number. The set of values of k for which the equation f (x) = 0 and f  f (x )
= 0 have same real solution set.
(d) If f (2x + 1) = 4x2 + 14x, then find the sum of the roots of the equation f (x) = 0.

(e) Let a and b be real numbers and let f (x) = a sin x + b 3 x + 4,  x  R. If f log10 (log 3 10)  = 5 then

find the value of f log10 (log10 3) . 
 1  1
4. Let [x] = the greatest integer less than or equal to x. If all the values of x such that the product  x  x  2 
 2   
2 2 2 2
is prime, belongs to the set [x1, x2)  [x3, x4), find the value of x1  x 2  x 3  x 4 .

5. Let f : R  R – {3} be a function with the property that there exist T > 0 such that

f (x )  5
f (x + T) = for every x  R. Prove that f (x) is periodic.
f (x)  3
6. If f (x) = 1 + x  2 , 0  x  4
g (x) = 2  x ,  1  x  3
Then find fog (x), gof (x), fof (x) & gog (x). Draw rough sketch of the graphs of fog (x) & gof (x) .

7. Let {x} & [x] denote the fractional and integral part of a real number x respectively. Solve 4{x}= x + [x]

9x  1   2   3   2005 
8. Let f (x) = x then find the value of the sum f  +f   +f   + ....+ f  
9 3  2006   2006   2006   2006 

3 4
9. The set of real values of 'x' satisfying the equality   +   = 5 (where [ ] denotes the greatest integer
x x
 b b
function) belongs to the interval  a ,  where a, b, c  N and is in its lowest form. Find the value of a
 c c
+ b + c + abc.
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10. f (x) and g (x) are linear function such that for all x,   f  g ( x ) and g  f ( x )  are Identity functions.
If f (0) = 4 and g (5) = 17, compute f (2006).

11. If for all real values of u & v, 2 f(u) cos v = f (u + v) + f (u  v), prove that, for all real values of x.

(i) f (x) + f ( x) = 2a cos x

(ii) f ( x) + f( x) = 0
(iii) f ( x) + f (x) =  2b sin x. Deduce that f (x) = a cos x  b sin x, a, b are arbitrary constants.

12. Find out for what integral values of n the number 3 is a period of the function :
f(x) = cos nx . sin (5/n) x.

sin x  sin 3x  sin 5x  sin7x

13. If f(x) = , then the fundamental period of f(x) is :
cos x  cos 3x  cos 5x  cos 7x

 
(A) (B) (C)  (D) None of these
4 2

x|x|
14. If f(x) = – then f –1(x) equals.
1  x2

|x| |x| x
(A) (B) (sgn(–x)) (C) – (D) None of these
1– | x | 1– | x | 1– x

15. If f(x) satisfies x + |f(x)| = 2 f(x) then f –1(x) satisfies.

(A) 3x + |f –1(x)| = 2f –1(x) (B) x + |f –1(x)| = 2f –1(x)
(C) f –1(x) – |x| = 2x (D) 3x – |f–1(x)| = 2f –1(x)

16. In the following, [ ] and { } are greatest integer function and fractional part function respectively.

Column-I Column -II

 4x 
(A) f 1(x) =   sgn(x2 – x + 1) (P) discontinuous at more than 3 points
  

but less than 6 points in [–2, 2]

  2x  1  
(B) f 2(x) = cos–1  sgn cos 2    (Q) non derivable at more than 2 points
  

but atmost 5 points in [–2, 2]

(C) f 3(x) = max. ({x + sin2x + cos2x}, {5 – x}) (R) range contains atleast one integer but not more
than seven and no irrational value in [–2, 2]

(D) f 4(x) = x 2 + [x]2 (S) Many one but not even function in [–2, 2]
(T) neither odd nor periodic in [–2, 2]

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17. Column I Column II
(A) The integral values of x for which (P) –1

 
 2[| sin x |  | cos x |] 
 
f(x) = cos–1  2 11  , is defined
 sin x  2 sin x  
 4 

(where [.] denotes greatest integer function) is/are

(B) The possible value(s) of tan a, such that [cos a] + [sin a + 1] = 0, (Q) 0

where [.] denote greatest integer function is/are

(C) The integers in the domain of function (R) 2

f(x) = log10(ax3 + (b + a)x2 + (b + c)x + c) if b2 < 2ac, a > 0 is/are

(D) The number of integers in the domain of function, (S) 4

f(x) = sgn(log(4 – |x|)(x2 + 4x + 4)) is

Comprehension (Q.18 to Q.20)

x 2  1,  1  x  1
Consider the function f(x) = 
nx, 1 x  e

Let f 1(x) = f(|x|)

f 2(x) = |f(|x|)|
f 3(x) = f(–x)
Now answer the following questions.

