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RD Sharma solutions for Class 11 Mathematics chapter 3 - Functions

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RD Sharma Mathematics Class 11

Mathematics Class 11 - Shaalaa.com

Chapter 3: Functions

Ex. 3.10Ex. 3.20Ex. 3.30Ex. 3.40Ex. 3.50Others

Chapter 3: Functions Exercise 3.10 solutions [Pages 7 - 8]

Ex. 3.10 | Q 1 | Page 7

Define a function as a set of ordered pairs.

 
Ex. 3.10 | Q 2 | Page 7

Define a function as a correspondence between two sets.

 
Ex. 3.10 | Q 3 | Page 7

What is the fundamental difference between a relation and a function? Is every relation a function?

Ex. 3.10 | Q 4.1 | Page 7

Let A = {−2, −1, 0, 1, 2} and f : A → Z be a function defined by f(x) = x2 − 2x − 3. Find:

(a) range of f, i.e. f(A).

Ex. 3.10 | Q 4.2 | Page 7

Let A = {−2, −1, 0, 1, 2} and f : A → Z be a function defined by f(x) = x2 − 2x − 3. Find:

(b) pre-images of 6, −3 and 5.

 
Ex. 3.10 | Q 5 | Page 7
\[f\left( x \right) = \begin{cases}3x - 2, & x < 0; \\ 1, & x = 0; \\ 4x + 1, & x > 0 .\end{cases}\]

find: f(1), f(−1), f(0) and f(2).

 

 

Ex. 3.10 | Q 6 | Page 7

A function f : R → R is defined by f(x) = x2. Determine (a) range of f, (b) {x : f(x) = 4}, (c) [yf(y) = −1].

Ex. 3.10 | Q 7.1 | Page 7

Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine

(a) the image set of the domain of f

Ex. 3.10 | Q 7.2 | Page 7

Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine

(b) {x : f(x) = −2}

Ex. 3.10 | Q 7.3 | Page 7

Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine

(c) whether f(xy) = f(x) : f(y) holds

 
Ex. 3.10 | Q 8.1 | Page 8

Write the following relations as sets of ordered pairs and find which of them are functions:

(a) {(xy) : y = 3xx ∈ {1, 2, 3}, y ∈ [3,6, 9, 12]}

Ex. 3.10 | Q 8.2 | Page 8

Write the following relations as sets of ordered pairs and find which of them are functions: 

(b) {(xy) : y > x + 1, x = 1, 2 and y = 2, 4, 6}

Ex. 3.10 | Q 8.3 | Page 8

Write the following relations as sets of ordered pairs and find which of them are functions:

{(xy) : x + y = 3, xy, ∈ [0, 1, 2, 3]}

 

 

Ex. 3.10 | Q 9 | Page 8

Let f : R → R and g : C → C be two functions defined as f(x) = x2 and g(x) = x2. Are they equal functions?

Ex. 3.10 | Q 10.1 | Page 8

fgh are three function defined from R to R as follow:

(i) f(x) = x2

Find the range of function.

 
Ex. 3.10 | Q 10.2 | Page 8

fgh are three function defined from R to R as follow:

(ii) g(x) = sin x

Find the range of function.

Ex. 3.10 | Q 10.3 | Page 8

fgh are three function defined from R to R as follow:

(iii) h(x) = x2 + 1

Find the range of function.

Ex. 3.10 | Q 11.1 | Page 8

Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}
Determine which of the set are functions from X to Y.

(a) f1 = {(1, 1), (2, 11), (3, 1), (4, 15)} 

Ex. 3.10 | Q 11.2 | Page 8

Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}
Determine which of the set are functions from X to Y.

(b) f2 = {(1, 1), (2, 7), (3, 5)}

Ex. 3.10 | Q 11.3 | Page 8

Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}
Determine which of the set are functions from X to Y.

(c) f3 = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

 

 

Ex. 3.10 | Q 12 | Page 8

et A = (12, 13, 14, 15, 16, 17) and f : A → Z be a function given by
f(x) = highest prime factor of x.
Find range of f.

Ex. 3.10 | Q 13 | Page 8

If f : R → R be defined by f(x) = x2 + 1, then find f−1 [17] and f−1 [−3].

 
Ex. 3.10 | Q 14 | Page 8

Let A = [pqrs] and B = [1, 2, 3]. Which of the following relations from A to B is not a function?

