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RD Sharma solutions for Class 12 Mathematics chapter 2 - Functions

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 2 : Functions

Page 46

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 .

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

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 .

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| .

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 .

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.

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 fogare both defined. Also, find fog and gof.

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.

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.

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.

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 find fog and gof. Are they equal functions ?

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.

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.

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.

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.

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.

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.

Q 13 | Page 46

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

Q 14 | Page 46

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

Pages 54 - 55

Q 1.1 | Page 54

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

Q 1.2 | Page 54

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

Q 1.3 | Page 54

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

Q 1.4 | Page 54

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

Q 1.5 | Page 54

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

Q 1.6 | Page 54

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

Q 1.7 | Page 54

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

Q 1.8 | Page 54

Find fog and gof  if : f(x) = c, c ∈ R, g(x) = sin x2.

Q 1.9 | Page 54

Find fog and gof  if : f(x) = x2 + 2 , g (x) = 1 − `1/ (1-x)`.

Q 2 | Page 54

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

Q 3 | Page 54

If f(x) = |x|, prove that fof = f.

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

Q 5 | Page 54

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

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).

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.

Q 8 | Page 54

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

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.

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.

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 ≠ f2 .

Q 12 | Page 55

Let

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

Find fof.

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" > x, ∀x ∈ R .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).

Pages 68 - 69

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)}

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)}

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)}

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.

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

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

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.

Q 5 | Page 68

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

Q 6 | Page 68

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

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.

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?

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 .`

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).

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).

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.

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]

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]

Q 15 | Page 69

Let f : NN be a function defined as f(x)=9x2+6x5. Show that f : NS, where S is the range of f, is invertible. Find the inverse of f and hence find f1(43) and f1(163).

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.

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.

Q 18 | Page 69

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

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)}.

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.

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.

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.

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.

Q 24.1 | Page 69

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

Q 24.2 | Page 69

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

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 .

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

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,
(a) S defines a function from A to B
(b) S0 defines a function from A to C
(c) S0 defines a function from A to B
(d) S defines a function from A to C

Q 2 | Page 75

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

(a) injective
(b) surjective
(c) bijective
(d) 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) \[A = \left\{ x \in R : - 1 < x < \infty \right\}, B = \left\{ x \in R : 2 < x < 4 \right\}\]

(b) \[A = \left\{ x \in R : - 3 < x < \infty \right\}, B = \left\{ x \in R : 2 < x < 4 \right\}\]

(c) \[A = \left\{ x \in R : - 2 < x < \infty \right\}, B = \left\{ x \in R : 2 < x < 4 \right\}\]

(d) None of these

Q 4 | Page 75

The function f : R → R defined by

`f (x) = 2^x + 2^(|x|)` is 

(a) one-one and onto
(b) many-one and onto
(c) one-one and into
(d) many-one and into

Q 5 | Page 75

Let the function

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

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

(a) f is one-one but not onto
(b) f is onto but not one-one
(c) f is both one-one and onto
(d) None of these

Q 6 | Page 75

The function 

f : A → B defined by 

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

(a) A = (- ∞ , 3] and B = ( - ∞, 1 ]

(b) A = [- 3 , ∞) and B = ( - ∞, 1 ]

(c) A = (- ∞ , 3] and B = [ 1 ,∞)

(d) 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 

(a) injective but not surjective
(b) surjective but not injective
(c) bijective
(d) 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
(a) many-one and onto
(b) many-one and into
(c) one-one and into
(d) 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
(a) one-one and onto
(b) neither one-one nor onto
(c) one-one but-not onto
(d) onto but not one-one

Q 10 | Page 76

The function

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

(a) one-one and onto
(b) one-one but not onto
(c) onto but not one-one
(d) onto but not one-one

Q 11 | Page 76

The range of the function

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

(a) {1, 2, 3, 4, 5}
(b) {1, 2, 3, 4, 5, 6}
(c) {1, 2, 3, 4}
(d) {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"  2 ),(-n/2,when  n  is  even ) :}`

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

(a) x
(b) y
(c) z
(d) none of these

Q 14 | Page 76

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

(a) \[f\left( x \right) = x^3\]

