Chapter

6
Functions
buprly
Quantity

Economics
economics lo analyze and model
Functions are tIsed in
such as supply and demand
various economic phenomena, utility functions, and
production and cost functions,
optimization problems.

Chapter Outline
Algebra of functions
6.2
i6.1 Function

Quick Review

Inverse of a function:
Xto set Y, if onto function, then
f: X’ Y’fisa function from set If f: X ’ Y be a one-one,
to cach clementxeX,a unique element
yE Y.
the mapping f : Y ’ X such that f ) =x,
of the function
Domain and Range: xe X, ye Y is called inverse
which fx)
Domain: All possible values of x for
f:X’ Y.
Numerical
exists.
for all values Modulus or Absolute value or
ii. Range: Allpossible values of f(r), function:
of x. =x, ifx >0
i.e., R= ly e Y:y= f(x)} =-X, ifx <0
iii. Rc Co-domain =0, ifx =0
One-One function or Injection: For f: R’R, D;= R, Rç= RU{0}
(injective) iff
Afunction f: X ’ Y is one-one Signum function:
i. x*y’fx) +f(y)
i. f(x) =fy) ’x*y sign (r) = H ifr 0
=0 ifx=0
Onto function or Surjection:
(surjective) iff Or, sign (r) =1 ifx >0
A function f: X ’Y is onto
Range of f= Co-domain of f. =-1 ifx <0
=0 ifx =0
Into Function:
function, if there For f: R ’R, D;=R
Afunction f: X Y is an into
pre-image in X. R(={-1, 0, 1}
exists an element in Y having no
or step
Many-One function: The Greatest Integer function
x, y e X function or floor function:
f: X ’ Y is a many-one function, if xeRLet [x]denotes the greatest Integer in x.
such thatx#y, but f(x) = fy)
[x]=x when xe I
Bijective function: ii. [x]=0 when 0<x<|
function both injective and surjective is called
A ii. x]<x when x I
bijectíve function.

118
x=kvhen ksx<k+
x]Sx<x] +1
1, ifk el
ii: (f- gx) = f(r) - g(x)
iii. (f g)) =f(x)ga)
Even and Odd Function:
Even function: If f(-)= f(x) iv. (f/g) (r) = f(«) gr) +0
gx)
Odd function: If Vxe domain
f-)=- xe domain
f() V. (kf)(x) = kf(r), k e R
Periodic function: A Some spccial functions:
A function f(x) is = kr, keR
If f(x +y)= f() + f), then f(x)
fx+ T) = f() xesaiddomain.
to be periodic
function if If f(xy) = f(r) + fy), then f(x) = log x
Here the
value ofT is called the period of the least +ve
i.

function. ii. If f(x + y) = f(x)f(v), then f(r) =e

Operations on functions:
(f+ g)(*) = f() + g()
iv.
Iff9 1(-)+),then f)*+1
Shortcuts

The number of functions from a finite set A Periodic functions

afinite set B=[n(B)A) into 10.
Period
Functions
The number of one-one functions that can be
defined from a set A into a finite set B is sin'x, cos'(); if (n = even)
n(B) PA ; if n(B) > n(A) sec'x,cosec'x; (ifn is odd and fraction) 2T
0 ; otherwise
The number of onto functions, that can be |sinx|, lcosx|, tanx, lcotel, (cosecxl, |secxl
1
defined from a finite set A, containing n
elements onto a finite set B, containing 2 x-[x], sin(x- [x), sin(x - [-x]), x-[-x]
sin (sin:), cos(cos:) 2
elements = 2" - 2

The number of onto functions from A to B cosx

2T
where, o(A) = m, o(B) = n and m 2n is
n

r=l
1+cosx 2T
finite set A Vcosx ,
The number of bijections from a 2

onto a finite set B is

n(A) ! ; ifn(A) = n(B) (sinr) + |cosx), sin'xt cosx 2
otherwise
x B)n (B>x A)]=n
If o(AnB) =n then o[(A
X-axis, cuts the graph of cosx +cos+ cos
If any line parallel to point, then function is 2
2
the function atmost one
one-one.
X-axis,
single line parallel to points,
+..+ cos co)
If there is even a
function atleast two
cuts the graph of the
then function ismany-one. cos(cosr) + cos(sinx) 2

For, Domain and Range: 2T

Defined if sin(sinx) t sin(cosr)
Functions (in the form)
f(x) >0
|sin x+ cos.x
f(r) >0
2Sinx + 2cosx 2