  • (a) R1 = [(p, 1), (q, 2), (r, 1), (s, 2)]

  • (b) R2 = [(p, 1), (q, 1), (r, 1), (s, 1)]

  • (c) R3 = [(p, 1), (q, 2), (p, 2), (s, 3)

  • (d) R4 = [(p, 2), (q, 3), (r, 2), (s, 2)].

     
Ex. 3.10 | Q 15 | Page 8

Let A = [9, 10, 11, 12, 13] and let f : A → N be defined by f(n) = the highest prime factor of n. Find the range of f.

Ex. 3.10 | Q 16 | Page 8

The function f is defined by \[f\left( x \right) = \begin{cases}x^2 , & 0 \leq x \leq 3 \\ 3x, & 3 \leq x \leq 10\end{cases}\] The relation g is defined by \[g\left( x \right) = \begin{cases}x^2 , & 0 \leq x \leq 2 \\ 3x, & 2 \leq x \leq 10\end{cases}\]

Show that f is a function and g is not a function.

 
 
Ex. 3.10 | Q 17 | Page 8

If f(x) = x2, find \[\frac{f\left( 1 . 1 \right) - f\left( 1 \right)}{\left( 1 . 1 \right) - 1}\]

 
Ex. 3.10 | Q 18 | Page 8

Express the function f : X → given by f(x) = x+ 1 as set of ordered pairs, where X = {−1, 0, 3, 9, 7}

Chapter 3: Functions Exercise 3.20 solutions [Pages 11 - 12]

Ex. 3.20 | Q 1 | Page 11

If f(x) = x2 − 3x + 4, then find the values of x satisfying the equation f(x) = f(2x + 1).

 
Ex. 3.20 | Q 2 | Page 11

If f(x) = (x − a)2 (x − b)2, find f(a + b).

 
Ex. 3.20 | Q 3 | Page 11

If  \[y = f\left( x \right) = \frac{ax - b}{bx - a}\] , show that x = f(y).

 

 

Ex. 3.20 | Q 4 | Page 11

If  \[f\left( x \right) = \frac{1}{1 - x}\] , show that f[f[f(x)]] = x.

 

 

Ex. 3.20 | Q 5 | Page 11

If  \[f\left( x \right) = \frac{x + 1}{x - 1}\] , show that f[f[(x)]] = x.

 

 

Ex. 3.20 | Q 6 | Page 11

If  \[f\left( x \right) = \begin{cases}x^2 , & \text{ when }  x < 0 \\ x, & \text{ when }  0 \leq x < 1 \\ \frac{1}{x}, & \text{ when }  x \geq 1\end{cases}\]

find: (a) f(1/2), (b) f(−2), (c) f(1), (d)

\[f\left( \sqrt{3} \right)\] and (e) \[f\left( \sqrt{- 3} \right)\]
 
Ex. 3.20 | Q 7 | Page 11

If  \[f\left( x \right) = x^3 - \frac{1}{x^3}\] , show that

\[f\left( x \right) + f\left( \frac{1}{x} \right) = 0 .\]
 

 

Ex. 3.20 | Q 8 | Page 11

If \[f\left( x \right) = \frac{2x}{1 + x^2}\] , show that f(tan θ) = sin 2θ.

 

 

Ex. 3.20 | Q 9 | Page 12

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

(i) \[f\left( \frac{1}{x} \right) = - f\left( x \right)\]

(ii) \[f\left( - \frac{1}{x} \right) = - \frac{1}{f\left( x \right)}\]

Ex. 3.20 | Q 10 | Page 12

If f(x) = (a − xn)1/na > 0 and n ∈ N, then prove that f(f(x)) = x for all x.

 
Ex. 3.20 | Q 11 | Page 12

If for non-zero xaf(x) + bf \[\left( \frac{1}{x} \right) = \frac{1}{x} - 5\] , where a ≠ b, then find f(x).

 

Chapter 3: Functions Exercise 3.30 solutions [Page 18]

Ex. 3.30 | Q 1.1 | Page 18

Find the domain of the real valued function of real variable: 

(i)  \[f\left( x \right) = \frac{1}{x}\]

 

Ex. 3.30 | Q 1.2 | Page 18

Find the domain of the real valued function of real variable: 

(ii)  \[f\left( x \right) = \frac{1}{x - 7}\]

 

Ex. 3.30 | Q 1.3 | Page 18

Find the domain of the real valued function of real variable: 

(iii) \[f\left( x \right) = \frac{3x - 2}{x + 1}\]

 

Ex. 3.30 | Q 1.4 | Page 18

Find the domain of the real valued function of real variable: 

