1

CPP: FUNCTIONS(1) Find the domain of the following functions.

1. ( ) 2f x x 1 x= − −

2. ( ) 1f x

x x=

−

3. ( ) ( )310 10log log 1f x x= +

4. ( ) ln lnf x x=

5. ( ) ( )1 24cos log x

f x e−

=

6. ( ) [ ]( )lnf x x x= − , [.] = G.I.F.

7. 2 2 2x y+ =

8. ( )2

1 21sin 1

2

xf x x

x− += + −

9. ( ) [ ]( )1sinf x x x−= + , [.] = G.I.F.

10. ( ) ( )2

1 12

1 xf x sin log x cos(sinx) cos

2x− − += + +

11. ( ) ( )10

12

log 1f x x

x= + +

−

12. ( ) ( )22

log 3

3 2

xf x

x x

+=

+ +

13. ( ) 1f x sin 2x6

− π= +

14. ( ) ( )2

1

4

sin 2

xf x

x−

−=−

15. ( ) ( )( )210 10log 1 log 5 16f x x x= − − +

16. ( ) ( )0.32

log 1

3 18

xf x

x x

−=

− −

17. ( ) 1

2

xf x

x

−=

−

18. ( )2

10

5log

4

x xf x

−=

19. ( ) 11

1 2cos log

3 x

xf x x−

−

−= +

20. ( ) ( )24 5 3log log log 18 77f x x x= − −

21. ( ) 10

1log

sinf x

x=

22. ( ) [ ]1, . . . .

1 7 6f x G I F

x x= = − + − −

23. ( ) [ ]1, . . . .

1 5f x G I F

x= =

− −

24. ( ) ( )1 2 2 1 !35cos

12

xf x x

x x− +

= − + +

, [.] =

G.I.F.

25. ( ) 10100

2log 1log x

xf x

x

+ = −

26. ( )12 9 4

1

1f x

x x x x=

− + − +

27. ( ) 7

3x

xf x P−−=

28. ( ) 10 10 10 10log log log logf x x=

29. ( ) 0.4

1log

5

xf x

x

− = +

30. ( ) [ ]1 2sec 1 , .f x x x− = − + = G.I.F.

31. ( ) ( )2 1log

xf x x

−=

2

32. ( ) { }logxf x x= , {.} = fractional part

33. ( ) 1 2sin 2 3 ,f x x− = − [.] = G.I.F.

34. ( ) ( )1 3 1

cos2 ln 4

xf x

x− −

= + −

35. ( ) [ ]log ,xf x x= [.] = G.I.F.

36. ( ) 1 22

1sin log

2f x x− =

37. ( )[ ] [ ]

[ ]2

1, .

6f x

x x= =

− −G.I.F.

38. ( ) 3 1 12 5

1log log 1 1f x

x

= − + −

39. ( ) ( )2

22

loglog

3 4 log

xf x

x

= −

40. ( ) 1 1cos

sinf x

x−=

ANSWERS

1. 1

,12

2. ( ),0−∞ 3. (0, ∞) 4. (0, 1) ∪ (1, ∞)

5. 1 1

2, ,22 2

− − ∪ 6. R – I 7. (-∞, 1) 8. {-1, 1}

9. [0, 1) 10. {1} 11. [-2, 1) – {0} 12. ( ) { }3, 2, 1− ∞ − − −

13. 1 1

,4 2

− 14. [ )1,2 15. ( )2,3 16. [ )2,6

17. ( ) [ ] ( ), 2 1,1 2,−∞ − ∪ − ∪ ∞ 18. [ ]1,4 19. ( ) ( )0,1 1,2∪ 20. ( )8,10

21. { }|R n n Zπ− ∈ 22. ( ] { } [ ){ }0,1 2,3,4,5,6 7,8R − ∪ ∪ 23. ( ] [ ), 7 7,−∞ − ∪ ∞

24. 1

2 −

25. 1 1 1

0, ,100 100 10

∪ 26. R

27. { }3,4,5 28. ( )1010 ,∞ 29. ( )1,∞ 30. ( ] [ ),0 1,−∞ ∪ ∞

31. ( )2,∞ 32. ( )0,1 33. [ ] { }1,1 0− −

34. [ ] [ ) ( )5, 1 1,3 3,4− − ∪ ∪ 35. [ )2,∞ 36. [ ] [ ]2, 1 1,2− − ∪ 37. ( ) [ ), 2 4,−∞ − ∪ ∞

38. ( )0,1 39. [ )8,16 40. ( )2 1 , 2

n n Zπ+ ∈

3

CPP: FUNCTIONS(2) Find the range of the following functions.