1 f(r) 0
f(r) 119
MHT-CET Triumph Maths (MCQS)
Classical Thinking
g(-4) and g'
6.1 Function 8. Using the graph ofy= g), find
1. Let A={1, 2, 3} and (-lt
B= (2, 3, 4}, then which of the following is a
function from A to B?
(A) (1. 2). (1, 3), (2, 3), (3, 3)}
(B) ((0, 3), (2, 4)} -6-5
(C) {(1, 3), (2, 3), (3, 3)}
(D) ((1,2), (2, 3), (3, 4), (3, 2)}
N is defined by
2. If the function f: N -6,4) (3,-5)
f(25) is equal to
f(r) =x, then f(16)+ f()
5 (A) 0,-6 (B) 0, -5
5
(A) (B) 7 (C) -6, 0 (D) -5, 0
6
for which f(r) =4
5 (D) 1 9. From the graph below, find x
(C) 3
=x-3x +2
3. If afunction f(r) is given as f(x) 4y= f(r)
for all x eR, then f(-1)=
(A) 6 (B) 0 y=4
(D) 8
(C) 2
4. Let f: R’R be defined by
(2x ; x>3
.fr)=x; l<1<3.
X
|3x ; xs1
(A) 3 and -1 (B) 4 and 0
Then f(-1) + f(2) + f(4) is (D) 0and -1
(B) 14 (C) 3 and 0
(A) 9 (D) 10 10. f(1) = 11, then a =
If f(x)= ax + 6 and 17
(C) 5 (C) 11 (D) 5
(A) 6 (B)
as f(r) =x- 3x + 2
5. If afunction f(r) is given 1) - f[a - 1) =
for allx e R, then f(a + h)= 11. Iff()= 4xx, then f(a+
(B) 2(4-a)
+2 +h (A) 4(2 - a)
(A) a+ (2a +3)h- 3a (C) 4(2+a) (D) 2(4 + a)
+h
(B) a + (2a-3)h + 3a + 2 period T, then
+h If f(r) is periodic function with> 0, is periodic
(C) a+ (2a - 3)h- 3a +2
12.
+h the function f(ar + b) where a
(D) a+ (2a +3)h + 3a + 2 with period
(A) Tb (B) aT
(D) T/a
6. If f(r) = x'+-,x*0then
X
(C) bT
13. Let f = {(1, 1), (2, 4), (0, -2), (-1, -5)} be a
(B) + linear function from Z into Z. Then,
fx) is
(A) * X

(A) f() = 3x- 2 (B) fr) = 6x-8

() (D) (C) f(r)= 5x-2 (D) f() = Zr +2
function, the domain is a
14. If in greatest integer
7. If f(r) =x- 6x +9, 0<x<4, then f(3) = set of realnumbers, then range will be set of
(A) 4 (A) Real numbers
Rational numbers
(B) 1 (B)
(C) 0 (C) Imaginary numbers
(D) Integers
(D) does not exist
 Chapter 6: Functions
15. Y
21. Domain of the function log-9| is
y=l (A) R (B) R--3, 3]
(C) R-{-3, 3} (D) (-3, 3}
X +X
22. Domain of the function Jlog{(5x-x*)/6} is
y=-1 is the graph of (A) (2, 3) (B) [2,3]
(C) [1, 2] (D) [1,3]
23. The domain of the function of f: R R defined
(A) Modulus function (B) Step function
(C) Signum function (D) Rational function by f(r) = *-7x+12 is
16. The diagram given below shows that (A) (-o, 3] U(4, o) (B) (3,4)
(C) (-o, 3] [4, o) (D) (-o, 3] n [4, o)
24. If the domain of function f() = - 6x +7 is
(-o0, o), then the range of function is
2
(A) (- 0, 0) (B) (-2, o)
(C) (-2, 3) (D) (-0, - 2)
4
25. The range of the function f(r) = 9- is
f:A + B
(A) (A) (0,3) (B) [0, 3]
fs a function from A to B (D) [0,3)
fis a one-one function from A to B (C) (0,3]
(B)
(C) fis a bijection from A to B x+2
26. The range of the function fx) = is
(D) fis not a function. |+2|
17. The diagram given below shows that (A) {0, 1} (B) {-1, 1}
: (C) R (D) R- (-2}
6.2 Algebra of functions
b
2 1. Inverse of the function y = 2x -3 is
x+3 x-3
(A) 2
(B) 2
1 1
f: A B (C) 2r-3 (D) X+3
(A) fis a function from A to B
(B) fis a one-one function from A to B 2. Iff: N’N, f() =x +3, then f)=
(C) fis an onto function from A to B (A) x+3 (B) does not exists
(D) fis not a function. (C) x-3 (D) 3-x
18. Iff: R R, then fx) =x is 3 If f: R’ R is defined as f (*) =-3x + 4 for
(A). One-one but not onto
(B) Onto but not one-one
all xe R,then f(2) is equal to
One-one and onto
(A) {1, 2} (B) (1, 2)
(C) (C) [1, 2] (D) none of these
(D) Many-one
4 Iff= {(1,4), (2, 5), (3, 6)} and
19. Iff (r) = V5x-7 then dom(f) = g= {(4, 8), ($, 7), (6, 9)}, then gof is
(A)
() R (B) (B) {(1, 8), (2, 7), (3, 9)}
(C) {(1,7), (2, 8), (3, 9)}
(C) (D) (D) ((1, 8), (2, 5), (3, 9)}
If f(r) = x,gr) = 5x -6, then g[fx)] =
20. Domain of function f(x) = sin 5x is (A) 25x-60x +36 (B) 5x +6
(C) 25x + 60x 36 (D) Sx-6
(A) (B)
6. If f(x) = 3x- 1, g(r)=+1 then f [g(¢)] =
(A) 3r +2
(C) R (D)
() (C) 3x-2
(B) 9x- 6x +2
(D) 9x + 6x -2
121
 both f().fy)
(B) (D) =2
f(-)
(B)
equals x 4(D)
1 and [f))
(B)3f()
(D) (-10, = then elements
1s 0.7 then
(C) (B)
x gof injective be
and and +a
a" (1+ y elements
element
f 1
Sr-78r)
5x-3
7x+4
and
the
f, of
which
g be