(iv)  \[f\left( x \right) = \frac{2x + 1}{x^2 - 9}\]

 

Ex. 3.30 | Q 1.5 | Page 18

Find the domain of the real valued function of real variable:  

(v)  \[f\left( x \right) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}\]

 

Ex. 3.30 | Q 2.1 | Page 18

Find the domain of the real valued function of real variable:

(i) \[f\left( x \right) = \sqrt{x - 2}\]

 

Ex. 3.30 | Q 2.2 | Page 18

Find the domain of the real valued function of real variable:

(ii) \[f\left( x \right) = \frac{1}{\sqrt{x^2 - 1}}\]

 

Ex. 3.30 | Q 2.3 | Page 18

Find the domain of the real valued function of real variable:

(iii) \[f\left( x \right) = \sqrt{9 - x^2}\]

 

Ex. 3.30 | Q 2.4 | Page 18

Find the domain of the real valued function of real variable:

(iv)  \[f\left( x \right) = \frac{\sqrt{x - 2}}{3 - x}\]

 

Ex. 3.30 | Q 3.01 | Page 18

Find the domain and range of the real valued function:

(i) \[f\left( x \right) = \frac{ax + b}{bx - a}\]

 

Ex. 3.30 | Q 3.02 | Page 18

Find the domain and range of the real valued function:

(ii) \[f\left( x \right) = \frac{ax - b}{cx - d}\]

 

 

Ex. 3.30 | Q 3.03 | Page 18

Find the domain and range of the real valued function:

(iii)  \[f\left( x \right) = \sqrt{x - 1}\]

 

Ex. 3.30 | Q 3.04 | Page 18

Find the domain and range of the real valued function:

(iv) \[f\left( x \right) = \sqrt{x - 3}\]

 

Ex. 3.30 | Q 3.05 | Page 18

Find the domain and range of the real valued function:

(v) \[f\left( x \right) = \frac{x - 2}{2 - x}\]

Ex. 3.30 | Q 3.06 | Page 18

Find the domain and range of the real valued function:

(vi) \[f\left( x \right) = \left| x - 1 \right|\] 

 

Ex. 3.30 | Q 3.07 | Page 18

Find the domain and range of the real valued function:

(vii)  \[f\left( x \right) = - \left| x \right|\]

 

Ex. 3.30 | Q 3.08 | Page 18

Find the domain and range of the real valued function:

(viii)  \[f\left( x \right) = \sqrt{9 - x^2}\]

 

Ex. 3.30 | Q 3.09 | Page 18

Find the domain and range of the real valued function:

(ix)  \[f\left( x \right) = \frac{1}{\sqrt{16 - x^2}}\]

Ex. 3.30 | Q 3.1 | Page 18

Find the domain and range of the real valued function:

(x)  \[f\left( x \right) = \sqrt{x^2 - 16}\]

Chapter 3: Functions Exercise 3.40 solutions [Page 38]

Ex. 3.40 | Q 1.1 | Page 38

Find f + gf − gcf (c ∈ R, c ≠ 0), fg, \[\frac{1}{f}\text{  and } \frac{f}{g}\] in :

(a) If f(x) = x3 + 1 and g(x) = x + 1

Ex. 3.40 | Q 1.2 | Page 38

Find f + gf − gcf (c ∈ R, c ≠ 0), fg, \[\frac{1}{f}\text{  and } \frac{f}{g}\] in : 

(b) If \[f\left( x \right) = \sqrt{x - 1}\]  and  \[g\left( x \right) = \sqrt{x + 1}\]

 

Ex. 3.40 | Q 2 | Page 38

Let f(x) = 2x + 5 and g(x) = x2 + x. Describe (i) f + g (ii) f − g (iii) fg (iv) f/g. Find the domain in each case.

 
Ex. 3.40 | Q 3 | Page 38

If f(x) be defined on [−2, 2] and is given by \[f\left( x \right) = \begin{cases}- 1, & - 2 \leq x \leq 0 \\ x - 1, & 0 < x \leq 2\end{cases}\]  and g(x)

\[= f\left( \left| x \right| \right) + \left| f\left( x \right) \right|\] , find g(x).