18. Number of positive solution of the equation 2f2(x) – 1 = 0 is (are)

(A) 4 (B) 3 (C) 2 (D) 1

19. Number of integral solution of the equation f1(x) = f 2(x) is (are)

(A) 1 (B) 2 (C) 3 (D) 4

20. If f 4(x) = log27(f 3(x) + 2), then range of f4(x) is

1   1
(A) [1, 9] (B)  ,   (C) 0,  (D) [1, 27]
3   3

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 EXERCISE–III
 4x  3   4x  3 
1. The period of the function f(x) = 4 sin4  2
 + 2 cos   is :
 6   32 

3 2 33 4 2 4 3
(A) (B) (C) (D)
4 4 3 3

 1 x 
2. If 2f(x – 1) – f   = x, then f(x) is :
 x 

1 1  (1  x ) 1 1 1 
(A) 2(1  x )   (B) 2(x – 1) – (C) x2 + 2
4 (D) ( x  2)  
3 (1  x )  x x 4 ( x  2) 

3. If f : R  R be a function satisfying f(2x + 3) + f(2x + 7) = 2, x R, then period of f(x) is :

(A) 2 (B) 4 (C) 8 (D) 16

1
4. If 2 < x2 < 3, then the number of positive roots of {x2} =   , (where {x} denotes the fractional part of x) is :
x 
(A) 0 (B) 1 (C) 2 (D) 3

5. f(x) =
 
2 e x  e  x sin x  tan x  is (where [] denotes the greatest integer function)
 x  2 
2  3
  

(A) an odd function (B) an even function (C) neither even nor odd (D) both even and odd

 x 2 
6. The domain of the function f(x) = sin–1  log2 is :
 2 
(A) [– 2, 2] – (–1, 1) (B) [– 1, 2] – {0} (C) [1, 2] (D) [– 2, 2] – {0}

7. The function f : [a, ) R, where R denotes the range corresponding to the given domain, with rule f(x) = 2x3
– 3x2 + 6, will have an inverse provided
(A) a  1 (B) a 0 (C) a  0 (D) a  1

8. If f : R  R, f(x) = ex & g : R  R, g(x) = 3x – 2, then the value of (fog)–1(x) is equal to

2  log x x3
(A) log (x – 2) (B) (C) log   (D) None of these
3  2 

| x | –3 
9. The function f(x) = cos–1   + (loge(4 – x))–1 is defined for
 2 
(A) [– 1, 0]  [1, 5] (B) [– 5, – 1]  [1, 4)
(C) [– 5, – 1]  [1, 4) – {3} (D) [1, 4) – {3}

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 x – [ x]
10. Let f(x) = , then range of f(x) is ([.] = G...F.) :
1 – [ x]  x
(A) [0, 1] (B) [0, 1/2] (C) [1/2, 1] (D) [0, 1/2)

11. If f(x) be a polynomial satisfying f(x) . f(1/x) = f(x) + f(1/x) and f(4) = 65 then f(6) = ?
(A) 176 (B) 217 (C) 289 (D) None of these

12. The range of the function f(x) = sin(cos x)  cos(sin x) is :

(A) 1, 1  cos1 (B)  cos1,1  cos1 (C)  cos1, 1  sin1 (D) 1, 1  sin1

13. Let f(x) = cot(5  3x)(cot(5)  cot(3x)) – cot 3x  1 , then domain is :

 n    n n – 5   n – 5 
(A) R –   , n  I (B) (2n + 1) ,nI (C) R –  , , n  I (D) R–  , n  I
3 6 3 3   3 

14. If f(x) is even, periodic function defined for all x  R and has period 1, then

 1 1  2 
(A) f  x   = f(x) (B) f   x  = f  – x  (C) f(x + 1) = f(2x + 1) (D) f(0) can not be zero
 2 3  3 

15. The number of bijective functions f : A  A, where A = {1, 2, 3, 4} such that f(1)  3, f(2)  1, f(3)  4, f(4)  2
is :
(A) 11 (B) 23 (C) 12 (D) 9

n(n  1)
16. The period of the function, f(x) = [x] + [2x] + [3x] + .... + [nx] – x, where n  N and [ ] denotes the
2
greatest integer function, is :

