RD Sharma solutions for Class 12 Maths chapter 2 - Functions [Latest edition]

Chapters Chapter 2: Functions

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Others
Exercise 2.1 [Pages 31 - 32]

RD Sharma solutions for Class 12 Maths Chapter 2 Functions Exercise 2.1 [Pages 31 - 32]

Exercise 2.1 | Q 1.1 | Page 31

Give an example of a function which is one-one but not onto ?

Exercise 2.1 | Q 1.2 | Page 31

Give an example of a function which is not one-one but onto ?

Exercise 2.1 | Q 1.3 | Page 31

Give an example of a function which is neither one-one nor onto ?

Exercise 2.1 | Q 2.1 | Page 31

Which of the following functions from A to B are one-one and onto?
f1 = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}

Exercise 2.1 | Q 2.2 | Page 31

Which of the following functions from A to B are one-one and onto?

f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {abc}

Exercise 2.1 | Q 2.3 | Page 31

Which of the following functions from A to B are one-one and onto ?

f3 = {(ax), (bx), (cz), (dz)} ; A = {abcd,}, B = {xyz}.

Exercise 2.1 | Q 3 | Page 31

Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto

Exercise 2.1 | Q 4 | Page 31

Let A = {−1, 0, 1} and f = {(xx2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.

Exercise 2.1 | Q 5.01 | Page 31

Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2

Exercise 2.1 | Q 5.02 | Page 31

Classify the following function as injection, surjection or bijection :  f : Z → Z given by f(x) = x2

Exercise 2.1 | Q 5.03 | Page 31

Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3

Exercise 2.1 | Q 5.04 | Page 31

Classify the following function as injection, surjection or bijection :  f : Z → Z given by f(x) = x3

Exercise 2.1 | Q 5.05 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = |x|

Exercise 2.1 | Q 5.06 | Page 31

Classify the following function as injection, surjection or bijection :

f : Z → Z, defined by f(x) = x2 + x

Exercise 2.1 | Q 5.07 | Page 31

Classify the following function as injection, surjection or bijection :

f : Z → Z, defined by f(x) = x − 5

Exercise 2.1 | Q 5.08 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = sinx

Exercise 2.1 | Q 5.09 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = x3 + 1

Exercise 2.1 | Q 5.1 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = x3 − x

Exercise 2.1 | Q 5.11 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = sin2x + cos2x

Exercise 2.1 | Q 5.12 | Page 31

Classify the following function as injection, surjection or bijection :

f : Q − {3} → Q, defined by f (x) = (2x +3)/(x-3)

Exercise 2.1 | Q 5.13 | Page 31

Classify the following function as injection, surjection or bijection :

f : Q → Q, defined by f(x) = x3 + 1

Exercise 2.1 | Q 5.14 | Page 31

Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4

Exercise 2.1 | Q 5.15 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = 3 − 4x

Exercise 2.1 | Q 5.16 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = 1 + x2

Exercise 2.1 | Q 5.17 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = x/(x^2 +1)

Exercise 2.1 | Q 6 | Page 31

If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.

Exercise 2.1 | Q 7 | Page 31

Show that the function f : R − {3} → R − {2} given by f(x) = (x-2)/(x-3) is a bijection.

Exercise 2.1 | Q 8.1 | Page 32

Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : f (x) = x/2

Exercise 2.1 | Q 8.2 | Page 32

Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|

Exercise 2.1 | Q 8.3 | Page 32

Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2

Exercise 2.1 | Q 9.1 | Page 32

Set of ordered pair of  a function? If so, examine whether the mapping is injective or surjective :{(xy) : x is a person, y is the mother of x}

Exercise 2.1 | Q 9.2 | Page 32

Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(ab) : a is a person, b is an ancestor of a

Exercise 2.1 | Q 10 | Page 32

Let A = {1, 2, 3}. Write all one-one from A to itself.

Exercise 2.1 | Q 11 | Page 32

If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.

Exercise 2.1 | Q 12 | Page 32

Show that the exponential function f : R → R, given by f(x) = ex, is one-one but not onto. What happens if the co-domain is replaced byR0^+ (set of all positive real numbers)?

Exercise 2.1 | Q 13 | Page 32

Show that the logarithmic function  f : R0+ → R   given  by f (x)  loga x ,a> 0   is   a  bijection.

Exercise 2.1 | Q 14 | Page 32

If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.

Exercise 2.1 | Q 15 | Page 32

If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.