 
 
 
Ex. 3.40 | Q 4.1 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(i) f + g

 
 
Ex. 3.40 | Q 4.2 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(ii) g − 

Ex. 3.40 | Q 4.3 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(iii) f g

Ex. 3.40 | Q 4.4 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(iv) \[\frac{f}{g}\]

 
Ex. 3.40 | Q 4.5 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(v) \[\frac{g}{f}\]

 
Ex. 3.40 | Q 4.6 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(vi)  \[2f - \sqrt{5} g\]

 
Ex. 3.40 | Q 4.7 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(vii) f2 + 7f

Ex. 3.40 | Q 4.8 | Page 38

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(viii) \[\frac{5}{8}\]

 
Ex. 3.40 | Q 5.1 | Page 38

If f(x) = loge (1 − x) and g(x) = [x], then determine function:

(i) f + g

 

Ex. 3.40 | Q 5.2 | Page 38

If f(x) = loge (1 − x) and g(x) = [x], then determine function:

(ii) fg

Ex. 3.40 | Q 5.3 | Page 38

If f(x) = loge (1 − x) and g(x) = [x], then determine function:

(iii) \[\frac{f}{g}\]

 
Ex. 3.40 | Q 5.4 | Page 38

If f(x) = loge (1 − x) and g(x) = [x], then determine function:

(iv) \[\frac{g}{f}\] Also, find (f + g) (−1), (fg) (0),

\[\left( \frac{f}{g} \right) \left( \frac{1}{2} \right), \left( \frac{g}{f} \right) \left( \frac{1}{2} \right)\]
 
 
Ex. 3.40 | Q 6 | Page 38

If fg and h are real functions defined by 

\[f\left( x \right) = \sqrt{x + 1}, g\left( x \right) = \frac{1}{x}\] and h(x) = 2x2 − 3, find the values of (2f + g − h) (1) and (2f + g − h) (0).
 
 
Ex. 3.40 | Q 7 | Page 38

The function f is defined by

\[f\left( x \right) = \begin{cases}1 - x, & x < 0 \\ 1 , & x = 0 \\ x + 1, & x > 0\end{cases}\] . Draw the graph of f(x).
 
 
Ex. 3.40 | Q 8 | Page 38

Let fg : R → R be defined, respectively by f(x) = x + 1 and g(x) = 2x − 3. Find f + gf − g and  \[\frac{f}{g}\] .

 

Ex. 3.40 | Q 9 | Page 38

Let f : [0, ∞) → R and g : R → R be defined by \[f\left( x \right) = \sqrt{x}\] and g(x) = x. Find f + gf − gfg and \[\frac{f}{g}\] .

 
 
Ex. 3.40 | Q 10 | Page 38

Let f(x) = x2 and g(x) = 2x+ 1 be two real functions. Find (g) (x), (f − g) (x), (fg) (x) and  \[\left( \frac{f}{g} \right) \left( x \right)\] .

 

Chapter 3: Functions Exercise 3.50 solutions [Pages 41 - 42]

Ex. 3.50 | Q 1 | Page 41

Write the range of the real function f(x) = |x|.

 
Ex. 3.50 | Q 2 | Page 41

If f is a real function satisfying \[f\left( x + \frac{1}{x} \right) = x^2 + \frac{1}{x^2}\]

for all x ∈ R − {0}, then write the expression for f(x).

 
 
Ex. 3.50 | Q 3 | Page 41

Write the range of the function f(x) = sin [x], where \[\frac{- \pi}{4} \leq x \leq \frac{\pi}{4}\] .

 
Ex. 3.50 | Q 4 | Page 41

If f(x) = cos [π2]x + cos [−π2x, where [x] denotes the greatest integer less than or equal to x, then write the value of f(π).

Ex. 3.50 | Q 5 | Page 41

Write the range of the function f(x) = cos [x], where \[\frac{- \pi}{2} < x < \frac{\pi}{2}\] .

 
Ex. 3.50 | Q 6 | Page 41

Write the range of the function f(x) = ex[x]x ∈ R.

 
Ex. 3.50 | Q 7 | Page 41

Let  \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] . Then write the value of α satisfying f(f(x)) = x for all x ≠ −1.

 

 

Ex. 3.50 | Q 8 | Page 42

If\[f\left( x \right) = 1 - \frac{1}{x}\] , then write the value of \[f\left( f\left( \frac{1}{x} \right) \right)\]

 

 

Ex. 3.50 | Q 9 | Page 42

Write the domain and range of the function  \[f\left( x \right) = \frac{x - 2}{2 - x}\] .

 
Ex. 3.50 | Q 10 | Page 41

If f(x) =  4x − x2x ∈ R, then write the value of f(a + 1) −f(a − 1).