1. ( ) 22

1

1f x x

x= +

+

2. ( ) 2f x 9 x= −

3. ( ) 1

2 cos3f x

x=

−

4. ( ) 21

xf x

x=

+

5. ( )2

23sin16

f x xπ= −

6. ( ) 7

3x

xf x P−−=

7. ( ) 1 1 1sin cos tanf x x x x− − −= + +

8. ( )2

2

x x 2f x

x x 1

+ +=+ +

9. ( ) ( )210log 3 4 5f x x x= − +

10. ( ) ( )23log 5 4f x x x= + −

11. ( ) 2

sin cos 3 2log

2

x xf x

− +=

12. ( ) 2 4sin cosf x x x= +

13. ( ) ( )1 2cot 2f x x x−= −

14. ( ) [ ][ ] ,

1

x xf x

x x

−=

− + [.] = G.I.F.

15. ( ) , 0x x

x x

e ef x x

e e

−

−

−= >+

16. ( ) 1 5f x x x= − + −

17. ( ) 1f x

1 2cos x=

−

18. ( )2

12

xf x cos

1 x−

= +

19. ( ) ( )1 2f x sin x x 1−= + +

20. ( ) 1 2 3f x x x x= − + − + −

21. ( ) 1 2f x x x= − − +

22. ( ) cos2 sin 281 27x xf x =

23. ( ) 4 2 1x xf x = − +

24. ( )2

2 2

x xf x

x x

−=+

25. ( ) 5

3

xf x

x=

+

26. ( )2

293sin

16f x x

π= −

1. [1, ∞) 2. [0, 3] 3. 1

,13

4. 1 1

,2 2

−

5. 3

0,2

6. { }1,2,3 7. 3

,4 4

π π

8. 7

1,3

9. 10

11log ,

3 ∞

10. (-∞, 2] 11. [ ]1,2 12. 3

,14

13. ,4

π π

14. 1

0,2

15. ( )0,1 16. 2,2 2

4

17. ( ] 1, 1 ,

3 −∞ − ∪ ∞

18. 0,2

π

19. ,3 2

π π

20. [ )2,∞

21. [ ]0,3 22. 55

1,3

3

23. 3

,4 ∞

24. 1

,12

R − −

25. ( )5,5− 26. [ ]0,3

Even/Odd Functions

1. ( ) ( )2af x log x 1 x , a 0, a 1= + + > ≠

2. ( )x

x

a 1f x x

a 1

+= −

3. ( ) x

x xf x 1

e 1 2= + +

−

4. ( ) ( )12 tan2

xf x eπ−= −

5. Let [ ]: 10,10f R− → , where

( )2

sin ,x

f x xa

= +

[.] = G.I.F. be an odd

function. Find the values of ‘a’.

6. Draw the graph of the following function and decide whether it is even or odd.

x | x |, x 1

f (x) [1 x] [1 x] 1 x 1

x | x |, x 1

≤ −= + + − − < <− ≥

, where

[.] = G.I.F.

7. If a function f satisfies

( ) ( ) 0f x f x a+ + = for all real x and

0,a > prove that the function is periodic. Also, find its period.

8. Let :f R R→ be a function given by

( ) ( ) ( ) ( )2f x y f x y f x f y+ + − = for all

real ,x y . If ( )0 0,f ≠ prove that f is an

even function. If ( )0 0,f = prove that f

is an odd function.

