Function ASSIGNMENT FOR IIT-JEE

FUNCTION L E V E L - 1 (Objective) 1. The period of the function, f(x) = [sin 3x] + |cos 6x| is : ( [.] denotes the greatest integer less than or equal to x) (a) π

2. The function f(x) =

(b)

2π 3

sec −1 x x − [ x]

(c) 2 π

(d)

π 2

, 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) = log x 2 ( x ) is defined for x belonging to : (a) (- ∞ , 0)

(b) (1, ∞ )

(c) (0, ∞ )

(d) none of these

π π   5 4. If f(x) = sin 2 x + sin 2  x +  + cos x. cos x +  and g  =1, then (gof)(x) = 3 3   4 (a) 1

(b) -1

(c) x

(d) none of these

4 − x2 is 5. Let [x] denote the greatest integer ≤ x . The domain of definition of function f ( x ) = [x ] + 2 (a) (−∞ , − 2) ∪ [ −1, 2]

(b) [0, 2]

(c) [-1, 2]

(d) (0, 2)

 5x − x 2   is 6. The domain of definition of the function f ( x ) = log10  4   (a) [1, 4]

(b) (1, 4)

7. The range of the function f ( x ) =  1  (a) − ,0  3 

(b) R

(c) (0, 5)

(d) [0, 5]

1  (c)  ,1 3 

(d) none of these

1 is 2 − cos 3x

8. The domain of definition of the function f ( x ) = (a) R

(b) (0, ∞ )

1 is | x | −x

(c) (- ∞ ,0)

(d) none of these

9. The function f ( x ) = log( x + x 2 + 1) is (a) an even function (b) an odd function

(c) periodic function

(d) none of these

(c) constant

(d) none of these

)

(

10. The function f ( x ) = cos log( x + x 2 + 1) is (a) even

(b) odd

11. Thr period of the function f(x) = sin4x+ cos4x is (a) π

(b) π /2

(c) 2 π

(d) none of these

12. Which of the following functions is an odd function (a) f(x) = constant (b) f(x) = sinx + cosx

(

(c) f ( x ) = sin log( x + x 2 + 1)

)

(d) f(x) = 1 + x + 2x3

13. The domain of definition of the functionf(x) = 7-xPx-3 is (a) [3, 7]

(b) {3,4,5,6,7}

14. The function f ( x ) = (a) even

(c) {3,4,5}

(d) none of these

sin 4 x + cos 4 x is x + x 2 tan x (c) periodic with period π (d) periodic with period 2 π

(b) odd

15. Range of f(x) = |x| + |x+1| is (a) [0, ∞ ) (b) (0, ∞ ) 16. Range of tan-1x - cot-1x is  − 3π π  , (b)   2 2 

(a) (0, π )

(c) (1, ∞ )

(d) [1, ∞ )

2  (c)  ,1 3 

2  (d)  ,1 3 

(c) (- ∞ ,0)

(d) (- ∞ ,0]

17. The range of | x | − x is (a) (0, ∞ )

(b) [0, ∞ )

 10 2 x − 1  18. If f ( x ) = x  2 x  then ‘f’ is  10 + 1  2

(a) an even function (b) an odd function 19. The domain of the function f(x) =

(c) neither even nor odd

1 + ( x + 2) is log10 (1 − x )

(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 e cos (a) 2

4

πx + x −[ x ]+ cos 2 πx

(d) cannot be determined

is ([.] denotes the greatest integer function)

(b) 1

(c) 0

(d) -1

2 −1 −1  1 + x   + + sin (log x ) cos(sin x ) sin 21. The domain of the function f(x) = 2  2x   

(a) {x : 1 < x < 2}

(b) {1}

(c) Not defined for any value x

(d) {-1, 1}

22. If [.] denotes the greatest integer function then the domain of the real valued function log[ x +1/ 2] | x 2 − x − 2 | is 3  (a)  , ∞  2 

3  (b)  , 2  ∪ ( 2, ∞ ) 2  6

1  (c)  , 2  ∪ ( 2, ∞ ) 2  2

23. The domain of the function f(x) = 4 x + 8 3

( x − 2)