injective
may
injective
them
=
fx)
2
then E
x
then (C) ,
k=
cos[-
2,
function
=
line
x
*0,
and
one
S
then two
f(2-x)-2) 3k,x f(-)};twothan
then
(B) thenshould
functions be be
should
surjective,fshould
of injective
function
= -y)2f(x).fiy)
(A)
10-x.
10+x
0.6
+ ]x
cos[r'
()-2 thethe=-f-) f(r 2 -
about f(x)=f-x)
(C) =
f)
exactly
exactly
moreset
3x+4 f 200x (B) (C) of
=|
f(r) = = empty
-41
(A) gand None f(x log +x) 2) Contains
Contains
Contains
Iff)= 0 11. -x
=
Iff(r) (C) two the + f(x) f() = [f(A)2f(:)
() graphsymmetrical
f() + R:
(A) for
g
Given+y) {) )If 0.5 =
Iff) f(2 f(r +
f) xe an
If and(A)(B)(C) (D) f(x (C) IF (A) .(A) Is
Thinking
Critical (A) The (B) (D) If S=(A)(B) (C) (D)
0. 12. 10. 11. 12.
7. 8. 9.
is -1!
() by real for (D)3 where
identity
1
(fo) +2r'+2
these given
(+1)
-(D) all
(D) x -1c=
b=4, 1c=
b=-1, =!-s, 0
thesatisfied,
for of (D) (D)
of be (r) -1 -b')
2(a
3(2b-3a)
none (fog)(B)
of f(y) value
1+x
value
(MCQs) R then: the 1
which -3 a+b -2
(D) ’
then (D) = 2
is 6
the (B) R =x+1, (B) f(y) 4, (C) then (C) for 3 (B) (D) (C) b+ (B) (D) = (C)
fiy)]
Mathsthen g: fthen fr) =
f(2)
+
3, is 1/3 c 8r are
that afy) = then
and satisfy + 1) and such (2) + ;
1, +'+1 g(r)
’R , If (B) 4
2x f(r+(B)
=
f() d, =4 cb=-1,=xt, 3(B) x,non-zerofthen -b') x), -iy)
+
Triumph y. b cx+1 2(a'
3(3a-b')
-2b)2(a' cos(log -1
(B)
(A)
+1 and+1 (-I)
(C) - R’R
and -
= of bx+ c= b,
3a)
3(2b
+
f)
If= 1
= value- 2, x
= f(r)1/2
xnumbers0 b=
f)fã)
R =x 6.1
Function 1) =
to
equal
MHT-CET (A)(C) f:Let f(r) (A)
Iffo) f: (A) fx)which (r) 1(A) fIfor a
where f{%)
(A) Let If (A) The+ =
f(x f(x)(A)(C) Iff (A) (C) If
7. 8. 9. 1. 2. 3. 4. 5.
6.
 Chapter 6: Functions

13. If [x] denotes the greatest integer < , then then fis
22. Iff:(0, o) ’ [0, o) andf ) =
(A) One-onc and onto
One-onc but not onto
(A) 99 (B) 98 (B) Ontobut not one-one
(C) 66 (D) 65 (C)
onc-one nor onto
(D) Neither
14. Let fbe a real valued function, satisfying defined by
f(r+y) = f() fo) for all x, ye R 23. If f: R - (3} ’ R - {1} be
x-2
Such that, f(1)= a. Then, f(¢) = f(x)= x-3
then fis
(A) a (B) ax
one-one into (B) one-one onto
(C) (D) log x (A) (D) many-one onto
(C) many-one into
15. Consider the function f(x) = cos x. Then numbers, then the
(A) fis of period 2 24. If R denotes the set of all real
(B) fis of period 2n function f: R’R defined by f(r) =(x] is
(A) One-one only
(C) fis not periodic Onto only
(D) fis of periodic n (B)
(C) Both one-one and onto
16. Which of the following is an even function? (D) Neither one-one nor onto
(A) * (B) x+ sin x defined as
25. Mapping f: R ’ R which is
(C) sin x (D) all of above f(x) = cos x, x e R willbe
Neither one-one nor onto
17. Which of the following functions is an odd (A)
function? (B) One-one
(C) Onto
(A) f() = V+x+*- I-x+* One-one onto
(D)
number
(B) 26. A is a set having 6 distinct elements. The
of distinct functions from A to A which are not
bijection is
(C) f() = log10 (B) 6-6
(A) 6!-6
(D) f(x) =k(constant) (C) 6-6! (D) 6!

If the real valued function f(x) =

a-1
is 27. IfA ={|xeN,xS5},
18. B={x|xe Z, x- 5x +6=0}.
then the number of onto functions from A to B
even, then n equals
(A) 2 (B) (D) 3
(A) 23 (B) 32
(C) 2 (D) 30