(b) \[f\left( x \right) = x + 2\]

(c) \[f\left( x \right) = 2x + 1\]
(d) \[f\left( x \right) = x^2 + x\]
Q 15 | Page 76

Which of the following functions from

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

to itself are bijections?
(a)  \[f\left( x \right) = \frac{x}{2}\]

(b) \[g\left( x \right) = \sin\left( \frac{\pi x}{2} \right)\]

(c) \[h\left( x \right) = |x|\]

(d)  \[k\left( x \right) = x^2\]

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) a bijection
(b) injective but not surjective
(c) surjective but not injective
(d) 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 =

(a) R
(b) [0, 1]
(c) [0, 1)
(d) [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 =
(a) R
(b) [1, ∞)
(c) [4, ∞)
(d) [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

Q 20 | Page 76

The function\[y = f\left( x \right)\] 
\[ \Rightarrow y = \left( x - 1 \right)\left( x - 2 \right)\left( x - 3 \right)\] 
\[\text{Sincey} \in R \text{ and } x \in R, \text{f is onto}.\]\[f\left( x \right) = \sin^{- 1} \left( 3x - 4 x^3 \right)\], is 

(a) bijection
(b) injection but not a surjection
(c) surjection but not an injection
(d) neither an injection nor a surjection

Q 21 | Page 76

Let

\[f : R \to R\]  be a function defined by

\[f\left( x \right) = \frac{e^{|x|} - e^{- x}}{e^x + e^{- x}} . \text{Then},\]
(a) f is a bijection
(b) f is an injection only
(c) f is surjection on only
(d) f is neither an injection nor a surjection
Q 22 | Page 77

Let

\[f : R - \left\{ n \right\} \to R\]

\[f\left( x \right) = \frac{x - m}{x - n}, where m \neq n .\]
(a) f is one-one onto
(b) f is one-one into
(c) f is many one onto
(d) 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
(a) one-one but not onto
(b) one-one and onto
(c) onto but not one-one
(d) neither one-one nor onto
Q 24 | Page 77

\[f : R \to R\] is defined by

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

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

Q 25 | Page 77

The function

\[f : R \to R, f\left( x \right) = x^2\]
(a) injective but not surjective
(b) surjective but not injective
(c) injective as well as surjective
(d) neither injective nor surjective
Q 26 | Page 77

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

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

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

Q 27 | Page 77

Which of the following functions from

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

(a) \[f\left( x \right) = |x|\]

(b)\[f\left( x \right) = \sin\frac{\pi x}{2}\]

(c) \[f\left( x \right) = \sin\frac{\pi x}{4}\]

(d) 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
(a) onto but not one-one
(b) one-one but not onto
(c) one-one and onto
(d) 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 

(a) one-one and onto
(b) many one and onto
(c) one-one and into
(d) 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 
(a) R
(b) {0}
(c) {0, 2}
(d) none of these
Q 31 | Page 77

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

(a) is given by  \[\frac{1}{3x - 5}\]  

(b) is given by \[\frac{x + 5}{3}\]

(c) does not exist because f is not one-one
(d) does not exist because f is not onto

Q 32 | Page 77

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

(a) \[f\left( x \right) = \sin^2 x, g\left( x \right) = \sqrt{x}\]

(b)  \[f\left( x \right) = \sin x, g\left( x \right) = |x|\]

(c)  \[f\left( x \right) = x^2 , g\left( x \right) = \sin \sqrt{x}\]

f and g cannot be determined.

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 

 (a)  \[\frac{1}{2} \log \frac{1 + x}{1 - x}\]

(b) \[\frac{1}{2} \log \frac{2 + x}{2 - x}\]

(c) \[\frac{1}{2} \log \frac{1 - x}{1 + x}\]

(d) 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

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

(a) \[\left( \frac{1}{2} \right)^{x \left( x - 1 \right)}\]

(b) \[\frac{1}{2} \left\{ 1 + \sqrt{1 + 4 \log_2 x} \right\}\]

(c) \[\frac{1}{2} \left\{ 1 - \sqrt{1 + 4 \log_2 x} \right\}\]