 
Ex. 3.50 | Q 11 | Page 42

If fgh are real functions given by f(x) = x2g(x) = tan x and h(x) = loge x, then write the value of (hogof)\[\left( \sqrt{\frac{\pi}{4}} \right)\] .

 

Ex. 3.50 | Q 12 | Page 42

Write the domain and range of function f(x) given by

\[f\left( x \right) = \frac{1}{\sqrt{x - \left| x \right|}}\] .
 
Ex. 3.50 | Q 13 | Page 42

Write the domain and range of  \[f\left( x \right) = \sqrt{x - \left[ x \right]}\] .

 
Ex. 3.50 | Q 14 | Page 42

Write the domain and range of function f(x) given by \[f\left( x \right) = \sqrt{\left[ x \right] - x}\] .

 
Ex. 3.50 | Q 15 | Page 42

Let A and B be two sets such that n(A) = p and n(B) = q, write the number of functions from A to B.

Ex. 3.50 | Q 16 | Page 42

Let f and g be two functions given by

f = {(2, 4), (5, 6), (8, −1), (10, −3)} and g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, −5)}.

Find the domain of f + g

Ex. 3.50 | Q 17 | Page 42

Find the set of values of x for which the functions f(x) = 3x2 − 1 and g(x) = 3 + x are equal.

Ex. 3.50 | Q 18 | Page 42

Let f and g be two real functions given by

f = {(0, 1), (2, 0), (3, −4), (4, 2), (5, 1)} and g = {(1, 0), (2, 2), (3, −1), (4, 4), (5, 3)}

Find the domain of fg.

Chapter 3: Functions solutions [Pages 42 - 45]

Q 1 | Page 42

Let A = {1, 2, 3} and B = {2, 3, 4}. Then which of the following is a function from A to B? 

 

  • (a) {(1, 2), (1, 3), (2, 3), (3, 3)}

  • (b) [(1, 3), (2, 4)]

  • (c) {(1, 3), (2, 2), (3, 3)}

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

     
Q 2 | Page 42

If f : Q → Q is defined as f(x) = x2, then f−1 (9) is equal to

  • (a) 3

  • (b) −3

  • (c) {−3, 3}

  • (d) ϕ

     
Q 3 | Page 42

Which one of the following is not a function?

  • (a) {(xy) : xy ∈ R, x2 = y}

  • (b) {(xy) : xy ∈, R, y2 = x}

  • (c) {(xy) : xy ∈ R, x2 = y3}

  • (d) {(xy) : xy ∈, R, y = x3}

     
Q 4 | Page 42

If f(x) = cos (log x), then the value of f(x2f(y2) −

\[\frac{1}{2}\left\{ f\left( \frac{x^2}{y^2} \right) + f\left( x^2 y^2 \right) \right\}\] is
 
  • (a) −2

  • (b) −1

  • (c) 1/2

  • (d) None of these

     
Q 5 | Page 43

If f(x) = cos (log x), then the value of f(xf(y) −\[\frac{1}{2}\left\{ f\left( \frac{x}{y} \right) + f\left( xy \right) \right\}\] is

 
  • (a) −1

  • (b) 1/2

  • (c) −2

  • (d) None of these

     
Q 6 | Page 43

Let f(x) = |x − 1|. Then,

  • (a) f(x2) = [f(x)]2

  • (b) f(x + y) = f(xf(y)

  • (c) f(|x| = |f(x)|

  • (d) None of these

     
Q 7 | Page 43

The range of f(x) = cos [x], for π/2 < x < π/2 is

  • (a) {−1, 1, 0}

  • (b) {cos 1, cos 2, 1}

  • (c) {cos 1, −cos 1, 1}

  • (d) [−1, 1]

     
Q 8 | Page 43

Which of the following are functions?

  • (a) {(xy) : y2 = xxy ∈ R}

  • (b) {(xy) : y = |x|, xy ∈ R}

  • (c) {(xy) : x2 + y2 = 1, xy ∈ R}

  • (d) {(xy) : x2 − y2 = 1, xy ∈ R}

     
Q 9 | Page 43

If  \[f\left( x \right) = \log \left( \frac{1 + x}{1 - x} \right) \text{ and}  g\left( x \right) = \frac{3x + x^3}{1 + 3 x^2}\] , then f(g(x)) is equal to

 

  • (a) f(3x)

  • (b) {f(x)}3

  • (c) 3f(x)

  • (d) −f(x)