Exercise 2.1 | Q 16 | Page 32

Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.

Exercise 2.1 | Q 17 | Page 32

Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.

Exercise 2.1 | Q 18 | Page 32

Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.

Exercise 2.1 | Q 19 | Page 32

Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.

Exercise 2.1 | Q 20 | Page 32

Suppose f1 and f2 are non-zero one-one functions from R to R. Is f_1 / f^2 necessarily one - one? Justify your answer. Here,f_1/f_2 : R → R   is   given   by   (f_1/f_2) (x) = (f_1(x))/(f_2 (x))  for all  x in R .

Exercise 2.1 | Q 21 | Page 32

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.

Exercise 2.1 | Q 22 | Page 32

Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.

Exercise 2.1 | Q 23 | Page 32

Let f : N → N be defined by

f(n) = { (n+ 1, if n  is  odd),( n-1 , if n  is  even):}

Show that f is a bijection.

[CBSE 2012, NCERT]

Exercise 2.2 [Pages 46 - 54]

RD Sharma solutions for Class 12 Maths Chapter 2 Functions Exercise 2.2 [Pages 46 - 54]

Exercise 2.2 | Q 1.1 | Page 46

Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and  g(x) = x2 + 5 .

Exercise 2.2 | Q 1.2 | Page 46

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = 2x + x2 and  g(x) = x3

Exercise 2.2 | Q 1.2 | Page 54

Find fog and gof  if : f (x) = x2 g(x) = cos x .

Exercise 2.2 | Q 1.3 | Page 46

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = x2 + 8 and g(x) = 3x3 + 1 .

Exercise 2.2 | Q 1.4 | Page 46

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = x and g(x) = |x| .

Exercise 2.2 | Q 1.4 | Page 54

Find fog and gof  if  : f (x) = ex g(x) = loge x .

Exercise 2.2 | Q 1.5 | Page 46

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = x2 + 2x − 3 and  g(x) = 3x − 4 .

Exercise 2.2 | Q 1.6 | Page 46

Find gof and fog when f : R → R and g : R → R is  defined by  f(x) = 8x3 and  g(x) = x1/3.

Exercise 2.2 | Q 2 | Page 46

Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.

Exercise 2.2 | Q 3 | Page 46

Let f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.

Exercise 2.2 | Q 4 | Page 46

Let A = {abc}, B = {u vw} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(av), (bu), (cw)}, g = {(ub), (va), (wc)}.
Show that f and g both are bijections and find fog and gof.

Exercise 2.2 | Q 5 | Page 46

Find  fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → Rg(x) = 3x3 + 1.

Exercise 2.2 | Q 6 | Page 46

Let R+ be the set of all non-negative real numbers. If f : R+ → R+ and g : R+ → R+ are defined as f(x)=x^2 and g(x)=+sqrtx , find fog and gof. Are they equal functions ?

Exercise 2.2 | Q 7 | Page 46

Let f : R → R and g : R → R be defined by f(x) = x2 and g(x) = x + 1. Show that fog ≠ gof.

Exercise 2.2 | Q 8 | Page 46

Let f : R → R and g : R → R be defined by f(x) = + 1 and (x) = x − 1. Show that fog = gof = IR.

Exercise 2.2 | Q 9 | Page 46

Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2xg(x) = 1/x and h(x) = ex.

Exercise 2.2 | Q 10 | Page 46

Consider f : N → Ng : N → N and h : N → R defined as f(x) = 2xg(y) = 3y + 4 and h(z) = sin z for all xyz ∈ N. Show that ho (gof) = (hogof.

Exercise 2.2 | Q 11 | Page 46

Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.

Exercise 2.2 | Q 12 | Page 46

Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.

Exercise 2.2 | Q 13 | Page 46

If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.

Exercise 2.2 | Q 14 | Page 46

If f : A → B and g : B → C are onto functions, show that gof is a onto function.

Exercise 2.3 [Pages 54 - 55]

RD Sharma solutions for Class 12 Maths Chapter 2 Functions Exercise 2.3 [Pages 54 - 55]

Exercise 2.3 | Q 1.3 | Page 54

Find fog and gof  if : f (x) = |x|, g (x) = sin x .

Exercise 2.3 | Q 1.4 | Page 54

Find fog and gof  if : f (x) = x+1, g(x) = e^x

.