1
(A) 1 (B) n (C) (D) Non periodic
n
17. Let set A consists of 5 elements and set B consists of 3 elements. Number of functions that can be de-fined
from A to B which are not surjective, is :
(A) 99 (B) 93 (C) 123 (D) None

x–3
18. Let f(x) = , x  –1. Then f2010(2014) [where fn(x) = fof.....of
  (x)] is :
x 1 n times

(A) 2010 (B) 4020 (C) 4028 (D) 2014

19. Let f(x) = log2(|sinx| + |cosx|). The range of f(x) is

(A) [– 1, 0] (B) [0, 1/2] (C) [– 1/2, 0] (D) [0, 1]

20. Let f(x) = sin x , then

(A) f(x) is periodic with period 2 (B) f(x) is periodic with period 
(C) f(x) is periodic with period 42 (D) None of these
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 EXERCISE–IV
3
1. Domain of definition of the function f(x) = + log10 (x3 – x), is- [AIEEE 2003]
4  x2
(A) (– 1, 0)  (1, 2)  (2, ) (B) (1, 2)
(C) ( – 1, 0) (1, 2) (D) (1, 2)  (2, )

2. If f : R  R satisfies f(x+ y) = f(x) + f(y), for all

n
x, y  R and f(1) = 7, then  f (r ) is- [AIEEE 2003]
r 1
7n (n  1) 7n 7(n  1)
(A) (B) (C) (D) 7n (n+1)
2 2 2
3. A function f from the set of natural numbers to integers defined by

 n 1
 , when n is odd
f(n) =  2n is [AIEEE 2003]
 , when n is even
 2
(A) neither one-one nor onto (B) one-one but not onto
(C) onto but not one-one (D) one-one and onto both

4. The range of the function f(x) = 7– xPx–3 is- [AIEEE 2004]

(A) {1, 2, 3} (B) {1, 2, 3, 4, 5, 6} (C) {1, 2,3,4} (D) {1, 2, 3, 4, 5}
5. If f : R  S, defined by f(x) = sin x – 3 cos x + 1, is onto, then the interval of S is- [AIEEE 2004]
(A) [0, 3] (B) [–1, 1] (C) [0, 1] (D) [–1, 3]

6. The graph of the function y = f(x) is symmetrical about the line x = 2, then- [AIEEE 2004]
(A) f(x+ 2) = f(x – 2) (B) f(2 + x) = f(2 – x) (C) f(x) = f(–x) (D) f(x) = – f(–x)

sin 1( x  3)
7. The domain of the function f(x) = is- [AIEEE 2004]
9  x2

(A) [2,3] (B) [2,3) (C) [1,2] (D) [1, 2)

8. A real valued function f(x) satisfies the functional equation f(x – y) = f(x) f(y) – f (a – x) f(a + y) where a is a
given constant and f(0) = 1, then f(2a – x) is equal to - [AIEEE-2005]
(A) –f(x) (B) f(x) (C) f(a) + f(a – x) (D) f(–x)

9. Let f : N  Y be a function defined as f(x) = 4x + 3 where Y = |y  N : y = 4x + 3 for some x  N|. Show that
f is invertible and its inverse is [AIEEE 2008]

y3 y3 y3 3y  4

(A) g(y) = 4 + (B) g(y) = (C) g(y) = (D) g(y) =
4 4 4 3

10. For real x, let f(x) = x3 + 5x + 1, then - [AIEEE 2009]

(A) f is one – one but not onto R (B) f is onto R but not one – one

(C) f is one – one and onto R (D) f is neither one – one nor onto R

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11. Let f(x) = (x + 1)2 –1, x  –1

Statement – 1 : [AIEEE 2009]

The set {x : f(x) = f–1(x)} = {0, –1}.

Statement – 2 :

f is a bijection.

(A) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement -1

(B) Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1.

(C) Statement -1 is true, Statement -2 is false. (D) Statement -1 is false, Statement -2 is true.

