Functions

1. CLASS XI FUNCTIONS WORKSHEET (T)
STHITPRAGYA SCIENCE CLASSES, GANDHIDHAM
ADVANCED MATHEMATICS FOR JEE | BITSAT| GUJCET| OLYMPIADS | IX, X, XI, XII, XII +
MISHAL CHAUHAN (M.Tech, IIT Delhi)
Address 1: Near Gayatri Mandir, Opp. PGVCL Office, Shaktinagar Address 2: Sec-5, G.H.B, Gandhidham
Contact: 9879639888 Email:sthitpragyaclasses@gmail.com
Q 1. If f(x) = x2 + x +  be integral function of the integral variable x then
(a)  is an integer and  is a rational fraction
(b)  and  are integers
(c)  is an integer and  is a rational fraction
(d)  and  are rational fractions
Q 2. Let f(x) = ax2 + bx + c, where a, b, c are rational, and f : Z  Z where Z is the set of
integers. Then a + b is
(a) a negative integer (b) an integer
(c) nonintegral rational number (d) none of these
Q 3. If f(x) = cos []x + cos [x], where [y] is the greatest integer function of y then f
2

 
 
 
is equal to
(a) cos 3 (b) 0 (c) cos 4 (d) none of these
Q 4. Let f(x) = sin (tan-1x). Then [f(- 3 )], where [.] denotes the greatest integer function, is
(a)
3
2
 (b) 0 (c) -1 (d) none of these
Q 5. If f(x) =
x 1
x 1


then f(ax) in term of f(x) is equal to
(a)
f(x) a
1 af(x)


(b)
(a 1)f(x) a 1
(a 1)f(x) a 1
  
  
(c)
(a 1)f(x) a 1
(a 1)f(x) a 1
  
  
(d) none of these
Q 6. Let f(1) = 1 and f(n) = 2
n 1
r 1
f(r)


 . Then
m
n 1
f(n)

 is equal to
(a) 3m – 1 (b) 3m (c) 3m-1 (d) none of these
Q 7. If f(x + 1) + f(x – 1) = 2f(x) and f(0) = 0 then f(n), n  N, is
(a) nf(1) (b) {f(1)}n (c) 0 (d) none of these
Q 8. If f(x + 1) + bf
1
x 1
 
 

 
= x, x  -1, a  b then f(2) is equal to
(a) 2 2
2a b
2(a b )


(b) 2 2
a
a b

(c) 2 2
a 2b
a b


(d) none of these
Q 9. Let f be a function satisfying f(x + y) = f(x) + f(y) for all x, y  R. If f(1) = k then f(n), n  N, is
equal to
(a) kn (b) nk (c) nk (d) none of these

2. CLASS XI FUNCTIONS WORKSHEET (T)
STHITPRAGYA SCIENCE CLASSES, GANDHIDHAM
ADVANCED MATHEMATICS FOR JEE | BITSAT| GUJCET| OLYMPIADS | IX, X, XI, XII, XII +
MISHAL CHAUHAN (M.Tech, IIT Delhi)
Address 1: Near Gayatri Mandir, Opp. PGVCL Office, Shaktinagar Address 2: Sec-5, G.H.B, Gandhidham
Contact: 9879639888 Email:sthitpragyaclasses@gmail.com
Q 10. Let f be a function satisfying f(x + y)= f(x).f(y) for all x, y  R. If f(1) = 3 then
n
r 1
f(r)

 is equal to
(a) n
3
(3 1)
2
 (b)
3
n(n 1)
2
 (c) 3n+1 – 3 (d) none of these
Q 11. If f(x + y) = f(x) + f(y) – xy – 1 for all x, y, and f(1) = 1 then the number of solutions of f(n) = n,
n  N, is
(a) one (b) two (c) four (d) none of these
Q 12. Let f(x) = 1 + | x |, x < -1
[x], x  -1, where [.] denotes the greatest integer function.
Then f{f(-2. 3)} is equal to
(a) 4 (b) 2 (c) -3 (d) 3
Q 13. The domain of the function
y = log10 log10 log10….log10 x is
(a) [10n, +) (b) (10n-1, +) (c) (10n – 2, +) (d) none of these
Q 14. The largest set of real values of x for which
2
1
f(x) (x 2)(5 x)
x 4
   

isreal functionis
(a) [1, 2)  (2, 5] (b) (2, 5] (c) [3, 4] (d) none of these
Q 15. Let f(x) =(x12
– x9
+ x4
– x + 1)-1/2
. The domainof the functionis
(a) (1, +) (b) (-,-1) (c) (-1, 1) (d) (-,+)
Q 16. The domainof the functionf(x) = 2
x 1 x
  is
(a)
1 1
1
, ,1
2 2
   
  
   
   
(b) [-1,1]
(c)
1 1
, ,
2 2
 
 
   
 

   
(d)
1
,1
2
 
 
 
Q 17. The domainof the function 2
f(x) 1 1 1 x
    is
(a) {x | x < 1} (b) {x | x > -1} (c) [0, 1] (d) [-1,1]
Q 18. The domainof the functionf(x) =log10 log10 (1+ x3
) is