Exercise 2.3 | Q 1.5 | Page 54

Find fog and gof  if : f(x) = sin−1 x, g(x) = x2

Exercise 2.3 | Q 1.6 | Page 54

Find fog and gof  if : f (x) = x+1, g (x) = sin x .

Exercise 2.3 | Q 1.7 | Page 54

Find fog and gof  if : f(x)= x + 1, g (x) = 2x + 3 .

Exercise 2.3 | Q 1.8 | Page 54

Find fog and gof  if : f(x) = c, c ∈ R, g(x) = sin x^2

Exercise 2.3 | Q 1.9 | Page 54

Find fog and gof  if : f(x) = x^2 + 2 , g (x) = 1 − 1/ (1-x).

Exercise 2.3 | Q 2 | Page 54

Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.

Exercise 2.3 | Q 3 | Page 54

If f(x) = |x|, prove that fof = f.

Exercise 2.3 | Q 4 | Page 54

If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2

Exercise 2.3 | Q 5 | Page 54

If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?

Exercise 2.3 | Q 6 | Page 54

Let fgh be real functions given by f(x) = sin xg (x) = 2x and h (x) = cos x. Prove that fog = go (fh).

Exercise 2.3 | Q 7 | Page 54

Let  f  be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.

Exercise 2.3 | Q 8 | Page 54

if f (x) = sqrt(1-x) and g(x) = log_e x are two real functions, then describe functions fog and gof.

Exercise 2.3 | Q 9 | Page 54

 if  f : (-π/2 , π/2) → R and g : [−1, 1]→ R be defined as f(x) = tan x and g(x) = sqrt(1 - x^2) respectively, describe fog and gof.

Exercise 2.3 | Q 10 | Page 54

if f (x) = sqrt (x +3) and  g (x) = x ^2 + 1 be two real functions, then find fog and gof.

Exercise 2.3 | Q 11 | Page 54

Let f be a real function given by f (x)=sqrt (x-2)
Find each of the following:

(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2

Also, show that fof ≠ f^2 .

Exercise 2.3 | Q 12 | Page 55

Let

f (x) ={ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}

Find fof.

Exercise 2.3 | Q 13 | Page 55

If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).

Exercise 2.4, Exercise 2.3 [Pages 68 - 69]

RD Sharma solutions for Class 12 Maths Chapter 2 Functions Exercise 2.4, Exercise 2.3 [Pages 68 - 69]

Exercise 2.4 | Q 1.1 | Page 68

State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}

Exercise 2.4 | Q 1.2 | Page 68

State with reason whether the following functions have inverse :

g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

Exercise 2.4 | Q 1.3 | Page 68

State with reason whether the following functions have inverse:

h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Exercise 2.4 | Q 2.1 | Page 68

Find f −1 if it exists : f : A → B, where A = {0, −1, −3, 2}; B = {−9, −3, 0, 6} and f(x) = 3 x.

Exercise 2.4 | Q 2.2 | Page 68

Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2

Exercise 2.4 | Q 3 | Page 68

Consider f : {1, 2, 3} → {abc} and g : {abc} → {apple, ball, cat} defined as f (1) = af (2) = bf (3) = cg (a) = apple, g (b) = ball and g (c) =  cat. Show that fg and gof are invertible. Find f−1g−1 and gof−1and show that (gof)−1 = f 1o g−1

Exercise 2.4 | Q 4 | Page 68

Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → Bg : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.

Exercise 2.4 | Q 5 | Page 68

Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1

Exercise 2.4 | Q 6 | Page 68

Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Exercise 2.4 | Q 7 | Page 68

Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 (x)= sqrt (x-4) where R+ is the set of all non-negative real numbers.

Exercise 2.4 | Q 8 | Page 68

if f(x) = (4x + 3)/(6x - 4), x ≠  2/3 show that fof(x) = x, for all x ≠ 2/3 . What is the inverse of f?

Exercise 2.4 | Q 9 | Page 68

Consider f : R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with f^-1 (x) = (sqrt (x +6)-1)/3 .

Exercise 2.4 | Q 10 | Page 69

If f : R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1. Hence, find f−1(24) and f−1 (5).

Exercise 2.4 | Q 11 | Page 69

A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).

Exercise 2.4 | Q 12 | Page 69

If f : Q → Qg : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.

Exercise 2.3 | Q 13 | Page 69

Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = (x-2)/(x-3).Show that f is one-one and onto and hence find f-1.