1+X then f(x) 1S

19. If f(x) = log,: 28. Number of bijective function from a set of
1-x 10 elements to itself is
(A) Even function (A) 5! (B) 10!
(B) fx)f(x) = f(x +xz) (C) 15! (D) 8!
f(x) = f(x -x) 29. Set Ahas 3 elements and set B has 4 elements.
(C)
f(x,) The number of injection that can be defined
(D) Odd function from A to B is
(A) 144 (B) 12
20. The function f(r) = (C) 24 (D) 64
(A) Odd 30. Which one of the following is a bijective
(B) Even function on the set of real numbers?
(C) Neither odd nor even (A) 2r-5
(D) Constant
(B)
(C) x (D) x+1
21,
The function f) =sin (log(xtNr+1)) is i 31. Let f: R ’Rbe defined by f() =x, then
(A) Even function (A) fis one-one and onto
(B) Odd function (B) fmay be one-one and onto
(C) Neither even nor odd (C) f is one-one but not onto
(D) f is neither one-one nor onto
(D) Periodic function
123
 coSx 1)-
(-1,
(0}; 1,1]-
{0} is
1(-,
-) 1]
(-9,- +6)
d d
IV J3
-/ 1] (B)[0,-1]
[1, (D) is
I+x-I-x (-1,9) -2,4]
(B) [2,
(B)
o) -6x
sinx is
-x',
R Jlog(r
= is
(B) (D) function(-o,3-V3)v(3+5,
o)(B)
C V C = is (r) (D) (B) (D) (B) (D) +8 (D) log,(x
+3) r+3x+2 is+r+2 -2.5 excluding
-3,-2],
f(x)S function function
f +y2r
IV by ofinterval o)(-,- function
function
1)U[0, [3,4] is1-3² o) -3}
-2, (-3, function 0excluding
(-l,-2}
o)-
R-{-I, 0excluding
b b defined 1]V[5,
o)(-o,
sin the -5x+6 U the
8.3*-2 [2, =
f(r) (1-x)
log,
0)U the these
of
none
the -1,3] the
1]
[0,
(C) the 1]
I, of -9,1]
(C) of of (-o,)
(A) (-2)
R- (-2,
to)
the
(A)2] domain (A)
S o)
[(C)
0 , 1)
(-1, domain[1,9]
(A) (-0,
0] of of of 1), 1],
’ then of of domain 3][2, -2, (-o, domain [0,
o) domain
Domain Domain sin
=
fx) domain
y= [-2, [0,
f(r)
=
R fo)
=
(C) (D) Iff: onto,(A) (A) (A)(C) The (C) The
The (C) The (C) (D) The (A) (B)(C) (D) The (A) (B)(C)(D)
37. 38. 39.. 40. 41. 42. !43. 44. 45.
IIi
List
to A of
numbers is anditems 24 1728 19683
0 12
=e
G
natural
fc) sets the II. IV. V.
(MCQs) by match d d II
the
is onto
then
fis onto onto defined
in then ofnonfrom
functions
bijective
bijectiveof to
number of functions
4 onto onto of is both A o IV
ontoone-one
Maths3x+ one-one one-onenor onto set elements to G C
by odd , even nor II.
onto respectively,
onto not
and One-one
and
onto onto f:R’Rinto A from numberxA
=x+ not and defined
and
ifnis
odd if
niseven one-one
not the is not one-one and
One-one List I
List of from
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defined
by when
n is and The
number
and one-one not but from n but notand of to number
G
GxG A is b b
Triumph
fa) many-one
one-one ’N but one-one
when
One-one
but One-one one of
Many-one those functions surjective
functions to
A
match
n+1 neither f function
Neither number Gx
A
f:R’R, not f:N 2 onto functionn-1
2 n
2 Onto (A)
Onto Many 4
andwith
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The correct V
MHT-CET f(n)
= f(n)
=
(A)(B) (C) (D) Let the I (B)
(A)(B) (C)(D)
A
(A)(B)(C) (D) The (B)(C)(D) If 3are List (a) (b) (c) (d) The (A)
32. 33. 34. 35. 36. 124
 Chapter 6: Functions
46. Domain of f(x) =lollog xis
(A) (0, oo) 155. Range of the functionfx) = is
(B) (1, o) 3x+2
(C) (0, 1)U0, o) (D) (-0, 1) (B) k- {0;
(A) R
47. The
domain of definition of f(1) is
(C) (0, )

(A) (-0, -1) V(2, o) +js

(B) (-1, 1]v(2, o) U(o,-2) 56. The range of thefunction, f(x)
(C) ((o, 1) U (2, o) (A) (0, 1) (B) [0, 1]
(D) -1, 1]v(2, o) (C) (1, o) (D) [1,)
48. The domain of the function t-3x+2
1 57. The range of the function f(X) = t+x-6 is
f()=log is
|sin x|
(A) (B) R
(A) R- {2n7, n e I}
(B) R-{n, ne I) (C) R- {1} (D) R-{-3, 2}
(C) R--, n) x-I+4
(D) (- 0, co) 58. Let f: R >Rbe defined as f(x) = +x+4
49. The domain of the function f(x)= log3+x(x-1) is Then the range of the function fr) is
(A) (-3, -1) v(1, o) (3 5
(B) [-3, -1)U[1, oo) (A) (B)
(C) -3, -2) U(-2,-1) U(1,o)
(D) -3, -2) U(-2, -1) U
[1, o) (C) (D)
50. The domain of the function
59. Range of the function f() = V+x+1 is equal
f() =exp(sx-3-2 ) is to
(A) (B)
(A) [0, o] (B) 2
(C) (-oo, 1] (D) V3 -V3
(C) 2 2 (D) (0, 0)
51. The domain of the function f(x)= sin(x-3) is
60. The range of f(x)= cos x- sin x is
V9-x
(A) [1, 2) (B) (2, 3) (A) (-1, 1) (B) -1, 1)
(C) [1,2] (D) [2, 3] (C) (D) -V2,2]
52. Domain of the function
f(x) = sin (1+3x+ 2x) is 61. The range of the function f(x) = log(3x + 4) is
(A) (-o, o) (B) (-1, 1) equal to
(A) [log. 2, o] (B) [loge 3, o)
(C) (D) (C) [2 loge 3, o) (D) [2log. 2, o)
53. The domain of the function 62. The range of function f(r) = loge v4-x is given
by
f(r) = (A) (0,0) (B) (-o,)
(C) (-o, log. 2] (D) (loge 2, o)
(A) (-3, 3)
(B) -3, 3] 63. If f: [0, T/2) ’R is defined as
(C) (-o, -3) (3,o) 1 tan

(D) (-,-3]U[3, o) f(0) =- tan 9 tan . Then the range of

-1
54. The domain of the function - tan

cos (log(r + 5x + 8)) is fis

(A) [2,3] (B) (-2, 2] (A) (2, o) (B) (-o, - 2]
(C) [3, 1] (D) (-3, -2] (C) [2, o) (D) (- 0, 2]
125
 MHT-CET Triumph Maths (MCas)
6.2 8. If the functions f, g. h are defined from
Algebra of functions
of real numbers R to R such that the
1.
If the function f: [1, «o) ’[1, oo) is defined by 0, if xso
f(r) =x-1, g(r) =Vx'+1, hr) =|x, if x>0'
fx) = 2M-D then f () is
then the composite function (hofog)\(r) =
r-)