1. Odd 2. Even 3. Even 4. Odd 5. (100, ∞) 6. Even 7. 2a Periodic Functions 1. ( ) ( ) ( )cos sin cos cosf x x x= +

2. ( ) 2 3sin tan sin tan2 2 2

x x xf x x= + + + +K

1sin tan2 2n n

x x−+ +

3. ( ) | sin x cos x |f x

| sin x | | cos x |

+=+

4. ( ) [ ] cos cos2 cosx x x x n xf x e π π π− + + + += K , [.] =

G.I.F.

5. ( ) ( )sin cosf x x x= +

6. ( ) ( )cos sin cosf x x x= −

7. ( ) [ ]tan ,2

f x xπ =

[.] = G.I.F.

8. ( ) 3sin sin 3f x x x=

9. ( ) ( )1 sin

cos 1 cos

xf x

x ecx

+=+

10. For what value of ‘a’ are the functions sin cosax ax+ and sin cosx x+ periodic

with the same period?

1. 2

π 2. 2n π 3. π 4. 1

5. 2π 6. 2

π 7. 2 8. π

9. π 10. a = 4

5

CPP: FUNCTIONS(3) One-One/Many-One/Onto/Into Functions 1. : ,f R R→ ( ) 3 26 11 6f x x x x= − + −

2. : ,f R R→ ( ) 2 sinf x x x= +

3. [ ) [ ): 0, 0, ,f ∞ → ∞ ( )1

xf x

x=

+

4. : ,f R R→ ( ) 3 2 3 sinf x x x x x= + + +

5. : ,f R R→ ( )2

2

4

1

xf x

x

−=+

6. : ,f R R→ ( )2

2

x 4x 30f x

x 8x 18

+ +=− +

7. : ,f N N→ ( ) ( )1x

f x x= − −

8. : ,f R R→ ( ) 4 4xxf x = +

9. : ,f R R→ ( ) 2f x x x= +

10. [ ] [ ]: 1,1 1,1 ,f − → − ( )f x x x=

11. { } { }: 3 1 ,f R R− → − ( ) 2

3

xf x

x

+=−

12. : ,f R R→ ( ) 0, x rationalf x

x, x irrational

∈= ∈

: ,g R R→ ( ) 0, x irrationalg x

x, x rational

∈= ∈

Then ( ) ( )f g x− is ...

13. Which of the following functions from Z to itself is a bijection?

(a) ( ) 3f x x= + (b) ( ) 5f x x=

(c) ( ) 3 2f x x= + (d) ( ) 2f x x x= +

14. Which of the following functions is one-one?

(a) ( ) 2 2, f x x x R= + ∈

(b) ( ) [ )2 , 2,f x x x= + ∈ − ∞

(c) ( ) ( ) ( )4 5 , f x x x x R= − − ∈

(d) ( ) { }2

2

4 3 5, , ,

4 3 5

x xf x x R

x xα β+ −= ∈ −

+ −

where ,α β are the roots of the equation 24 3 5 0.x x+ − =

15. If the function : ,f A B→

( ) 2 6 8f x x x= − + − is bijective, find the

maximum interval for B. 16. Find the number of bijective functions

from a set A to itself when A contains 100 elements.

17. Find the number of surjections from

{ }1,2,3, ,A n= K onto { }, .B a b=

18. Set A has 3 elements and set B has 4 elements. Find the number of injections that can be defined from A to B.

19. Find two distinct linear functions which map [-1, 1] onto [0, 2].

20. If : ,f R R→ ( ) sinf x px x= + is one-one

and onto, find the interval in which p lies. 21. Let f be a one-one function with domain

{ }, ,x y z and range { }1,2,3 such that

exactly one of the following statements is true and the rest are false:

( ) ( ) ( )1, 1, 2.f x f y f z= ≠ ≠ Find the

value of ( )1 1 .f −

22. Find the interval of values of a for which the function :f R R→ defined by

( ) ( )3 24 6 6f x x a x ax= + + + + is one-one.

23. Find the interval of values of a for which the function :f R R→ defined by

( )2

2

6 8

6 8

ax xf x

a x x

+ −=+ −

is onto.

1. Many-one and Onto 2. One-one and Onto 3. One-one and Into 4. One-one and Onto 5. Many-one and Into 6. Many-one and Into 7. One-one and Onto 8. Many-one and Into 9. Many-one and Into 10. One-one and Onto 11. One-one and Onto 12. One-one and Onto 13. (a) 14. (b) 15. ( ],1B = −∞ 16. 100!