(d) None of these

− 52 − 2 2 ( x −1) is

(b) [3, ∞ ) (c) (1, 0) 24. A function whose graph is symmetrical about the y-axis is given by (a) (0, 1)

(d) none

(a) f ( x ) = log e ( x + x 2 + 1)

(b) f(x + y) = f(x) + f(y) for all x, y ∈ R

(c) f(x) = cos x + sin x

(d) none of these

1  25. If f ( x ) = x 2 − x + 1, x ∈  , ∞  then value of ‘x’ satisfying f(x) = f -1(x) is 2 

(a) 1

(b) 2

(c)

1 2

(d) none of these

 x −1  26. If f ( x ) = log0.4   and g(x) = x2 - 36, then Df/g is  x −5 (a) (−∞, 0) ~ {−6}

(b) (0, ∞) ~ {1, 6}

(c) (1, ∞ ) ~ {6}

(d) [1, ∞ ) ~ {6}

27. The domain of the real-valued function f(x) = loge |loge x| is (a) (0,1) ∪ (1, ∞ )

(b) (0, ∞ )

(c) (e, ∞ )

(d) (1, ∞ )

28. If f(x) and g(x) are two functions of ‘x’ such that f(x) + g(x) = ex 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 y = log a ( x + x 2 + 1) (a > 0, a ≠ 1) is (a)

1 x (a − a − x ) 2

(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 π , then (a) a ∈ [ 4,5]

(b) a = [ 4,5]

(c) a ∈ [ 4,5)

(d) none of these

x −[ x ]+ |cos πx |+|cos 2 πx |+........+|cos nπx | , then period of f(x) is 31. If f(x) = e

(a) 1

(b)

1 n

(c)

n 2 −1 n2 +1

(d)

1 1.2.3......n

2− | x |  −1  2 − | x |  −1  2 − | x |   + cos   + tan   , then Df is  4   4   4 

32. Let f ( x ) = sin −1 

(b) [6, ∞)

(a) [-6, 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 π (c) The range of the function f(x) = cos (3 sin x) is [- 1, 1] (d) None of these π2 34. Let f ( x ) = 4 cos x − . Then 9 2

π

π

  (a) D f =  , ∞, , R f = [ −1, 1] 3  

  (b) D f =  , ∞, , R f = [ −2, 2] 3  

π π (c) D f =  − ∞, −  ∪  , ∞  and R f = [ −4, 4] 3  3 

π  (d) D f =  − ∞, − , R f = (0, 4] 

3

35. The range of the function f ( x ) = sin( xe[ x ] + x 2 − x ), x ∈ (−1, ∞) where [x] denotes the greatest integer function is: (a) φ

(b) [0, 1]

(c) [-1, 1]

(d) R

36. The domain of f(x) = log2 log3 log 4 / π (tan-1 x)-1 = (b) ( 4 / π, ∞ )

(a) R

(c) (0, 1)

(d) None of these

37. If f(x) is a periodic function of the period ‘ λ ’ then f( λ x + a), where a is a constant, is a periodic function of the period (b) λ

(a) 1



38. If f ( x ) = cos −1 ( x − x 2 ) + 1 − 



(a)  2 , 

1+ 5   2 

(c)

λ a

(d) none of these

1  1 + then domain of f(x) is (where [.] is the greatest integer) | x |  [ x 2 − 1]

 1+ 5  (b)  2 ,  

2



 1− 2   (c)  − 2 ,  

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 R1 and R2 respectively, then (a) R 1 = (−1,1), R 2 = (−∞,0)

(b) R 1 = (−∞, 0], R 2 = [−1,1]

(c) R 1 = (−1,1), R 2 = (−∞,0]

(d) R 1 = [−1,1], R 2 = (−∞,0]

40. If f ( x ) = log

1+ x , then 1− x

(a) f(x) is even

(b) f(x1).f(x2) = f(x1 + x2)

(c)

f ( x1 ) = f ( x1 − x 2 ) f (x 2 )

(d) f(x) is odd

L E V E L - 2 (Subjective)

1. Find the value of x for which, f ( x ) =

(2x − 1)(x − 1) 2 ( x − 2)3 >0 ( x − 4) 4

2. Find the values of x for which f ( x ) =

( x − 2) 2 (1 − x ) ( x − 3) 3 ( x − 4) 2 ≤0. ( x + 1)