     
Q 10 | Page 43

If A = {1, 2, 3} and B = {xy}, then the number of functions that can be defined from A into B is

  • (a) 12

  • (b) 8

  • (c) 6

  • (d) 3

     
Q 11 | Page 43

If \[f\left( x \right) = \log \left( \frac{1 + x}{1 - x} \right)\] , then \[f\left( \frac{2x}{1 + x^2} \right)\]  is equal to

 

 

  • (a) {f(x)}2

  • (b) {f(x)}3

  • (c) 2f(x)

  • (d) 3f(x)

     
Q 12 | Page 43

If f(x) = cos (log x), then value of \[f\left( x \right) f\left( 4 \right) - \frac{1}{2} \left\{ f\left( \frac{x}{4} \right) + f\left( 4x \right) \right\}\] is 

  • (a) 1

  • (b) −1

  • (c) 0

  • (d) ±1

     
Q 13 | Page 43

If  \[f\left( x \right) = \frac{2^x + 2^{- x}}{2}\] , then f(x + yf(x − y) is equal to

 

  • (a) \[\frac{1}{2}\left[ f\left( 2x \right) + f\left( 2y \right) \right]\]

     

  • (b)  \[\frac{1}{2}\left[ f\left( 2x \right) - f\left( 2y \right) \right]\]

     

  • (c)  \[\frac{1}{4}\left[ f\left( 2x \right) + f\left( 2y \right) \right]\]

     

  • (d) \[\frac{1}{4}\left[ f\left( 2x \right) - f\left( 2y \right) \right]\]

     

Q 14 | Page 43

If 2f (x) − \[3f\left( \frac{1}{x} \right) = x^2\] (x ≠ 0), then f(2) is equal to

 
  • (a)  \[- \frac{7}{4}\]

     

  • (b)  \[\frac{5}{2}\]

     

  • (c) −1

  • (d) None of these

     
Q 15 | Page 43

Let f : R → R be defined by f(x) = 2x + |x|. Then f(2x) + f(−x) − f(x) =

  • (a) 2x

  • (b) 2|x|

  • (c) −2x

  • (d) −2|x|

     
Q 16 | Page 43

The range of the function  \[f\left( x \right) = \frac{x^2 - x}{x^2 + 2x}\]  is 

 
  • (a) R

  • (b) R − {1}

  • (c) R − {−1/2, 1}

  • (d) None of these

     
Q 17 | Page 43

If x ≠ 1 and \[f\left( x \right) = \frac{x + 1}{x - 1}\] is a real function, then f(f(f(2))) is

 
  • (a) 1

  • (b) 2

  • (c) 3

  • (d) 4

     
Q 18 | Page 43

If f(x) = cos (loge x), then \[f\left( \frac{1}{x} \right)f\left( \frac{1}{y} \right) - \frac{1}{2}\left\{ f\left( xy \right) + f\left( \frac{x}{y} \right) \right\}\] is equal to

 
  • (a) cos (x − y)

  • (b) log (cos (x − y))

  • (c) 1

  • (d) cos (x + y)

     
  • (e) 0

Q 19 | Page 44

Let f(x) = x, \[g\left( x \right) = \frac{1}{x}\]  and h(x) = f(xg(x). Then, h(x) = 1

  • (a) x ∈ R

  • (b) x ∈ Q

  • (c) x ∈ R − Q

  • (d) x ∈ R, x ≠ 0

     
Q 20 | Page 44

If  \[f\left( x \right) = \frac{\sin^4 x + \cos^2 x}{\sin^2 x + \cos^4 x}\] for x ∈ R, then f (2002) = 

  • (a) 1

  • (b) 2

  • (c) 3

  • (d) 4

     
Q 21 | Page 44

The function f : R → R is defined by f(x) = cos2 x + sin4 x. Then, f(R) =

  • (a) [3/4, 1)

  • (b) (3/4, 1]

  • (c) [3/4, 1]

  • (d) (3/4, 1)

     
Q 22 | Page 44

Let A = {x ∈ R : x ≠ 0, −4 ≤ x ≤ 4} and f : A ∈ R be defined by  \[f\left( x \right) = \frac{\left| x \right|}{x}\] for x ∈ A. Then th (is

  • (a) [1, −1]

  • (b) [x : 0 ≤ x ≤ 4]

  • (c) {1}

  • (d) {x : −4 ≤ x ≤ 0}

     
  • (e) 