[CBSE 2012, 2014]

Exercise 2.4 | Q 14 | Page 69

Consider the function f : R→  [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with -1 (y) = (sqrt(54 + 5y) -3)/5             [CBSE 2015]

Exercise 2.4 | Q 15 | Page 69

Let f : N→N be a function defined as f(x)=9x^2+6x−5. Show that f : N→S, where S is the range of f, is invertible. Find the inverse of f and hence find f^-1(43) and f^−1(163).

Exercise 2.4 | Q 16 | Page 69

Let f : R {- 4/3} - 43 →">→ R be a function defined as f(x) = (4x)/(3x +4) . Show that f : R - {-4/3}→ Rang (f) is one-one and onto. Hence, find f -1.

Exercise 2.4 | Q 17 | Page 69

If f : R → (−1, 1) defined by f (x) = (10^x- 10^-x)/(10^x + 10 ^-x) is invertible, find f−1.

Exercise 2.4 | Q 18 | Page 69

If f : R → (0, 2) defined by f (x) =(e^x - e^(x))/(e^x +e^(-x))+1is invertible , find f-1.

Exercise 2.4 | Q 19 | Page 69

Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.

Exercise 2.4 | Q 20 | Page 69

Let A = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → Ag : A → A be two functions defined by f(x) = x2 and g(x) = sin (π x/2). Show that g−1 exists but f−1 does not exist. Also, find g−1.

Exercise 2.4 | Q 21 | Page 69

Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.

Exercise 2.4 | Q 22 | Page 69

If A = {1, 2, 3, 4} and B = {abcd}, define any four bijections from A to B. Also give their inverse functions.

Exercise 2.4 | Q 23 | Page 69

Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.

Exercise 2.4 | Q 24.1 | Page 69

If f : A → Ag : A → A are two bijections, then prove that fog is an injection ?

Exercise 2.4 | Q 24.2 | Page 69

If f : A → Ag : A → A are two bijections, then prove that fog is a surjection ?

[Pages 72 - 74]

RD Sharma solutions for Class 12 Maths Chapter 2 Functions [Pages 72 - 74]

Q 1 | Page 72

Which one of the following graphs represents a function? Q 2 | Page 73

Which of the following graphs represents a one-one function? Q 3 | Page 73

If A = {1, 2, 3} and B = {ab}, write the total number of functions from A to B.

Q 4 | Page 73

If A = {abc} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.

Q 5 | Page 73

Write the total number of one-one functions from set A = {1, 2, 3, 4} to set B = {abc}.

Q 6 | Page 73

If f : R → R is defined by f(x) = x2, write f−1 (25)

Q 7 | Page 73

If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.

Q 8 | Page 73

If f : R → R is given by f(x) = x3, write f−1 (1).

Q 9 | Page 73

Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x3. Write f−1(1).

Q 10 | Page 73

Let f  be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).

Q 11 | Page 73
If f : R → R be defined by f(x) = x4, write f−1 (1).
Q 12 | Page 73

If f : C → C is defined by f(x) = x4, write f−1 (1).

Q 13 | Page 73

If f : R → R is defined by f(x) = x2, find f−1 (−25).

Q 14 | Page 73

If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).

Q 15 | Page 73

If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).

Q 16 | Page 73

Let $f : \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \to R$  be a function defined by f(x) = cos [x]. Write range (f).

Q 17 | Page 73

If f : R → R defined by f(x) = 3x − 4 is invertible, then write f−1 (x).

Q 18 | Page 73

If f : R → Rg : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).

Q 19 | Page 73

Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by $f\left( x \right) = \frac{\left| x \right|}{x}$Write the range of f.

Q 20 | Page 73

Let $f : \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \to$ A be defined by f(x) = sin x. If f is a bijection, write set A.

Q 21 | Page 73

Let f : R → R+ be defined by f(x) = axa > 0 and a ≠ 1. Write f−1 (x).

Q 22 | Page 73

Let f : R − {−1} → R − {1} be given by$f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)$

Q 23 | Page 74

Let f : R - {- 3/5} → R be a function defined as f  (x) = (2x)/(5x +3).

f-1 : Range of f → R -{-3/5}.

Q 24 | Page 74

Let f : R → Rg : R → R be two functions defined by f(x) = x2 + x + 1 and g(x) = 1 − x2. Write fog (−2).

Q 25 | Page 74

Let f : R → R be defined as  f (x) = (2x - 3)/4. write fo f-1 (1) .