3. Solve

| x + 3 | +x >1. x+2

  3 4. Find the domain of the function; f ( x ) = log  log |sin x| ( x 2 − 8x + 23) −  log | sin x |   2

 x −1  5. Find domain for f ( x ) = log 0.4  .  x +5 1− 2 | x |  6. Find domain for y = cos −1   + log |x −1| x . 3  

 π  7. Find domain for f ( x ) = [sin x ] cos   where [ ] denotes greatest integer function.  [ x − 1]  8. Find the range for y =

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

9. Find domain and range of the function y = loge(3x2 - 4x + 5).  x +1  10. If f is an even function, find the real values of x satisfying the equation f ( x ) = f  .  x+2

11. Find whether the given function is even or odd function, where f ( x ) =

greatest integer function.

x (sin x + tan x ) .where [ ] denotes x + π 1  π  − 2

12. If f : [1, ∞ ) → [ 2, ∞ ) is given by f ( x ) = x +

1 then find f −1 ( x ) . (assume bijective). x

13. Let f : [1 / 2, ∞ ) → [3 / 4, ∞ ) , where f(x) = x2 - x + 1. Find the inverse of f(x). 14. f ( x ) =

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

15. Find the domain of f ( x ) =

1 log1/ 2 ( x − 7 x + 13) 2

16. Find the domain of single valued function y = f(x) given by the equation 10x + 10y = 10.  π 1 17. Let x ∈  0,  , then find the solution of the function f ( x ) = .  2 − log sin x tan x

18. Find the range of log3(log1/2(x2 + 4x + 4)). 19. Find the domain & range of : f ( x ) = 3 sin

π2 − x2 . 9

20. Find the inverse of following functions: (i) f ( x ) = sin −1 ( x / 3), x ∈ [−3, 3] [assuming bijective] (ii) f ( x ) = ln ( x 2 + 3x + 1), x ∈ [1, 3] . [assuming bijective] x − 1, 21. f ( x ) =  2  x ,

−1 ≤ x ≤ 0 and g(x) = sinx. Find h(x) = f(|g(x)|) + |f(g(x))|. 0 ≤ x ≤1

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

1 1 23. A cubic expression f(x) satisfies the condition f ( x ) + f   = f ( x )f   , then prove that f(x) = 1 + x3or1 - x3. x x

If f(3) = 28. Then prove that f(2) = 9. 24. Let f(x) be a polynomial function satisfying, f ( x )f ( y) = f ( x ) + f ( y) + f ( xy ) − 2 ∀ x , y ∈ R . If f(2) = 5 then prove that f(5) = 26. 1 1 25. If for non-zero x, af ( x ) + bf   = − 5 where a ≠ b then find f(x). x x

LEVEL - 3

(Questions asked from previous Engineering Exams)

1. The domain of the function f ( x ) = log(1 − x ) + x 2 − 1 is (b) (1, ∞)

(a) [-1, 1]

2. The range of the function f ( x ) = (a) [0, 1]

(d) ( −∞, − 1]

(c) (0, 1) 1+ x2 is equal to x2 (c) (1, ∞ )

(b) (0, 1)

(d) [1, ∞)

3. The curves y = | x |3 + 3 | x |2 + 2 and y = x 3 + 3x 2 + 2 have the same graph for (b) x ≥ 0

(a) x > 0

4. Domain of the function f ( x ) = (a) φ

(c) all x except 0 −1 1 1 + 2sin x + + 7 is 2 x x −3

(b) R - {0}

(c) R

5. The domain of definition of the function y = 3e (a) (1, ∞)

(b) [1, ∞)

x 2 −1

(b) {cos 1, 1, cos 2}

(d) None of these log ( x − 1) is (d) (−∞ , − 1) ∪ (1, ∞ )