    {-1,1} 

Q 23 | Page 44

If f : R → R and g : R → R are defined by f(x) = 2x + 3 and g(x) = x2 + 7, then the values of x such that g(f(x)) = 8 are

  • (a) 1, 2

  • (b) −1, 2

  • (c) −1, −2

  • (d) 1, −2

     
Q 24 | Page 44

If : [−2, 2] → R is defined by \[f\left( x \right) = \begin{cases}- 1, & \text{ for }  - 2 \leq x \leq 0 \\ x - 1, & \text{ for }   0 \leq x \leq 2\end{cases}\] , then
{x ∈ [−2, 2] : x ≤ 0 and f (|x|) = x} =

 
  • (a) {−1}

  • (b) {0}

  • (c) \[\left\{ - \frac{1}{2} \right\}\]

  • (d) ϕ

     
Q 25 | Page 44

If  \[e^{f\left( x \right)} = \frac{10 + x}{10 - x}\] , x ∈ (−10, 10) and \[f\left( x \right) = kf\left( \frac{200 x}{100 + x^2} \right)\] , then k =

 
  • (a) 0.5

  • (b) 0.6

  • (c) 0.7

  • (d) 0.8

Q 26 | Page 44

f is a real valued function given by \[f\left( x \right) = 27 x^3 + \frac{1}{x^3}\] and α, β are roots of \[3x + \frac{1}{x} = 12\] . Then,

 
 
  • (a) f(α) ≠ f(β)

  • (b) f(α) = 10

  • (c) f(β) = −10

  • (d) None of these

     
Q 27 | Page 44

If  \[f\left( x \right) = 64 x^3 + \frac{1}{x^3}\] and α, β are the roots of \[4x + \frac{1}{x} = 3\] . Then,

 
  • (a) f(α) = f(β) = −9

  • (b) f(α) = f(β) = 63

  • (c) f(α) ≠ f(β)

  • (d) none of these

     
Q 28 | Page 44

If \[3f\left( x \right) + 5f\left( \frac{1}{x} \right) = \frac{1}{x} - 3\]  for all non-zero x, then f(x) =

  • (a)  \[\frac{1}{14}\left( \frac{3}{x} + 5x - 6 \right)\]

  • (b)  \[\frac{1}{14}\left( - \frac{3}{x} + 5x - 6 \right)\]

  • (c) \[\frac{1}{14}\left( - \frac{3}{x} + 5x + 6 \right)\]

  • (d) None of these

     
Q 29 | Page 44

If f : R → R be given by for all \[f\left( x \right) = \frac{4^x}{4^x + 2}\]  x ∈ R, then

 
  • (a) f(x) = f(1 − x)

  • (b) f(x) + f(1 − x) = 0

  • (c) f(x) + f(1 − x) = 1

  • (d) f(x) + f(x − 1) = 1

     
Q 30 | Page 44

If f(x) = sin [π2x + sin [−π]2 x, where [x] denotes the greatest integer less than or equal to x, then

  • (a) f(π/2) = 1

  • (b) f(π) = 2

  • (c) f(π/4) = −1

  • (d) None of these

     
Q 31 | Page 45

The domain of the function

\[f\left( x \right) = \sqrt{2 - 2x - x^2}\] is
 
  • (a)  \[\left[ - \sqrt{3}, \sqrt{3} \right]\]

     

  • (b)  \[\left[ - 1 - \sqrt{3}, - 1 + \sqrt{3} \right]\]

     

  • (c) [−2, 2]

  • (d)  \[\left[ - 2 - \sqrt{3}, - 2 + \sqrt{3} \right]\]

     

Q 32 | Page 45

The domain of definition of  \[f\left( x \right) = \sqrt{\frac{x + 3}{\left( 2 - x \right) \left( x - 5 \right)}}\] is 

  
  • (a) (−∞, −3] ∪ (2, 5)

  • (b) (−∞, −3) ∪ (2, 5)

  • (c) (−∞, −3) ∪ [2, 5]

  • (d) None of these

     
Q 33 | Page 45

The domain of the function \[f\left( x \right) = \sqrt{\frac{\left( x + 1 \right) \left( x - 3 \right)}{x - 2}}\] is

  
  • (a) [−1, 2) ∪ [3, ∞)

  • (b) (−1, 2) ∪ [3, ∞)

  • (c) [−1, 2] ∪ [3, ∞)

  • (d) None of these

     
Q 34 | Page 45

The domain of definition of the function  \[f\left( x \right) = \sqrt{x - 1} + \sqrt{3 - x}\] is