(A)
|0, x= 0
(A) x, x>0 (B)
(B) +/+4log, 1) |-r,x<0
(C) -l+ 4log, x) (C)
0, xs0
(D) none of these
(D) Not defined
9 If f be the greatest integer function and gbe the
2. The inverse of the function f(x) = e-e + 2 is
e+e modulus function, then (gof)
(A) 1 (B) -1
(A) loge (D) 4
(C) 2

() log. 2-x
(D) loge 10. If f(x) = sin x +

3. If f:R ’ R is defined by f(x)=, then and ig-1,then gof:) is

(A) f)=-* (A) apolynomial of first degree in sin x and cosy
(B) f')= (B) a constant function
(C) a polynomial of second degree in sinr
(C) The function f(x) does not exist and cos x
(D) none of these
(D) f')=
11. If g) =x+x-2 and-(gof)(r)=2r-
2
Sx +2,
4. If f: R ’ R be a mapping defined by
f() =*+5,then f() is equal to then f) is equal to
(A) 2x+3 (B) 2r-3
(A) x+5
(B) (*+5)i (C) 2x+3*+ 1 (D) 2r-3x-1
1

(C) (5-x): (D) (-5); 12. If x# 1and f(r) = X+l

X-1
is a real function, then

2x-1
(x+5), then f() is equal to f(f({(2) is
5. If f() = x+5 (A) 1 (B) 2
x+5 5x+1 (C) 3 (D) 4
(A) 2x-1, (B) 2-x
5x-1 X-5 1 13. Iffr) l-x then f[f(cos 20)] =
1+x
(C) -,X*2 (D) 2x+1*
2-x (A) tan 20 (B) sec 20
(C) COs 20 (D) cot 20
10*-10 is
The inverse of the
6.
functiono-g010-% 14. If f(r) = (25 - x for 0 < x < 5, then
1+x
(A) logio(2 -x) (B) 10gio-x
2
2x
(C) log,(2x -1) (D) (A) 2-4 (B) 23
2-x (C) 22 (D) 2l
7. f:R’ R and g: [0, o) ’ Ris defined by 15. If f(x) = , X*-1, then for what value of a
f(x) =x and g () = Vx. Which one of the
following is not true? is f(f(r)) x
(A) fog(-4) =4 (B) gof(-2) =2 (A) V2 (B) -2
(C) gof(4) = 4 (D) fog(2) =2 (C) 1 (D) -1
-
126
 uOMfi Chapter 6: Functions
16. If f(r) =ar + b and g(x) = cx + d, then 21. Let fr)=x and g() - sin xfor allx e R. Then
flg(r)) = g(f)) is cquivalent to the set of all x satisfying
(A) (¢) =g(a) (fogogo) (r) = (gogof) (x), where
(B) f(d) = g(b) (fog) (*) = f(g(x)) is
(C) f(a) = g(c) (A) t /nn,ne {0, 1, 2, ...}
(D) f(b) = g(b) (B) Vnn,ne {1, 2, ...}
17. If f(r) = 2x + 1 and g(x) = for all real x, (C)+2
2nn , n e {.., -2, -1, 0, 1, 2, ...}
(D) 2nT, n e {.., -2, -1, 0, 1, 2, ..}
then (fog)" is equal to
22. Let f, g:R ’ R be two functions defined as
1 f(r) =|x| +x and g) = -xtxe R. Then
(A) x (B)
(fog) (r) for x<0 is
00 (A) 0 (B) 4x
(C) -x (D) (C) -4x (D) 2x

23. Let S, T, Ube three non-void sets and f: S ’T,

18. If f) =sin'x+sin' x++ and
g:T’U be so that go f:S’Uis surjective.
Then
g-1, then (gof)\%) is equal to: (A) gand fare both surjective
and 1 (B) g is surjective, f may not be so
(A) (B) 0 (C) fis surjective, g may not be so
2
(D) fand g both may not be surjective
(C) Sin x (D) none of these
24. If f: R ’ R be defined by f(x) = e and
19. Let g(x) =1+x- [x] and
g:R’R be defined by g(r) =x. The mapping
-1,x<0 gof: R ’R be defined by
f(x) =0,x=0, then for all x, f(g(x)) is equal to
(gof)()=g [f ()] eR. Then
|1,x>0 (A) gof is bijective but fis not injective
(A) x (B) 1 (B) gof is injective and g is injective
(C) f(¢) (D) g)
(C) gof is injective butg is not bijective
20. Two functions f:R ’ R and g:R’ R are (D) gof is surjective and g is surjective
defined as follows: 25. If f: A ’ Bis a bijection and g: B’A is the
0; (x rational) inverse of f, then fog is equal to
f(x) = |1; (x irrational)
(A) IA (B) Ig (C) f (D) 8
g)=-; (rrational) 26. A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6} are two
|0; (x irrational), sets and function f : A ’ B is defined by
then (gof)(e) + (fog)(r) = f(x) =x + 2; Vr e A, then the function fis
(A) -1 (B) 0 (A). Bijective (B) Onto
(C) 1 (D) 2 (C) One-one (D) Many-one