(c) R ~ {1}

6. The range of the function f(x) = cos [x], where − (a) {-1, 1, 0}

(d) all x

π π < x < , is 2 2

(c) {cos 1, − cos 1, 1}

(d) none of these

7. If b2 - 4ac = 0 and a > 0, then domain of the function y = log (ax3 + (a + b)x2 + (b + c)x + c) is  b2  (a) R ~ −   2a 

 b   (b) R ~  −  ∪ {x | x ≥ −1}   2a  

 b   (c) R ~  −  ∪ (−∞, − 1]    2a  

(d) none of these

8. Which of the following functions is an even function? (a) f ( x ) =

a x +1 a x + a −x f ( x ) = (b) a x −1 a x − a −x

(c) f ( x ) = x

a x −1 a x +1

(

(d) f ( x ) = log 2 x + x 2 + 1

9. If (log3 x) (logx 2x) (log2x y) = logx x2, then y is equal to (a) 9

(b) 18

(c) 27

(d) 81

)

x 2 − x +1 is 10. The range of the function f ( x ) = 2 x + x +1 1  (c)  , 3 3 

(b) [3, ∞ )

(a) R

(d) none of these

11. The domain of the function f (x) = 2 − 2x − x2 is (a) − 3 ≤ x ≤ 3

(b) −1 − 3 ≤ x ≤ −1+ 3 (c) − 2 ≤ x ≤ 2

(d) none of these

12. If the domain of the function f ( x ) = x 2 − 6 x + 7 is (−∞, ∞), then range of the function is (a) ( −∞, ∞ )

(b) [ −2, ∞ )

(d) ( −∞, − 2)

(c) (-2, 3)

x2  −  13. The domain of the function f ( x ) = sin 1  log 2  is 2   (b) 0 ≤ x ≤ 1

(a) − 1 ≤ x ≤ 1

(c) 1 ≤ x ≤ 2

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

(a) 1

(b)

1/ 2

 x   f   + f (xy)  has the value  y     

1 2

(c) -2

15. The inverse of the function f ( x ) =  x −2  (a) log e   x −1 

1 2

1 ≤ y≤1 3

 x −1   (b) log e   3− x 

1/ 3

1 (b) − ≤ y < 1 3

17. The domain of the function (a) [1, 2]

(d) 0

e x − e− x + 2 is given by e x + e−x

16. The range of the function for real x of y =

(a)

(d) none of these

1/ 2

 x   (c) log e  2−x 

 x −1   (d) log e   x +1

−2

1 is 2 − sin 3x 1 (c) − > y > 1 3

(d)

1 > y>1 3

1 + 2 x − x 2 , where [ . ] denotes greatest integer function. [x ]

(b) [0, 2]

(c) [0, 1)

(d) [1, 2)

(c) [-9, 1]

(d) [-9, -1]

(c) {1, 2, 3, 4}

(d) {1, 2, 3, 4, 5}

18. Domain of sin-1(log3(x/3)) is (a) [1, 9]

(b) [-1, 9]

19. The range of f(x) = 7-xPx - 3 is (a) {1, 2, 3}

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

20. The value of n ∈ Z for which the function f ( x ) =

(a) 2

(b) 3

sin nx has 4π as its period is x sin  n

(c) 5

(d) 4

21. The domain of the function f(x) = log2(log3(log4 x)) 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 + αx , ( α ≠ 0 ) is the inverse of itself then the value of α 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))) is

3 , then the function is 4

(a) g ( x ) = ± 1 − x 2

(b) g ( x ) = − 1 − x 2

(c) g ( x ) = 1 − x 2

(d) g(x) = 1+ x 2

25. Which of the following functions is periodic

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

26. f ( x ) =

1  x sin , x ≠ 0 f ( x ) =  x (b)  0, x=0

(c) f(x) = x cos x

(d) none of these

( x + 1)( x − 3) is real valued in the domain x−2

(a) (−∞ , − 1] ∪ [3, ∞ )

(b) ( −∞, − 1] ∪ ( 2, 3] (c) [ −1, 2) ∪ [3, ∞) (d) none of these

27. 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

28. If the function f : [1, ∞ ) → [1, ∞ ) is defined by f(x) = 2x(x - 1) then f-1(x) is 1 (a)   2

x ( x −1)

(b)

(

)

(

)

1 1 1 + 1 + 4 log 2 x (c) 1 − 1 + 4 log 2 x (d) not defined 2 2

29. If log0.3(x - 1) < log0.09(x - 1), then x lies in the interval (a) (2, ∞ )

(b) (1, 2)