 
  • (a) [1, ∞)

  • (b) (−∞, 3)

  • (c) (1, 3)

  • (d) [1, 3]

     
Q 35 | Page 45

The domain of definition of the function \[f\left( x \right) = \sqrt{\frac{x - 2}{x + 2}} + \sqrt{\frac{1 - x}{1 + x}}\] is 

 
  • (a) (−∞, −2] ∪ [2, ∞)

  • (b) [−1, 1]

  • (c) ϕ

  • (d) None of these

     
Q 36 | Page 45

The domain of definition of the function f(x) = log |x| is

  • (a) R

  • (b) (−∞, 0)

  • (c) (0, ∞)

  • (d) R − {0}

     
Q 37 | Page 45

The domain of definition of  \[f\left( x \right) = \sqrt{4x - x^2}\] is 

 
  • (a) R − [0, 4]

  • (b) R − (0, 4)

  • (c) (0, 4)

  • (d) [0, 4]

     
Q 38 | Page 45

The domain of definition of  \[f\left( x \right) = \sqrt{x - 3 - 2\sqrt{x - 4}} - \sqrt{x - 3 + 2\sqrt{x - 4}}\] is 

 

  • (a) [4, ∞)

  • (b) (−∞, 4]

  • (c) (4, ∞)

  • (d) (−∞, 4)

     
Q 39 | Page 45

The domain of the function \[f\left( x \right) = \sqrt{5 \left| x \right| - x^2 - 6}\] is

 
  • (a) (−3, − 2) ∪ (2, 3)

  • (b) [−3, − 2) ∪ [2, 3)

  • (c) [−3, − 2] ∪ [2, 3]

  • (d) None of these

     
Q 40 | Page 45

The range of the function \[f\left( x \right) = \frac{x}{\left| x \right|}\] is

  • (a) R − {0}

  • (b) R − {−1, 1}

  • (c) {−1, 1}

  • (d) None of these

     
Q 41 | Page 45

The range of the function \[f\left( x \right) = \frac{x + 2}{\left| x + 2 \right|}\],x ≠ −2 is

 
  • (a) {−1, 1}

  • (b) {−1, 0, 1}

  • (c) {1}

  • (d) (0, ∞)

     
Q 42 | Page 45

The range of the function f(x) = |x − 1| is

  • (a) (−∞, 0)

  • (b) [0, ∞)

  • (c) (0, ∞)

  • (d) R

     
Q 43 | Page 45

Let  \[f\left( x \right) = \sqrt{x^2 + 1}\ ] . Then, which of the following is correct?

 

  • (a)  \[f\left( xy \right) = f\left( x \right)f\left( y \right)\]

     

  • (b)  \[f\left( xy \right) \geq f\left( x \right)f\left( y \right)\]

     

  •   (c)  \[f\left( xy \right) \leq f\left( x \right)f\left( y \right)\]

     

  • (d) none of these                        

     
Q 44 | Page 45

If  \[\left[ x \right]^2 - 5\left[ x \right] + 6 = 0\], where [.] denotes the greatest integer function, then 

 

  • (a) ∈ [3, 4]   

  •    (b) ∈ (2, 3]           

  •   (c) ∈ [2, 3]      

  •   (d) ∈ [2, 4)

Q 45 | Page 45

The range of  \[f\left( x \right) = \frac{1}{1 - 2\cos x}\] is 

 

  • (a) [1/3, 1]    

  •   (b) [−1, 1/3]    

  •   (c) (−∞, −1) ∪ [1/3, ∞)   

  •    (d) [−1/3, 1]   

Chapter 3: Functions

Ex. 3.10Ex. 3.20Ex. 3.30Ex. 3.40Ex. 3.50Others

RD Sharma Mathematics Class 11

Mathematics Class 11 - Shaalaa.com

RD Sharma solutions for Class 11 Mathematics chapter 3 - Functions

RD Sharma solutions for Class 11 Maths chapter 3 (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 Class 11 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 11 Mathematics chapter 3 Functions are Cartesian Product of Sets, Brief Review of Cartesian System of Rectanglar Co-ordinates, Relation, Functions, Some Functions and Their Graphs, Algebra of Real Functions, Ordered Pairs, Equality of Ordered Pairs, Pictorial Diagrams, Graph of Function, Pictorial Representation of a Function, Exponential Function, Logarithmic Functions.

Using RD Sharma Class 11 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 11 prefer RD Sharma Textbook Solutions to score more in exam.

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