1
12. The domain of the function f(x) =
| x |  x is :
(A) (–, ) (B) (0, ) (C) (–, 0) (D) (–, ) – {0}

13. If a  R and the equation –3(x – [x])2 + 2(x – [x]) + a2 = 0 [IIT Mains 2014]
(where [x] denotes the greatest integer  x) has no integral solution, then all possible values of a lie in the
interval
(A) (– , –2)  (2, ) (B) (–1, 0)  (0, 1) (C) (1, 2) (D) (–2, –1)

14. If X = {4n – 3n – 1 : n  N} and Y = {9(n – 1) : n N}, where N is the set of natural numbers, then X Y is equal
to : [IIT Mains 2014]
(A) Y (B) N (C) Y – X (D) X
 1
15. If f(x) + 2f   = 3x, x  0, and S = {x R : f(x) = f(–x)} ; then S : [IIT Main 2016]
x
(A) is an empty set (B) contains exactly one element
(C) contains exactly two elements (D) contains more than two elements

16. Let a, b, c  . If f(x) = ax2 + bx + c is such that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy,  x, y ,
10
then  f (n) is equal to :
n1
[IIT Main 2017]

(A) 190 (B) 255 (C) 330 (D) 165

 1 1 x
17. The function f :    – ,  defined as f(x) = , is : [IIT Main 2017]
 2 2 1  x2
(A) surjective but not injective (B) neither injective nor surjective
(C) invertible (D) injective but not surjective

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 EXERCISE–V
1. If the function f : [1, )  [1, ) is defined by f(x) = 2x (x  1), then f 1(x) is [JEE '99, 2]
x ( x  1)
 1 1 1
(A)  
 2
(B)
2

1  1  4 log2 x  (C)
2
1  1  4 log2 x  (D) not defined

2. The domain of definition of the function, y (x) given by the equation, 2x + 2y = 2 is

(A) 0 < x  1 (B) 0  x  1 (C)  < x  0 (D)  < x < 1
[JEE 2000  Scr.), 1 out of 35]

 1 , x  0

3.(a) Let g (x) = 1 + x  [ x ] & f (x) =  0 , x  0 . Then for all x , f (g (x)) is equal to
1 , x0
(A) x (B) 1
 (C) f (x) (D) g (x)
where [ ] denotes the greatest integer function.

(b) If f : [1 , )  [2 , ) is given by , f (x) = x + 1/x, then f 1 (x) equals

x  x2  4 x x  x2  4
(A) (B) (C) (D) 1  x2  4
2 1  x2 2
log (x  3)
(c) The domain of definition of f (x) = 2 2 is :
x  3x  2
(A) R \ { 1,  2} (B) ( 2, ) (C) R\{ 1,  2,  3} (D) ( 3, ) \ { 1,  2}
(d) Let E = {1, 2, 3, 4 } & F = {1, 2}. Then the number of onto functions from E to F is
(A) 14 (B) 16 (C) 12 (D) 8
x
(e) Let f (x) = , x  1 . Then for what value of  is f (f (x)) = x ?
x 1
[JEE 2001 (Screening) 5  1 = 5]

(A) 2 (B)  2 (C) 1 (D)  1.

4.(a) Suppose f(x) = (x + 1)2 for x  –1. If g(x) is the function whose graph is the reflection of the graph of f (x) with
respect to the line y = x, then g(x) equals
1
(A) – x – 1, x > 0 (B) , x > –1 (C) x  1 , x > –1 (D) x – 1, x > 0
(x  1) 2
(b) Let function f : R R be defined by f (x) = 2x + sinx for x  R. Then f is [JEE 2002 (Screening), 3 + 3]
(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

x2  x  2
5. (a) Range of the function f (x) = 2 is
x  x 1
(A) [1, 2] (B) [1,  ) (C) [2, 7/3] (D) (1, 7/3]
x
(b) Let f (x) = defined from (0, )  [ 0, ) then f (x) is [JEE 2003 (Scr),3+3]
1 x
(A) one- one but not onto (B) one- one and onto (C) Many one but not onto (D) Many one and onto

6. Let f (x) = sin x + cos x, g (x) = x2 – 1. Thus g ( f (x) ) is invertible for x  [JEE 2004 (Screening)]
(A) [–/2, 0] (B) [–/2, ] (C) [–/4, /4] (D) [0, /2]

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7. If the functions f (x) and g (x) are defined on R  R such that
0, x  irrational
 0, x  rational

f (x) =  , g (x) =  , then (f – g)(x) is
x, x  irrational x, x  rational
(A) one-one and onto (B) neither one-one nor onto
(C) one-one but not onto (D) onto but not one-one [JEE 2005 (Scr.)]