3. CLASS XI FUNCTIONS WORKSHEET (T)
STHITPRAGYA SCIENCE CLASSES, GANDHIDHAM
ADVANCED MATHEMATICS FOR JEE | BITSAT| GUJCET| OLYMPIADS | IX, X, XI, XII, XII +
MISHAL CHAUHAN (M.Tech, IIT Delhi)
Address 1: Near Gayatri Mandir, Opp. PGVCL Office, Shaktinagar Address 2: Sec-5, G.H.B, Gandhidham
Contact: 9879639888 Email:sthitpragyaclasses@gmail.com
(a) (-1, +) (b) (0, +) (c) [0, +) (d) (-1,0)
Q 19. The domainof the function 2 2
f(x) x [x]
  ,where [x] = the greatestintegerlessthanor
equal tox, is
(a) R (b) [0, +) (c) (-,0] (d) none of these
Q 20. The domainof
1
f(x)
| cosx | cosx


is
(a) [-2n,2n] (b) (2n, 2n 1
 ) (c)
(4n 1) (4n 3)
,
2 2
   
 
 
 
(d)
(4n 1) (4n 1)
,
2 2
   
 
 
 
Q 21. The domainof 2
x 1
f(x) log (x)

 is
(a) ( 2, )
 (b) (0, )
 (c) (1
, )
 (d) none of these
Q 22. The domainof the functionf(x) = 16 x 20 3x
2x 1 4x 5
C P
 
 
 , where the symbolshave theirusual
meanings,isthe set
(a) {1, 2, 3, 4, 5} (b) {2, 3, 4} (c) {2, 3} (d) none of these
Q 23. The domainof
2
1 2
1 x
f(x) sin 1 x
2x
  

  
 
 
is
(a) {1} (b) (-1,1) (c) {1, -1} (d) none of these
Q 24. The domainof the function 1 1 | x |
f(x) sec
2
 
 
  
 
is
(a) (-,-3]  [3, +) (b) [3, +) (c)  (d) R
Q 25. The function
1 2
4
cos (log x )
f(x) e

 isreal valued.Itisdefinedif
(a)
1
x ,2
2
 
  
 
(b)
1 1
x 2, ,2
2 2
   
   
   
   
(c)
1
x 2,
2
 
  
 
 
(d) none of
these
Q 26. The domainof the real-valuedfunctionf(x)=loge | loge x | is
(a) (1, +) (b) (0, +) (c) (e,+) (d) none of these
Q 27. If [.] denotesthe greatestintegerfunctionthenthe domainof the real valuedfunction
2
[x 1/ 2]
log | x x 2 |
   is

4. CLASS XI FUNCTIONS WORKSHEET (T)
STHITPRAGYA SCIENCE CLASSES, GANDHIDHAM
ADVANCED MATHEMATICS FOR JEE | BITSAT| GUJCET| OLYMPIADS | IX, X, XI, XII, XII +
MISHAL CHAUHAN (M.Tech, IIT Delhi)
Address 1: Near Gayatri Mandir, Opp. PGVCL Office, Shaktinagar Address 2: Sec-5, G.H.B, Gandhidham
Contact: 9879639888 Email:sthitpragyaclasses@gmail.com
(a)
3
,
2
 


 
(b)
3
,2 (2, )
2
 
 


 
(c) (e, )
 (d) none of these
Q 28. The domainof the functionf(x) =loge (x – [x]),where [.] denotesthe greatestinteger
function,is
(a) R (b) R - Z (c) (0, +) (d) none of these
Q 29. The domainof the functionf(x) =sin-1
(x + [x]),where [.] denotethe greatestinteger
function,is
(a) [0, 1) (b) [-1,1] (c) (-1, 0) (d) none of these
Q 30. Let f(x) = 2
x
log 25 and g(x) = logx 5 thenf(x) = g(x) holdsforx belongingto
(a) R (b) (0, 1)  (1,+) (c)  (d) none of these
Q 31. Let f(x) = 2 2
x x
sin cos
2 2
 and g(x) = sec2
x – tan2
x.The two functionsare equal overthe set
(a)  (b) R (c) R x | x (2n 1) ,n Z
2

 
   
 
 
(d) none of these
Q 32. The range of the function 2
2
1
f(x) x
x 1
 

is
(a) [1, +) (b) [2, +) (c)
3
,
2
 


 
(d) none of these
Q 33. Let f(x) =
2
1
2
x
cos
1 x
  
 

 
. The range of f is
(a) 0,
2

 
 
 
(b) ,
2 2
 
 

 
 
(c) ,0
2

 

 
 