Q 26 | Page 74

Let f be an invertible real function. Write ( f-1  of ) (1) + ( f-1  of ) (2) +..... +( f-1 of ) (100 )

Q 27 | Page 74

Let A = {1, 2, 3, 4} and B = {ab} be two sets. Write the total number of onto functions from A to B.

Q 28 | Page 74

Write the domain of the real function

f (x) = sqrtx - [x] .

Q 29 | Page 74

Write the domain of the real function

f (x) = sqrt([x] - x) .

Q 30 | Page 74

Write the domain of the real function

f (x) = 1/(sqrt([x] - x).

Q 31 | Page 74

Write whether f : R → R, given by f(x) = x + sqrtx^2 is one-one, many-one, onto or into.

Q 32 | Page 74

If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).

Q 33 | Page 74

What is the range of the function

f (x) = ([x - 1])/(x -1) ?

Q 34 | Page 74

If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).

Q 35 | Page 74

If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).

Q 36 | Page 74

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.

Q 37 | Page 74

If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog.    [NCERT EXEMPLAR]

Q 38 | Page 74

Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f .   [NCERT EXEMPLAR]

Q 39 | Page 74

Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)}                                                                                                        [NCERT EXEMPLAR]

Q 40 | Page 74

Write the domain of the real function f defined by f(x) = sqrt (25 -x^2)   [NCERT EXEMPLAR]

Q 41 | Page 74

Let A = {abcd} and f : A → A be given by f = {( a,b ),( b , d ),( c , a ) , ( d , c )} write f^-1. [NCERT EXEMPLAR]

Q 42 | Page 74

Let fg : R → R be defined by f(x) = 2x + l and g(x) = x2−2 for all x

∈ R, respectively. Then, find gof.  [NCERT EXEMPLAR]

Q 43 | Page 74

If the mapping f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}, given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, then write fog. [NCERT EXEMPLAR]

Q 44 | Page 74

If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = $\alpha x + \beta$  then find the values of $\alpha$ and $\beta$ . [NCERT EXEMPLAR]

Q 45 | Page 74

If f(x) = 4 −( x - 7)3 then write f-1 (x).

[Pages 75 - 79]

RD Sharma solutions for Class 12 Maths Chapter 2 Functions [Pages 75 - 79]

Q 1 | Page 75

Let$A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = \text{B and C} = \left\{ x \in R : x \geq 0 \right\} and$$S = \left\{ \left( x, y \right) \in A \times B : x^2 + y^2 = 1 \right\} \text{and } S_0 = \left\{ \left( x, y \right) \in A \times C : x^2 + y^2 = 1 \right\}$

Then,

•  S defines a function from A to B

•  S_0 defines a function from A to C

• S0 defines a function from A to B

•  S defines a function from A to C

Q 2 | Page 75

$f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }$

• injective

• surjective

• bijective

• None of these

Q 3 | Page 75

$f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4$ is a bijection, then

• $A = \left\{ x \in R : - 1 < x < \infty \right\}, B = \left\{ x \in R : 2 < x < 4 \right\}$

• $A = \left\{ x \in R : - 3 < x < \infty \right\}, B = \left\{ x \in R : 2 < x < 4 \right\}$

• $A = \left\{ x \in R : - 3 < x < \infty \right\}, B = \left\{ x \in R : 2 < x < 4 \right\}$

• None of these

Q 4 | Page 75

The function f : R → R defined by

f (x) = 2^x + 2^(|x|) is

• one-one and onto

• many-one and onto

• one-one and into

• many-one and into

Q 5 | Page 75

Let the function

$f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}$

$f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},$

•  f is one-one but not onto

• f is onto but not one-one

• f is both one-one and onto

• None of these

Q 6 | Page 75

The function

f : A → B defined by

f (x) = - x2 + 6x - 8 is a bijection if

• A = (- ∞ , 3] and B = ( - ∞, 1 ]

•  A = [- 3 , ∞) and B = ( - ∞, 1 ]

• A = (- ∞ , 3] and B = [ 1 ,∞)

• A = [3 ,∞ ) and B = [ 1 ,∞ )

Q 7 | Page 75

Let

$A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B$ Then, the mapping$f : A \to \text{B given by} f\left( x \right) = x\left| x \right|$ is