8. Let f(x) = x2 and g(x) = sinx for all x  R. Then the set of all x satisfying (f o g o g o f) (x) = (g o g o f) (x), where
(f o g) (x) = f(g(x)), is [JEE 2011]
(A)  n, n  {0 ,1, 2....} (B)  n, n  {1, 2,....}

(C)  2n, n  {..., – 2, – 1, 0 ,1, 2,....} (D) 2n, n  {..., – 2, – 1, 0 ,1, 2,....}
2

b–x
9. Let f : (0, 1)  R be defined by f(x) = , where b is a constant such that 0 < b < 1. Then
1 – bx
1
(A) f is not invertible on (0, 1) (B) f  f –1 on (0, 1) and f(b) = [JEE 2011]
f ' (0)
1
(C) f = f –1 on (0, 1) and f(b) = (D) f –1 is differentiable on (0, 1)
f ' (0)

10. The function f : [0, 3]  [1, 29], defined by f(x) = 2x3 – 15x2 + 36x + 1, is [JEE 2012]
(A) one-one and onto (B) onto but not one-one.
(C) one-one but not onto. (D) neither one-one nor onto.

  
11. Let f :  – ,   R be given by [IIT Advance 2014]
 2 2
f(x) = (log(sec x + tan x))3.
Then
(A) f(x) is an odd function (B) f(x) is a one-one function
(C) f(x) is an onto function (D) f(x) is an even functionx

12. Let f 1 : R  R, f 2 [0, )  R, f 3 : R  R and f4 : R  [0, ) be defined by [IIT Advance 2014]

| x | if x  0 sin x if x  0
f 1 (x) =  x ; f 2(x) = x2 ; f 3(x) =  ;
e if x  0  x if x  0

f2 ( f1( x )) if x  0
and f 4(x) = f ( f ( x ))  1 if x  0
2 1
List-I List-II
(P) f 4 is (1) onto but not one-one
(Q) f 3 is (2) neither continuous nor one-one
(R) f 2of 1 is (3) differentiable but not one-one
(S) f 2 is (4) continuous and one-one
Codes :
P Q R S P Q R S
(A) 3 1 4 2 (B) 1 3 4 2
(C) 3 1 2 4 (D) 1 3 2 4

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    
13. Let f(x) = sin  sin  sin x   for all x  R and g(x) = sin x for all x R. Let (fog) (x) denote f(g(x)) and
 6  2  2
(gof) (x) denote g(f(x)). Then which of the following is (are) true ? [IIT Advance 2015]
 1 1  1 1
(A) Range of f is   ,  (B) Range of fog is   , 
 2 2  2 2
f(x) 
(C) xlim  (D) There is an x  R such that (gof) (x) = 1
0 g(x) 6

14. Let X be a set with exactly 5 elements and Y bet a set with exactly 7 elements. If  is the number of one-one
1
functions from X to Y and  is the number of onto functions from Y to X, then the value of ( – ) is
5!
_______. [JEE Advance 2018]

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 ANSWER KEY
FUNCTIONS
PROFICIENCY TEST-01
1. D

 5   3     3 5   1
2. (i)  4 , 4    4 , 4    4 , 4  (ii)   4 ,  2  (2, )(iii) (–  , – 3]
     

 1   1 1 
(iv) (– , – 1) [0, ) (v) (3  2 < x < 3  ) U (3 < x  4) (vi)  0,   , 
 100   100 10 
1  5  1  5 
(vii) (1 < x < 1/2) U (x > 1) (viii)  , 0   ,  (ix) (3, 1] U {0} U [ 1,3 )
 2   2 

 1    5 
(x) { 4 }  [ 5, ) (xi) (0 , 1/4) U (3/4 , 1) U {x : x  N, x  2} (xii)   ,  ,6 
 6 3   3 
 1 
(xiii) [– 3,– 2)  [ 3,4) (xiv) R –  , 0
 2 
(xv) 2K < x < (2K + 1) but x  1 where K is nonnegative integer

3. (i) D : x R ; R : [0 , 2] (ii) D : x R ; R [ –1 , 1 ]

(iii) D : {xx R ; x  3 ; x  2} R : {f(x)f(x)R , f(x)  1/5 ; f(x)  1}
(iv) D : R ; R : (–1, 1) (v) D : x R ; R : [5, 6)
(vi) D : x [–1, 0] ; R : [/3, /2] (vii) D : x (–1, 5) ; R : (–  2]
4. B
5. D
6. (i) (a) 2K  x  2K +  where K I (b) [3/2 , 1]
(ii) (a) Range : [– 1/3, 3], Domain = [4, 7] ; (b) Range [–1, 9] and domain [11, 14]