(d) none of these
Q 34. The range of the real-valuedfunction 2
f(x) 9 x
  is
(a) [0, 3] (b) [-3,3] (c) [-3, 0] (d) none of these
Q 35. The range of the functionf(x) =| x – 1 | + | x - 2|, -1 ≤ x ≤ 3, is
(a) [1, 3] (b) [1, 5] (c) [3, 5] (d) none of these
Q 36. The range of the function 2
3
y log (5 4x x )
   is
(a) (0, 2] (b) (-,2] (c) (0, 9] (d) none of these
Q 37. Let f:{x,y,z}  {a, b, c} be a one-one functionandonlyone of the conditions(i) f(x)  b,(ii)
f(y) = b, (iii) f(z) ais true thenthe functionf is givenbythe set

5. CLASS XI FUNCTIONS WORKSHEET (T)
STHITPRAGYA SCIENCE CLASSES, GANDHIDHAM
ADVANCED MATHEMATICS FOR JEE | BITSAT| GUJCET| OLYMPIADS | IX, X, XI, XII, XII +
MISHAL CHAUHAN (M.Tech, IIT Delhi)
Address 1: Near Gayatri Mandir, Opp. PGVCL Office, Shaktinagar Address 2: Sec-5, G.H.B, Gandhidham
Contact: 9879639888 Email:sthitpragyaclasses@gmail.com
(a) {(x,a),(y,b),(z, c)} (b) {(x,a),(y,c),(z, b)} (c) {(x,b),(y,a),(z, c)} (d) {(x,c),(y,b),(z,
a)}
Q 38. Let f : R  R be a functionsuchthatf(x) = x3
– 6x2
+ 11x – 6. Then
(a) f isone-one andinto (b) f ismany-one andinto
(c) f is one-one andonto (d) f ismany-one andonto
Q 39. Let f : R  R be a functionsuchthatf(x) = x3
+ x2
+ 3x + sinx.Then
(a) f isone-one andinto (b) f isone-one andonto
(c) f is many-one andinto (d) f ismany-one andonto
Q 40. The functionf : R  R definedbyf(x) =6x
+ 6|x|
is
(a) one-one andonto (b) many-one andonto
(c) one-one andinto (d) many-one andinto
Q 41. If the real-valuedfunctionf(x)=px + sinx isa bijectivefunctionthenthe setof possible
value of p  R is
(a) R  {0} (b) R (c) (0, +) (d) none of these
Q 42. Let f(x) =2x + |cos x|.Thenf is
(a) one-one andinto (b) one-one andonto (c) many-one andinto (d) many-one and
onto
Q 43. Let f be a functionfromR to R givenby
2
2
x 4
f(x)
x 1



. Thenf(x) is
(a) one-one andinto (b) one-one andonto (c) many-one andinto (d) many-one and
onto
Q 44. Let f : R  A = y | 0 y
2

 
 
 
 
be a functionsuchthat f(x) =tan-1
(x2
+ x + k),where k isa
constant.The value of k for whichf isan ontofunction,is
(a) 1 (b) 0 (c)
1
4
(d) none of these
Q 45. 2
f(x) x x
  isa functionfromR R. Then f(x) is
(a) injective (b) surjective (c) bijective (d) none of these
Q 46. Whichof the followingisanevenfunction?

6. CLASS XI FUNCTIONS WORKSHEET (T)
STHITPRAGYA SCIENCE CLASSES, GANDHIDHAM
ADVANCED MATHEMATICS FOR JEE | BITSAT| GUJCET| OLYMPIADS | IX, X, XI, XII, XII +
MISHAL CHAUHAN (M.Tech, IIT Delhi)
Address 1: Near Gayatri Mandir, Opp. PGVCL Office, Shaktinagar Address 2: Sec-5, G.H.B, Gandhidham
Contact: 9879639888 Email:sthitpragyaclasses@gmail.com
Here [.]denotesthe greatestintegerfunctionandf isany function.
(a) [x] – x (b) f(x) – f(-x) (c) e3-2x
. tan2
x (d) f(x) + f(-x)
Q 47. Let f(x) =|x – 2| + |x – 3| + |x – 4| and g(x) = f(x + 1). Then
(a) g(x) isan evenfunction (b) g(x) isan oddfunction
(c) g(x) isneitherevennorodd (d) g(x) isperiodic
Q 48. 2
10
f(x) log (x x 1)
   is
(a) an odd function (b) a periodicfunction (c) an evenfunction (d) none of these
Q 49. A functionwhose graphissymmetrical aboutthe y-axisisgivenby
(a) 2
e
f(x) log (x x 1)
   (b) f(x y) f(x) f(y)
   for all x,y  R
(c) f(x) cosx sinx
  (d) none of these
Q 50. A functionwhose graphissymmetrical aboutthe originisgivenby
(a) x x
f(x) e e
  (b) e
f(x) log x
 (c) f(x y) f(x) f(y)
   (d) none of these
1b 2b 3c 4c 5c 6c 7a 8a 9b 10a
11a 12d 13d 14b 15d 16d 17d 18b 19d 20d
21a 22c 23c 24a 25b 26d 27b 28b 29a 30b
31c 32a 33d 34a 35b 36b 37c 38d 39b 40c
41a 42b 43c 44c 45d 46d 47c 48a 49d 50c