Concept Fusion
2. Let the function f:R ’ R be defined by
1. The range of the function f(x) = tan,-x f(x) =2x+ sinx, xeR. Then fis
is (A) One-to-one and onto
(B) (B) One-to-one but not onto
(A) [0, 3] (C) Onto but not one-to-one
(C) [3.] (0) V53] (D) Neither one-to-one nor onto
127
 Triumph Maths (MCQs)
MHT-CET Preious Years' Questions
1.
If f: R - xe
f() -4
{2} >R ’ is a function defined by 11. If|3r - 2|s.then
2
(2020%
. then its |2018|
range is (A)
(B)
(A) R
(C) R- (4} (B) R- (2}
(D) R--2, 2} (C) (D)
2.
Which of the following is an even function?
[2019] 12. If f: R ’R is given by f(x) = 7x + 8 an
(A) f()=3cosx+4 (B) fo) = 2sin x+3 f(12) = , then the value of kis
k
(C) f() =x+x (D) fr) = sin
(B) 7
[2020)
3. The (A) 1
domain of the function (D) 8
f(r) = sin x+ 16-r is (C) 4
[2019] If a function f : R ’ Riis defined
(A) -4,0) w (0, ] (B) (4, 4)
13.
(C) 4, -]U[0, ] (D) [4, a] f() =5 +3, then f)= [20201
4. If f()= 3x-2 and g(x) =x, then fog(<) = S(*-3) (B) 4(x +3)
(A) 5
[2019]
(A) 3x-2 (B) 2-3x2 4(x-3) 5(x+3)
(C) 3x2 (C) (D)
(D) 3x+2 5
4