(d) 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 

(a)  \[1 + \sqrt{1 - x}\]

(b)  \[1 - \sqrt{1 - x}\]

(c)  \[\sqrt{1 - x}\]

(d)  \[1 \pm \sqrt{1 - x}\]

Q 36 | Page 78

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

(a) \[\text{x for all x} \in R\]

(b) \[\text{x for all x} \in R - \left\{ 1 \right\}\]

(c) \[\text{x for all x} \in R - \left\{ 0, 1 \right\}\]

(d) 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 \[f^{- 1} \left( x \right)\]

(a)   \[\frac{1}{x - \left[ x \right]}\]

(b) [x] − x
(c) not defined
(d) none of these

Q 38 | Page 78

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

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

(a)  \[\frac{x + \sqrt{x^2 - 4}}{2}\]

(b)  \[\frac{x}{1 + x^2}\]

(c)  \[\frac{x - \sqrt{x^2 - 4}}{2}\]

(d)  \[1 + \sqrt{x^2 - 4}\]

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
(a) x
(b) 1
(c) f(x)
(d) 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 \[f \left( f\left( x \right) \right) = x?\]

(a) \[\sqrt{2}\]

(b) \[- \sqrt{2}\]

(c) 1
(d) −1

Q 41 | Page 78

The distinct linear functions that map [−1, 1] onto [0, 2] are
(a)  \[f\left( x \right) = x - 1, g\left( x \right) = x + 1\]

(b) \[f\left( x \right) = x - 1, g\left( x \right) = x + 1\]

(c) \[f\left( x \right) = - x - 1, g\left( x \right) = x - 1\]

(d) 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 =

(a) \[[2, \infty )\]

(b) \[( - \infty , 2]\]

(c) \[( - \infty , 4]\]

(d) \[[4, \infty )\]

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

(a) \[\sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}\]

(b) \[\text{ Sgn } \left( x \right) \sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}\]

(c) \[- \sqrt{\frac{x}{1 - x}}\]

(d) None of these

Q 44 | Page 79

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

 (a)  \[\text{fogoh}\left( x \right) = \frac{\pi}{2}\]

(b)  \[\text{fogoh}\left( x \right) = \pi\]

(c) \[\text{ho f og = hogo f}\]

(d) \[\text{hofog \neq hogof}\]

Q 45 | Page 79

If  \[g\left( x \right) = x^2 + x - 2 and \frac{1}{2} gof\left( x \right) = 2 x^2 - 5x + 2\] is equal to
(a)  \[2 x - 3\]

(b) \[2 x + 3\]
(c) \[2 x^2 + 3x + 1\]
(d) 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
(a) \[\sqrt{x - 1}\]

(b)  \[\sqrt{x}\]

(c) \[\sqrt{x + 1}\]

(d) \[- \sqrt{x}\]

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

(a) \[x^{1/3} - 3\]

(b) \[x^{1/3} + 3\]

(c) \[\left( x - 3 \right)^{1/3}\]

(d) \[x + 3^{1/3}\]

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 

(a) {3, 2, 1, 0}
(b) {0, −1, −2, −3}
(c) {0, 1, 8, 27}
(d) {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 

(a) \[\sqrt{x + 3}\]

(b)  \[\sqrt{x} + 3\]

(c)  \[x + \sqrt{3}\]

(d) 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

(a) \[\frac{\pi}{4}\]   

(b) \[\left\{ n\pi + \frac{\pi}{4}: n \in Z \right\}\]

(c) does not exist    

(d) 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)

(a) 9 

(b) 14                              

(c) 5                             

(d) 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
(a) nP2 

(b) 2n - 2

 (c) 2n - 1

  (d) 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

(a) 720                                                

(b) 120                                               

(c) 0                                                 

(d) 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
(a) 10C7 

(b) 10C7\[\times\] 7!

(c) 710 

(d)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,

(a) f-1 (x) = f (x)

(b)  `f^-1 (x) = - f(x)`

(c) fo f(x) = - x 

(d) `f^-1(x) = 1/19f(x)`

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma solutions for Class 12 Mathematics 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 Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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

Using RD Sharma Class 12 solutions Functions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

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