• injective but not surjective

• surjective but not injective

• bijective

• none of these

Q 8 | Page 75

Let

f : R → R be given by

$f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3$

where [x] denotes the greatest integer less than or equal to x. Then, f(x) is

(d) one-one and onto

• many-one and onto

• many-one and into

• one-one and into

• one-one and onto

Q 9 | Page 76

Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is

• one-one and onto

• neither one-one nor onto

• one-one but-not onto

• onto but not one-one

Q 10 | Page 76

The function $f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is$

• one-one and onto

• one-one but not onto

• onto but not one-one

• onto but not one-one

Q 11 | Page 76

The range of the function

$f\left( x \right) =^{7 - x} P_{x - 3}$

• {1, 2, 3, 4, 5}

• {1, 2, 3, 4, 5, 6}

• {1, 2, 3, 4}

• {1, 2, 3}

Q 12 | Page 76

A function f  from the set of natural numbers to integers defined by

{([n-1]/2," when  n is  odd"   is ),(-n/2,when  n  is  even ) :}

• neither one-one nor onto

• one-one but not onto

• onto but not one-one

• one-one and onto both

Q 13 | Page 76

Let f be an injective map with domain {xyz} and range {1, 2, 3}, such that exactly one of the following statements is correct and the remaining are false.

$f\left( x \right) = 1, f\left( y \right) \neq 1, f\left( z \right) \neq 2 .$

The value of

$f^{- 1} \left( 1 \right)$ is

•  x

• y

• z

• none of these

Q 14 | Page 76

Which of the following functions form Z to itself are bijections?

• $f\left( x \right) = x^3$

• $f\left( x \right) = x + 2$

• $f\left( x \right) = 2x + 1$

• $f\left( x \right) = x^2 + x$

Q 15 | Page 76

Which of the following functions from

$A = \left\{ x : - 1 \leq x \leq 1 \right\}$

to itself are bijections?

• $f\left( x \right) = \frac{x}{2}$

• $g\left( x \right) = \sin\left( \frac{\pi x}{2} \right)$

• $h\left( x \right) = |x|$

• $k\left( x \right) = x^2$

Q 16 | Page 76

Let

$A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|$

• a bijection

• injective but not surjective

• surjective but not injective

• neither injective nor surjective

Q 17 | Page 76

If the function$f : R \to \text{A given by} f\left( x \right) = \frac{x^2}{x^2 + 1}$ is a surjection, then A =

• R

• [0, 1]

• [0, 1]

• [0, 1]

Q 18 | Page 76

If a function$f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5$ is a bijection, then B =

• R

• [1, ∞)

• [4, ∞)

• [5, ∞)

Q 19 | Page 76

The function

$f : R \to R$ defined by$f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)$

(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto

• one-one but not onto

• onto but not one-one

• both one and onto

• neither one-one nor onto

Q 20 | Page 76

The  function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = sin^-1 (3x - 4x^3), is

• bijection

• injection but not a surjection

• surjection but not an injection

• neither an injection nor a surjection

Q 21 | Page 76

Let

$f : R \to R$  be a function defined by

$f\left( x \right) = \frac{e^{|x|} - e^{- x}}{e^x + e^{- x}} . \text{Then},$

•  f is a bijection

• f is an injection only

•  f is surjection on only

• f is neither an injection nor a surjection

Q 22 | Page 77

Let

$f : R - \left\{ n \right\} \to R$

$f\left( x \right) = \frac{x - m}{x - n}, \text{where} \ m \neq n .$ Then,

•  f is one-one onto

•  f is one-one into

•  f is many one onto

• f is many one into

Q 23 | Page 77

Let

$f : R \to R$
$f\left( x \right) = \frac{x^2 - 8}{x^2 + 2}$
Then,  f is

• one-one but not onto

• one-one and onto

• onto but not one-one

• neither one-one nor onto

Q 24 | Page 77

$f : R \to R$ is defined by

$f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is$

• one-one but not onto

• many-one but onto

• one-one and onto

• neither one-one nor onto

Q 25 | Page 77

The function

$f : R \to R, f\left( x \right) = x^2$

• injective but not surjective

• surjective but not injective

• injective as well as surjective

• neither injective nor surjective

Q 26 | Page 77

A function f from the set of natural numbers to the set of integers defined by

$f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}$

• neither one-one nor onto

• one-one but not onto

• onto but not one-one

• one-one and onto

Q 27 | Page 77

Which of the following functions from

$A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}$

• $f\left( x \right) = |x|$

• $f\left( x \right) = \sin\frac{\pi x}{2}$

• $f\left( x \right) = \sin\frac{\pi x}{4}$

• None of these

Q 28 | Page 77

$f : Z \to Z$  be given by

 f (x) = {(x/2, ", if  x is even" ) ,(0 , ", if  x  is  odd "):}

Then,  f is

• onto but not one-one

• one-one but not onto

• one-one and onto

• neither one-one nor onto

Q 29 | Page 77

The function $f : R \to R$ defined by

$f\left( x \right) = 6^x + 6^{|x|}$ is

• one-one and onto

• many one and onto

• one-one and into

• many one and into

Q 30 | Page 77

Let  $f\left( x \right) = x^2 and g\left( x \right) = 2^x$ Then, the solution set of the equation