PROFICIENCY TEST-02
1 (a) neither surjective nor injective (b) surjective but not injective
(c) neither injective nor surjective (d) neither injective nor surjective

(e) neither injective nor surjective (f) Both injective and surjective
(g) neither injective nor surjective
2. B 3. A 4. D 5. A 6. B 7. C 8. B

e x  e x log2 x 1 1 x
9. C 10. (a) ; (b) ; (c) log
2 log2 x  1 2 1 x

PROFICIENCY TEST-03
1 (a) odd, (b) even, (c) neither odd nor even, (d) odd, (e) neither odd nor even, (f) even,
(g) even, (h) even
2. A 3.(i) B (ii) A (iii) B 4. C 5. 64 6. 30 7. 102 8. B

9. (a) y = log (10  10x) ,   < x < 1 (b) y = x/3 when   < x < 0 & y = x when 0  x < + 
10. (i) period of fog is , period of gof is 2 ; (ii) range of fog is [1 , 1] , range of gof is [tan1, tan1]

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 EXERCISE–I
Q 1.
(i) {x 1000  x < 10000} (ii) (–2, –1) U (–1, 0) U (1, 2) (iii) (1, 2)  (2, 5/2);
 3 
(iv) x  {4, 5} (v) x (3, 5) ~ , 
 2 

Q.2

(i) D : 1  x  2 R:  3, 6 
(ii) 
D : x  (2n, (2n + 1))  2 n  6 , 2 n  2 , 2 n  56 , n  I  and
R : loga 2 ; a  (0, )  {1}  Range is (–, ) – {0}

 1   1 1
(iii) D : [– 4, ) – {5}; R :  0,   ,
 6   6 3 

Q.3 (A) P, Q, R; (B) P, Q, R; (C) P, Q, R, S; (D) P, R, S

Q.4 (a) – 3/4; (b) 5050
Q.6 Domain = [–1, 5]; Range = [0, 3]
Q.7 a  {0, – 4}

x if x0 x2 if x0
  x 2 if 0  x  1  1 x if 0  x  1
Q.8 (gof)(x) = ; (fog)(x) =
  x if x 1
1  x 2 if x 1
log ( x 8 )
Q.9 f –1 (x) = 10 2 ; x = 10 Q.10 6016 Q.11 (ii) B Q.13 21
Q.14 5049 Q.16 (A) S; (B) R; (C) P; (D) Q
Q.17 (A) R; (B) S; (C) P; (D) Q

EXERCISE–II
1
Q 1. f 1(1) = y Q.3 (a) , (b) 1, (c) [0, 4), (d) – 5, (e) 3 Q.4 11
1002
x 1 , 0  x  1
 (1  x) ,  1  x  0 3 x , 1  x  2
Q.6 fog (x) = ; gof (x) = ;
x1 , 0x2 x 1 , 2  x  3
5x , 3  x  4

x , 1  x  0
x , 0 x 1
fof (x) = ; gog (x) = x , 0x2
4x , 3x4
4x , 2  x  3

Q.7 x = 0 or 5/3 Q.8 1002.5 Q.9 20 Q.10 122

Q.12 ± 1, ± 3, ± 5, ± 15 Q.13 C Q.14 B Q.15 D
Q.16 (A) – P, Q, R, S, T (B) – P, Q, S (C) – P, (D) – P, Q, S, T
Q.17 (A) – Q, R ; (B) – P, Q; (C) – Q, R, S; (D) – S
Q.18 C Q.19 D Q.20 C

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 EXERCISE–III

1. B 2. A 3. C 4. B 5. A 6. A 7. D 8. B 9. C

10. D 11. B 12. C 13. B 14. B 15. D 16. A 17. B 18. D

19. B 20. D

EXERCISE–IV

1. A 2. A 3. D 4. A 5. D 6. B 7. B

8. A 9. C 10. C 11. B 12. C 13. B 14. A

15. C 16. C 17. A

EXERCISE–V

Q.1 B Q.2 D

Q.3 (a) B, (b) A, (c) D, (d) A, (e) D Q.4 (a) D; (b) A

Q.5 (a) D , (b) A Q.6 C Q.7 A 8. A 9. A

10. B 11. A, B, C 12. D 13. ABC 14. 119

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