17. 2 2n − 18. 24 19. 1x + and 1 x− 20. ( )1,1R − −

21. y 22. [ ]2,8a ∈ 23. [ ]2,14a ∈

6

Inverse Function 1. Find the inverse of the function

( ) , 0, 1.x x

x x

a af x a a

a a

−

−

−= > ≠+

2. Let [ ) [ ): 4, 1,f ∞ → ∞ be defined by

( ) ( )45 .x xf x −= Find ( )1 .f x−

3. Let ( ] ( ]: ,2 ,4f −∞ → −∞ be defined by

( ) 24 .f x x x= − Find ( )1 .f x−

4. Let [ ) [ ): 1, 2,f ∞ → ∞ be defined by

( ) 1.f x x

x= + Find ( )1 .f x−

5. Let 1 3

f : , ,2 4 ∞ → ∞

be defined by

( ) 2 1.f x x x= − + Find the inverse of

( )f x if it exists. Hence or otherwise,

solve the equation 21 3

1 .2 4

x x x− + = + −

6. Find the values of the parameter a for which the function ( ) 1f x ax= + is the

inverse of itself. 7. Let :f R R→ be a function given by

( ) , .f x ax b x R= + ∈ Find ‘a’ and ‘b’ such

that ( ) .fof x I=

8. If ( ) 2

, 1

, 1 4,

8 , 4

x x

f x x x

x x

<

= ≤ ≤ >

find ( )1 .f x−

9. Let :f X Y→ be defined by

( ) sin cos .4

f x a x b x cπ = + + +

If f is

bijective, find X and .Y 10. If ( ) ( ) ( )3 / 2 1, ,xf x x e g x f x−= + = find

the value of ( )' 1 .g

1. ( )1 1 1log

2 1a

xf x

x− + = −

2. 52 4 log x+ + 3. 2 4 x− − 4. 2 4

2

x x+ −

5. ( )1 1 3f x x

2 4− = + − , x = 1 6. 1a = − 7. 1, 0a b= = or 1, a b R= − ∈

8. ( )1

2

, 1

, 1 16

, 1664

x x

f x x x

xx

−

<= ≤ ≤ >

9. [ ], , , ,2 2

X Y c r c rπ πα α = − − − = − +

where

1 2tan

a b

aα − +=

and 2 2 2r a b ab= + +

10. 2 Composite Functions

1. If ( ) x 1, x 1f x ,

2x 1, 1 x 2

+ ≤= + < ≤

( )2x , 1 x 2

g x ,x 2, 2 x 3

− ≤ <= + ≤ ≤

find ( )fog x

and ( ) .gof x

2. If ( ) 1 1 , 1 3f x x x= − + − − ≤ ≤ and

( ) 2 1 , 2 2g x x x= − + − ≤ ≤ , find

( )fog x and ( ) .gof x

3. If ( ) 2

2 , 1

2 1, 1

x xf x

x x

<= − ≥

and

( ) 2, 0,

2 , 0

x xg x

x x

+ <= ≥

find ( ).fog x

4. If ( ) 1 , 0 2,

3 , 2 3

x xf x

x x

+ ≤ ≤= − < ≤

find ( ) .fof x

5. If ( ) , 1,

2 , 1

x xf x

x x

≤=

− > find ( ) .fof x

7

6. Let f be a function defined on [ ]2,2−

and given by ( ) 1, 2 0

1, 0 2

xf x

x x

− − ≤ ≤= − < ≤

and ( ) ( ) ( ) .g x f x f x= + Find ( ) .g x

7. If ( ) [ ]1 , 3,3 ,f x x x= − ∈ − find the

domain of ( ) .fof x

8. If ( ) 2f x x= − and ( ) 1 2 ,g x x= − find

the domain of ( ).fog x

1. ( )2

2

x 1, 1 x 1fog x ,

2x 1, 1 x 2

+ − ≤ ≤= + < ≤

( ) ( )2gof x x 1 , 2 x 1= + − ≤ ≤

2. ( )1, 2 1

1, 1 0

1, 0 2

x x

fog x x x

x x

+ − ≤ < −= − − − ≤ ≤ − < ≤

, ( ) 1 , 1 1

3 , 1 3

x xgof x

x x

+ − ≤ ≤= − < ≤

3. ( )