$fog \left( x \right) = gof \left( x \right)$ is

• R

• {0}

• {0, 2}

• none of these

Q 31 | Page 77

If $f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)$

• is given by  $\frac{1}{3x - 5}$

• is given by $\frac{x + 5}{3}$

• does not exist because f is not one-one

• does not exist because f is not onto

Q 32 | Page 77

If $g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}$

•  $f\left( x \right) = \sin^2 x, g\left( x \right) = \sqrt{x}$

• $f\left( x \right) = \sin x, g\left( x \right) = |x|$

• $f\left( x \right) = x^2 , g\left( x \right) = \sin \sqrt{x}$

•  f and g cannot be determied

Q 33 | Page 78

The inverse of the function

$f : R \to \left\{ x \in R : x < 1 \right\}$ given by

$f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}$ is

• $\frac{1}{2} \log \frac{1 + x}{1 - x}$

•  $\frac{1}{2} \log \frac{2 + x}{2 - x}$

• $\frac{1}{2} \log \frac{1 - x}{1 + x}$

• none of these

Q 34 | Page 78

Let

$A = \left\{ x \in R : x \geq 1 \right\}$ The inverse of the function,

$f : A \to A$ given by

$f\left( x \right) = 2^{x \left( x - 1 \right)} , is$

• $\left( \frac{1}{2} \right)^{x \left( x - 1 \right)}$

• $\frac{1}{2} \left\{ 1 + \sqrt{1 + 4 \log_2 x} \right\}$

•  $\frac{1}{2} \left\{ 1 - \sqrt{1 + 4 \log_2 x} \right\}$

• not defined

Q 35 | Page 78

Let

$A = \left\{ x \in R : x \leq 1 \right\} and f : A \to A$ be defined as

$f\left( x \right) = x \left( 2 - x \right)$ Then,

$f^{- 1} \left( x \right)$ is

• $1 + \sqrt{1 - x}$

• $1 - \sqrt{1 - x}$

• $\sqrt{1 - x}$

• $1 \pm \sqrt{1 - x}$

Q 36 | Page 78

Let  $f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)$

• $\text{x for all x} \in R$

•  $\text{x for all x} \in R - \left\{ 1 \right\}$

•  $\text{x for all x} \in R - \left\{ 0, 1 \right\}$

• none of these

Q 37 | Page 78

If the function

$f : R \to R$  be such that

$f\left( x \right) = x - \left[ x \right]$ where [x] denotes the greatest integer less than or equal to x, then $f^{- 1} \left( x \right)$

•  $\frac{1}{x - \left[ x \right]}$

• [x] − x

• not defined

• none of these

Q 38 | Page 78

If  $F : [1, \infty ) \to [2, \infty )$ is given by

$f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)$

• $\frac{x + \sqrt{x^2 - 4}}{2}$

• $\frac{x}{1 + x^2}$

• $\frac{x - \sqrt{x^2 - 4}}{2}$

• $1 + \sqrt{x^2 - 4}$

Q 39 | Page 78
Let
$g\left( x \right) = 1 + x - \left[ x \right] \text{and} f\left( x \right) = \begin{cases}- 1, & x < 0 \\ 0, & x = 0, \\ 1, & x > 0\end{cases}$ where [x] denotes the greatest integer less than or equal to x. Then for all $x, f \left( g \left( x \right) \right)$ is equal to

• x

• 1

• f(x)

•  g(x)

Q 40 | Page 78

Let  $f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1$ Then, for what value of α is $f \left( f\left( x \right) \right) = x?$