5. If f(x) = 3x + 6, g(*) = 4x + k and fog() = gofx) x-3

then k= 14. The range of the function f()=,x*5is
[2019]
1 (2020|
(A) -9 (B) 18 (C) G (D) 9
(A) R- 1} (B) R- {1}
6. The domain of the real valued function (C) R- {5} (D) R--5}
If R={(a, b) /b=a-1, a e Z,5<a<9},then
V3-x 2is
f() = [2019] 15.
the range of R is [2020]
(A) [2,3] (B) (2, 3] (A) {5, 6, 7} (B) {5, 6,7, 8, 9}
(C) [2, 3) (D) (2, 3) (C) {7, 8, 9} (D) {6,7, 8}
1. The range of function f(r) = sin x + cosec x is 16. If f(x) =ax + bx + 2 and f(1) =4, f(3) =38, then
[2019] a-b= [2020]
(A) -1, 1] (B) (-1, 1) (A) 8 (B) 2
(C) R-2,2] (D) R2,2) (C) 2 (D) 15
8. Function f: R ’ R defined by f(x) =+5 is 17. If f: R ’ R, g:R R defined by
[2019] f(x) =x-3x +4 and gr) = 2x + 1, then the
(A) many-one and onto value of x for which f(r) = (fog) () is [2020]
(B) one-one and onto (A) 1. 2 (B) 1,
(C) one-one and into 3
(D) many-one and into
(C) 1, 3
(D) -1, 3
9. Function f:R’ R is defined by f(r) =| + x.!
Which of the following statement is true? [2019] i. 18. x+2
Domain of the real valued function f(r) = 9-x
(A) fis many-one
(B) fis constant function ;21 is [2020]
(C). f is one-one (A) R (B) R- {3}
(D) fis onto (C) R- (-3, 3) (D) -3<xs3
1 4x +7 then the value of f{ff(2)]} =
10. The range of the function f(x) = 19. Iff(r) = 7x-4
xe(3, o) is [2019] (2020]
3 39
(A) (-3, 3) (B) [-3,3] 35
(D) 35
(C) 3, o) (D) (0, o)
(A) 39
(B) 3 (C) 2
128
 Chapter 6: Functions
0. Iff:R’ R, g: R -’ R are two functions
defined by fl) - 2r 3, g() 29. The domain and range of the relation R given by
(fog)'() =+ 5, then 6
R-)/y-x*,xye Nand r <6 [20201
|20201
(A) (A) Domain - (2, 3}. Range (5}
(B)
( (B)
(C)
Domain {1, 2}, Range (5, 7}
Domain = {1,2, 3,4}, Range (5, 7}
(C) (D) .(D) Domain = {1, 2, 3, 4,5}. Range (7, 5}
3x +2
21. Iff:R’R, such that f(r) =e+e 30. If f(r)= 5x-3 xeR-.then (2020]
e-e
then f is
(A) f) =-f(r)
(A) an odd function |2020] (B) [x)] =-x
(B) a neither even nor odd (C) f ) = f()
(C) an even function
function (D) f() does not exist
(D) a periodic function
31. The domain of the function f(x) =
3r+4 7
22. If f() = 7x+4
and g(r) = Sx-3** 3
5 [2021|
then (gof) (3) = [2020] (A) (o, 0) (B) (2, 5)
(A) 3 (B) -3 (C) (0, o) (D) (-o, o)
1
(C) 3 (D) 32. Let A= [a, b, c, d], B= [1, 2, 3]. Relation RË,
3
R2, R3, Ry are as follows:
23. Given A={1,2, 3, 4, 5}, B ={1,4, 5}. IfR isa R=[(a, 1), (b, 2), (c, 1), (d, 2)]
relation from A to B such that (x, ) e R with Ry =[(a, 1), (b, 1), (c, 1), (d, 1)]
x>y, then range of Ris [2020] Rs =[(a, 2), (b, 3), (c, 2), (d, 2)]
(A) {1, 4, 5} (B) {4, 5} R4=[(a, 1), (b, 2), (a, 2), (d, 3)], then
(C) {2,4} (D) {1, 4} [2021]
(A) only R3 and Rg are not functions.
20 24. The domain of a function f(y) = cos (y-5) is (B)
(C)
only RË and R2 are not functions.
only R3 is not a function.
9
J25-y'
[2020]
(D) only Ry is not a function.
(A) (4, 5] (B) (4, 6] 33. If f(r) = 3[x] + 5{x + 1}, where ] is greatest
(C) (5, 5) (D) 4,5) integer function of x and } is fractional part
202 function ofx, then f(-1.32) = [2021|
25. If f(r) = 2r + bx + c, f(0) =3 and f(2) = 1, then (A) 4.6 (B) -2.6
(fof) (1) = [2020] (C) -7.4
(A) 0 (D) -3.4
(B) 2
(C) 1 (D) 3 ! 34. Let A={10, 11, 12, 14, 26} and let f: A ’N
db be such that f(a) = highest prime factor of a,
nt 2x+3 2
26. If f(r) = 3x-2*, ,then the function fof is where a e A, then range of f= [2021]
(A) {5,7, 13} (B) (5,7, 11, 13}
[2020]
(A) a constant function 1s0 (C) {3,5, 7, 11, 13} (D) 3,7, 11, 13}
(B) an identity function 35. The domain of the function fr)= Vr-l+6-x
(C) an even function is [2021]
an exponential function
(D) (A) [l, ) (B) [1, 6]
27. For f(r) = x], where [] is the greatest integer (C) (-o, 1) (D) (6, o)
function, which of the following is true, for 36. If f(x) = [8x] 3, where [x] is greatest integer
every xe R. [2020] function of x, then f(r) =
(A) x]+1=x (B) [x]+1>x (where =3.14) [2021]
(C) ]+ 1sx (D) x]+ 1<x (A) 21 (B) 25
(C) 23 (D) 22
28. The domain of the function f(r)= is
|2020] i 37. Range of the function(r) =3 +2 + 4' is [2021]
(A) R- {0} (B) R*U {0} (A) (3, o) (B) (-, o)
3
(C) R (D) R* (C) [3, o) (D) (o, 3]
129
MHT-CET Triumph Maths (MCQS)
38. The domain of the If the function f: R-{-1, 1} > A
(A) ( , )
function logo ( 5x + 6) is
|2021|
47.
f()= is surjective, then A is equal to defincd by
(C) (2, 3) (B) (-0, 2) U(3,0)
R-(-1,0)(2022
(D) None of these
39, If f) = (A) R- (-1, 0) (B)
and g) = then (fog)) = (C) R-{) (D) [0, o)
|2021] 48. For a suitable chosen real constant a, lct a
(A) 2r-1 function f:R- {-a} -’ R be defincd
(B)
3x+1 a-x
f() = Further, suppose that for any real
(C) x-1 a +x
+2 (D)
2.r+1 number x#aand f) -a, (fof) (:) =x. Then
40. If f(r) = 2fx} + 5x, where fx} is
fractional part is equalto (20221
function, then f(-1.4) is [2021]
(A) 8.2 (B) -8.2
(A) 3 (B). 3 (C) 3 (D)
(C) -5.8 (D) -S 3
41. If f(r) = e,g(r) = log x, then g[f()]= [2022]
(A) (B) x| (C) 2x (D) 1 49: If x] is greatest integer function and
2[2x -5]-1=7,then xlies in
42. a'-a*
(2022)
If f(x) = where a, x satisfy the (A) (B)
a' +a*
necessary conditions, then f()= [2022] (D)
(C)
(A) log. (B) 8 ) 25
(C) (D) 50. If logzx + logr t logx + log16* = and
36
x= 2%, then k is (2023)
43. The domain and range for the function
1 1 1
fr) = e Sin are (2022] (A) (B) 2 (C) 3 (D)
8
(A) domain=R, range =R
(B) domain =R, range = [1, o) 51. The domain of the function f(:) = sin
(C) domain =R, range = (0, o)
(D) domain = R, range=0, o) (Co, -a] [a, o). Then a is equal to [2023)
44. If R denotes the set of all real numbers then the
(A) (B) V17-1
functionf:R’ R defined by f(r) =k is (2022] 2 2
(A) injective and surjective. 1+17
(B) injective. (C) 2
(D) 2
(C) surjective.
(D) neither injective nor surjective. 52. If f() =
3x+4
and g(r) =
7x+4
then f(g)) =
5x-7 5x-3
45. The domain of the definition of the function
[2023]
f(x) = 4-x -+ log,. (r-x) is [2022]
(A)
x+1
(B) 41x
x+2
(A) (-1, 0) w(1, 2) U3,o) 5x-7
(B) (-2, -1)U(-I, 0) w(2, o) (C) g(f)) (D) 41
(C) (I, 0) w(1,2) U(2, o)
(D) (1, 2) w (2, o) 53. If g(r) = 1+ /x and f(gr)) = 3 + 2/x tx,
46. Let A = {xe R/x is not a positive integer}. Let a then f(f(r)) is |2023|
function f be defined as f: A ’ R so that (A) x+ 4x +6 (B) x++6
2x (C) x +x+6 (D) x+ 4x' +6
f(x) = then f is [2022]
X-1
(A) surjective but not injective. 54. If f(x) =x +land g(r) = , then the value of
(B) not injective. [2023|
(C) injective but not surjective. f(e(e(r(>) at x= 1is
(D) neither injective nor surjective. (A) 4 (B) 1 (C) 5 (D) 3