(c) (-2, -1)

(d) none of these

30. If g(f(x)) = |sin x| and f(g(x)) = (sin x ) 2 , then (a) f ( x ) = sin 2 x , g ( x ) = x

(b) f ( x ) = sin x , g ( x ) = | x |

(c) f ( x ) = x 2 , g ( x ) = sin x

(d) f and g cannot be determined

− 1, x < 0  31. Let g( x ) = 1 + x − [ x ] and 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)

32. The domain of definition of f ( x ) = (a) R ~ {-1, -2} 33. If f ( x ) = (a)

2

(b) ( −2, ∞ )

(d) g(x).

log 2 ( x + 3) is x 2 + 3x + 2

(c) R ~ {-1, -2, -3}

(d) (−3, ∞ ) ~ {−1, − 2}

αx , x ≠ − 1 , then for what value of α is f(f(x)) = x ? x +1

(b) − 2

(c) 1

(d) -1

34. The set of all real numbers x for which x2 - |x + 2| + x > 0, is (a) (−∞, − 2) ∪ (2, ∞ )

(b) ( −∞, − 2 ) ∪ ( 2 , ∞)

(c) (−∞ , − 1) ∪ (1, ∞ )

(d)

35. Range of the function f ( x ) =

(a) (1, ∞ )

(

2, ∞

)

x2 + x + 2 , x ∈ R is x2 + x +1

 11  (b) 1,   7

 7 (c) 1,   3

 7 (d) 1,   5

ANSWER KEY LEVEL - 1 (Objective) 1.

b

21.

b

2.

b

22.

b

3.

b

23.

b

4.

a

24.

d

5.

a

25.

a

6.

a

26.

c

7.

c

27.

a

8.

c

28.

c

9.

b

29.

a

10.

a

30.

c

11.

b

31.

a

12.

c

32.

a

13.

c

33.

a

14.

b

34.

c

15.

d

35.

c

16.

b

36.

c

17.

b

37.

a

18.

b

38.

a

19.

d

39.

d

20.

b

40.

d

ANSWER KEY LEVEL -2 (Subjective) 1.

x ∈ ( −∞, 1 / 2) ∪ (2, ∞) ~ {4}

14.

φ

2.

x ∈ ( −1, 1] ∪ [3, ∞ )

15.

(3, 4)

3.

x ∈ ( −5, − 2) ∪ ( −1, ∞ )

16.

x ∈ ( −∞ , 1)

4.

 3π   3π  f ( x ) ∈ (3, π) ∪  π,  ∪  , 5 2   2  

17.

 π π  ,  4 2

5.

f ( x ) ∈ (1, ∞ )

18.

Range∈ R

6.

x ∈ (0, 1) ∪ (1, 2]

19.

 3 3  π π Domain ∈  − ,  , Range ∈ 0,  2   3 3 

20.

(i) 3 sin x (ii)

21.

sin 2 x − sin x + 1, − 1 < x < 0  2 sin 2 x, 0 ≤ x ≤1 

7.

R ~ [1, 2)

8.

 1 Range = 0,   2

9.

  11   Range is log , ∞   3 

 −1+ 5 −1 − 5 − 3 + 5 − 3 − 5  , , ,  10. x =  2 2 2 2  

11.

12.

13.

f(x) is an odd function (if x ≠ nπ ) and f(x) is an even function if ( x = nπ ).

x + x2 − 4 2 1 3 + x− 2 4

− 3 ± 5 + ex 2

π π 1 22. f   = − 1, f ( π) = 0, f ( − π) = 0, f   = 2 2 4

23. 24.

25.

f (x ) =

1 5 a  − bx  − 2  a − b x  (a + b ) 2

ANSWER KEY LEVEL - 3 (Questions asked from previous Engineering Exams) 1.

d

19.

a

2.

c

20.

a

3.

b

21.

b

4.

a

22.

a

5.

a

23.

b

6.

b

24.

b

7.

c

25.

a

8.

c

26.

c

9.

a

27.

d

10.

c

28.

b

11.

b

29.

a

12.

b

30.

a

13.

d

31.

b

14.

d

32.

d

15.

b

33.

d

16.

a

34.

b

17.

a

35.

c

18.

a