2

2

2 4, 1

2 8 7, 1 0

14 , 0

21

8 1, 2

x x

x x x

fog x x x

x x

+ < − + + − ≤ <= ≤ <

− ≥

4. ( )2 , 0 1

2 , 1 2

4 , 2 3

x x

fof x x x

x x

+ ≤ ≤= − < ≤ − < ≤

5. ( )2 , 1

, 1 1

2 , 1

x x

fof x x x

x x

− < −

= − ≤ ≤ − >

6. ( ), 2 0

0, 0 1

2 2, 1 2

x x

g x x

x x

− − ≤ <= ≤ < − ≤ ≤

7. [ ]2,3− 8. 3 1

,2 2

−

Functional Equations

1. If 22

1 1, 0,f x x x

x x + = + ≠

prove that

( ) 2 2, 2.f x x x= − ≥

2. If ( ) ( )cos ln ,f x x= find the value of

( ) ( ) ( )1.

2

xf x f y f f xy

y

− +

3. ( )f x is a polynomial function satisfying

( ) ( )1 1,f x f f x f

x x = +

{ }0x R∈ −

and ( )3 28.f = Find ( )4 .f

4. g is a function satisfying

( ) ( ) ( ) ( ) ( ) 2g x g y g x g y g xy= + + − for

all real ,x y and ( )2 5.g = Find ( )5 .g

5. If ( ) , 0,x

x

af x a

a a= >

+ prove that

( ) ( )1 1.f x f x+ − = Also, find the value of

1 2 1996.

1997 1997 1997f f f + + +

K

2. 0 3. 65 4. 26 5. 998

8

Miscellaneous 1. The function ( )f x is defined for [ ]0,1 .x ∈ Find the domain of (i) ( )2 3f x + , (ii) ( )tanf x , and (iii)

( )sin .f x

2. Solve for x : { } [ ] [ ]6 2 5 3 .x x x x+ + + = +

3. f is an even function defined on ( )5,5− . Find the real values of x satisfying ( ) 1.

2

xf x f

x

+ = +

4. The function ( )f x satisfies ( ) ( ) ( )( ) ( )( )( )1

2 3 31 2 3 3f x p f x f x f x+ = + − + − for all real x and

0p > is a constant. Prove that f is periodic and find its period.

5. The function ( )f x satisfies ( ) ( ) ( )1 1 3f x f x f x+ + − = for all real x . Prove that the function is

periodic and find its period.

6. The function ( )f x satisfies the relation ( ) 1 15, 0, .af x bf x a b

x x + = − ≠ ≠

Find ( )f x .

7. The function ( ),f x y satisfies the relation ( )2 , 2 .f x y x y xy+ − = Find ( ), .f x y

8. If ( ) ( ) ( )f x y f x f y+ = for , ,x y N∈ ( )1 2f = and ( ) ( )1

16 2 1n

n

k

f a k=

+ = −∑ , find the value of .a

1. (i) 3

, 12

− − (ii) ,

4n n

ππ π + (iii) ( )2 , 2 1n nπ π + 2.

12

3

3. 1 5 3 5

,2 2

− ± − ± 4. 2p 5. 12

6. ( ) 2 2

1 5af x bx

a b x a b = − − − +

7. ( )2 2

,8

x yf x y

−= 8. 3a =

Number of solutions 1. Find the number of solutions of

( )sin sin sinx y x y+ = + and 1.x y+ =

2. Solve for x: 2

3 40.

3 4

x x

x x

−≥

− −

3. Solve for x: ( )1 1 2 .x x xe e e+ − = −

4. cos2 sin ,x x= [ ]2 ,5x π π∈ −

5. sin lnx xπ =

6. 7 5x x− = −

1. 6 2. ( ) [ ), 1 0,4x ∈ −∞ − ∪ 3. ln 3x = 4. 14

5. 6 6. 4