130
 Chapter 6: Functions

55. Ir R)- -8log,', *> 0,then (2), {4),. (A) -24-P)

t8)are in |2023]
(A) A.P. (B) 4-p(2p-4)
(B) G.P.
(C) H.P.
(C) fla-2p)(24-p)
D) Arithmetico Geometric Progression. (D) (2p-)(24-p)
56.
The range of the function f() = is (2023] The domain of the definition of the function f(x)
+1 62.
(A) (0, 1) (B) [0, 1) is given by the equation 2 + 2' = 2is[2024, 2023]
(C) (0, 1] (D) [0, 1] 0Sxs1
(A) 0<xs1 (B)
2x-3 (C) -o<xs0 (D) - o<x<1
3r-4#. then the value of f ()
$7. Iff) = 3
is [2023] 63. If f:1,o)[2,) is given by f(z)=x+*
4x-3 3x-2
(A) (B) then f(x) equals (2024]
3x-2 4r+3
3x-4 2.x +3 x+*-4 2
(C) (D) 4x-3
(4) (B) 1+x²
.4x-2 2
x-V-4 (D) 1+V-4
log, (*+3)
58. Th» domain of definitionof f(x)=
is (C)
+3x +2 2
[2024] 64. Let f(r) =(r+1)' -1,-1, then the set
(A) R-{12) [2024]
(B) (-2,) s/f(:) =f(a} is
(C) R-(-,-2,-3) (A) (0.1-1}
(D) (3,o)-{-1,-2) (B) {0.-1}
3-iV3, where i= I
lx < 1, then is (C) 2
$9. If f) = log.
[2024) (D)
equal to
(A) 2() (B) -2fc)
65. If x-s[x]+6 =0, where (1 denotes the
(C) (f) (D) 2f() [2024, 2020]
greatest integer function, then
,then (gogof) ()= (A) xe(2,4] (B) x[2,4]
2-x x+2
(C) xe24) (D) xe(2,4)
[2024]
(A)
6+x
(B)
6-x
10+2x
66. Let P=0/sin@- cos@ = 2cos and
10-2x
6+x 6-x Q=0/ sin 9+cos0 = V2sin0 be two sets, then
(C) (D) 10-2x [2024]
10+2x
(A) Pc Qand Q-P
61, Let a.B be the roots of the equation (B) Qa P
x-px +r=0 and 2B
2
be the roots of the (C) P0)
(D) P=Q
equation x-qx +r=0. Then the value of r is
[2024]

at the end of the book.

Answer Key of the chapter: Functions is given
to the relevant da
Evaluation test can be accessed
SOLUTIONS by scanning the adjacent QR code in
questions of this chapter & Evaluation Test Quill - The Padhai App.
Can be accessed by scanning the adjacent
QR code in Ouill-The Padhai App
131
 10.
(D) 20. (D) (D) 30. (A)
20. (A)
10. (B) 40. (D)
50. (D)
60.
(B)
10.
9.
(A) 19.
(D) (C) 9. (D)
(A) 29. (D)
19. (C) 39. (C)
49. (B)
59,
9.
(D)
18. (D)
8. 8. (D)
(C) 28. (A)
18. (B) 3S. (B)
48. (A)
58.
8.(B)
7. (D)
(C) 17. (B)
7. 7. (A)
(A) 27. (A)
17. (D) 37. (B)
47. (C)
57.
Chapter
6:Functions (A)
36. (C)
46. (C)
56.
6. (D)
(A) 16. (B)
26. (A)
6. 6. (B)
(D) 16. (C)
26.
(C)
5, (C) (D)
5. 5. (C)
15. (A)
25. (C)
35. (A)
45. (B)
55.
15. (B)
25. (B)
(A)
4. 14.
(D) (B)
24. (B)
4. 4. (A)
(B) 14. (D) 44. (D)
(C) (D)
24. 34. 54.
(A)
3. 13.
(A) (C)
23. (A)
3. 3. (C)
(B) 23. (A)
13. (B) 33. (C)
43. (B)
53. (C)
63.
(D) (B)
12. 22. 2. 12.
(B) (A) (A)
2. 12.
(B) (B) 32. (C)
22. (C) 42. (C)
(C)
52. (C)
52. 62.
Thìnking(D)
Classical 2. Thinking
Critical
(A) (C)
1.(©) 11. 21. 1.(A) (B)
11. (B)
1. (B) (B)
11. 21. (D) 41. (B)
31. (C) 51. 61.
(D)
6.1: 6.2: 6.1:
 Answer Vey

62: 1. (B) 2. (B) 3.. (C) 4. (D) 5. (B) 6. (B) 7.(A) 2. (3) 2A) 16. (5)
11. (B) 12. (C) 13. (C) 14. (D) 15. (D) 16. (9) 17. (B) 18.(9) 19.(B) 20. (A)
21. (A) 22. (C) 23. (B) 24.(C) 25. (B) 26. (C)

Concept Fusion

1. (B) 2. (A)

MHT-CET Previous Years Questions

1. (C) 2. (A) 3. (C) 4. (A) 5. (D) 6. (C) 7. (D) %. (D) 9. (A) 10. (D)
11. (C) 12. (C) 13. (A) 14. (A) 15. (A) 16. (A) 17. (A) 18.(C)
21. (A) 19. (C) 20. (B)
22. (A) 23. (D) 24. (D) 25. (D) 26. (B) 27. (B) 2%. (B) 29. (D)
31. (C) 32. (D) 33. (B) 34. (C) 35. (B) 36. (D)
30. (C)
41. (B) 42. (C) 37. (A) 38. (B) 39. (B) 40. (C)
43. (C) 44. (D) 45. (C) 46. (C) 47. (A) 48. (B)
51. (C) 52. (C) 53. (D) 54. (C) 55. (A) 49. (C) 50. (C)
56. (B) 57. (A) 58. (D) 59. (D)
61. (D) 62.,(D) 63. (A) 64. (B) 65. (C) 66. (D) 60. (D)