• $\sqrt{2}$

• $- \sqrt{2}$

• 1

• -1

Q 41 | Page 78

The distinct linear functions that map [−1, 1] onto [0, 2] are

• $f\left( x \right) = x + 1, g\left( x \right) = - x + 1$

• $f\left( x \right) = x - 1, g\left( x \right) = x + 1$

• $f\left( x \right) = - x - 1, g\left( x \right) = x - 1$

• None of these

Q 42 | Page 78

Let

$f : [2, \infty ) \to X$ be defined by

$f\left( x \right) = 4x - x^2$ Then, f is invertible if X =

•  $[2, \infty )$

•  $( - \infty , 2]$

•  $( - \infty , 4]$

• $[4, \infty )$

Q 43 | Page 78

If  $f : R \to \left( - 1, 1 \right)$ is defined by

$f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)$ equals

• $\sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}$

• $\text{ Sgn } \left( x \right) \sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}$

•  $- \sqrt{\frac{x}{1 - x}}$

• None of these

Q 44 | Page 79

Let [x] denote the greatest integer less than or equal to x. If $f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}$

• $\text{fogoh}\left( x \right) = \frac{\pi}{2}$

• fogoh (x) = π

•  $\text{ho f og = hogo f}$

• $\text{ho f og ≠ hogo f}$

Q 45 | Page 79

If  $g\left( x \right) = x^2 + x - 2\text{ and} \frac{1}{2} gof\left( x \right) = 2 x^2 - 5x + 2$ is equal to

• $2 x - 3$

• $2 x + 3$

• $2 x^2 + 3x + 1$

• 2   $x^2 - 3x - 1$

Q 46 | Page 79

If  $f\left( x \right) = \sin^2 x$ and the composite function   $g\left( f\left( x \right) \right) = \left| \sin x \right|$ then g(x) is equal to

• $\sqrt{x - 1}$

• $\sqrt{x}$

• $\sqrt{x + 1}$

• $- \sqrt{x}$

Q 47 | Page 79

If $f : R \to R$ is given by $f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)$ is equal to

•  $x^{1/3} - 3$

•  $x^{1/3} + 3$

• $\left( x - 3 \right)^{1/3}$

• $x + 3^{1/3}$

Q 48 | Page 79

Let   $f\left( x \right) = x^3$be a function with domain {0, 1, 2, 3}. Then domain of $f^{- 1}$ is

• {3, 2, 1, 0}

• {0, −1, −2, −3}

• {0, 1, 8, 27}

• {0, −1, −8, −27}

Q 49 | Page 78

Let
$f : R \to R$  be given by $f\left( x \right) = x^2 - 3$ Then, $f^{- 1}$ is given by

• $\sqrt{x + 3}$

• $\sqrt{x} + 3$

•  $x + \sqrt{3}$

• None of these

Q 50 | Page 79

Mark the correct alternative in the following question:

Let f : → R be given by f(x) = tanx. Then, f-1(1) is

•  $\frac{\pi}{4}$

• $\left\{ n\pi + \frac{\pi}{4}: n \in Z \right\}$

• does not exist

• none of these

Q 51 | Page 79

Mark the correct alternative in the following question:
Let f : R→ R be defined as, f(x) =  $\begin{cases}2x, if x > 3 \\ x^2 , if 1 < x \leq 3 \\ 3x, if x \leq 1\end{cases}$

Then, find f( $-$1) + f(2) + f(4)

• 14

• 5

• none of these

Q 52 | Page 79

Mark the correct alternative in the following question:
Let A = {1, 2, ... , n} and B = {a, b}. Then the number of subjections from A into B is

•  nP2

• 2n - 2

• 2n - 1

•  nC2

Q 53 | Page 79

Mark the correct alternative in the following question:

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

• 720

• 120

• none of these

Q 54 | Page 79

Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is

• 10C7

• 10C7$\times$ 7!

• 710

• 107

Q 55 | Page 79

Mark the correct alternative in the following question:
Let f :  $-$ $\left\{ \frac{3}{5} \right\}$ $\to$  R be defined by f(x) = $\frac{3x + 2}{5x - 3}$ Then,

• f-1 (x) = f (x)

• f^-1 (x) = - f(x)

• fo f(x) = - x

• f^-1(x) = 1/19f(x)

Chapter 2: Functions

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Others RD Sharma solutions for Class 12 Maths chapter 2 - Functions

RD Sharma solutions for Class 12 Maths chapter 2 (Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 2 Functions are Composition of Functions and Invertible Function, Types of Functions, Types of Relations, Introduction of Relations and Functions, Concept of Binary Operations, Inverse of a Function.

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