# RD Sharma solutions for Class 11 Mathematics Textbook chapter 3 - Functions [Latest edition]

## Chapter 3: Functions

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Others
Exercise 3.1[Pages 7 - 8]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 3 FunctionsExercise 3.1[Pages 7 - 8]

Exercise 3.1 | Q 1 | Page 7

Define a function as a set of ordered pairs.

Exercise 3.1 | Q 2 | Page 7

Define a function as a correspondence between two sets.

Exercise 3.1 | Q 3 | Page 7

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

Exercise 3.1 | 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).

Exercise 3.1 | 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.

Exercise 3.1 | 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).

Exercise 3.1 | 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].

Exercise 3.1 | 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

Exercise 3.1 | 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}

Exercise 3.1 | 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

Exercise 3.1 | 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]}

Exercise 3.1 | 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}

Exercise 3.1 | 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]}

Exercise 3.1 | 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?

Exercise 3.1 | 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.

Exercise 3.1 | 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.

Exercise 3.1 | 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.

Exercise 3.1 | 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)}

Exercise 3.1 | 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)}

Exercise 3.1 | 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)}

Exercise 3.1 | 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.

Exercise 3.1 | Q 13 | Page 8

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

Exercise 3.1 | 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)].

Exercise 3.1 | 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.

Exercise 3.1 | 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.

Exercise 3.1 | Q 17 | Page 8

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

Exercise 3.1 | 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}

Exercise 3.2[Pages 11 - 12]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 3 FunctionsExercise 3.2[Pages 11 - 12]

Exercise 3.2 | Q 1 | Page 11

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

Exercise 3.2 | Q 2 | Page 11

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

Exercise 3.2 | Q 3 | Page 11

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

Exercise 3.2 | Q 4 | Page 11

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

Exercise 3.2 | Q 5 | Page 11

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

Exercise 3.2 | 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) $f\left( \sqrt{- 3} \right)$

Exercise 3.2 | Q 7 | Page 11

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

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

Exercise 3.2 | Q 8 | Page 11

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

Exercise 3.2 | 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) $f\left( - \frac{1}{x} \right) = - \frac{1}{f\left( x \right)}$

Exercise 3.2 | 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.

Exercise 3.2 | 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).

Exercise 3.3[Page 18]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 3 FunctionsExercise 3.3[Page 18]

Exercise 3.3 | Q 1.1 | Page 18

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

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

Exercise 3.3 | Q 1.2 | Page 18

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

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

Exercise 3.3 | Q 1.3 | Page 18

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

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

Exercise 3.3 | Q 1.4 | Page 18

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

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

Exercise 3.3 | Q 1.5 | Page 18

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

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

Exercise 3.3 | Q 2.1 | Page 18

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

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

Exercise 3.3 | Q 2.2 | Page 18

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

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

Exercise 3.3 | Q 2.3 | Page 18

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

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

Exercise 3.3 | Q 2.4 | Page 18

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

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

Exercise 3.3 | Q 3.01 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.02 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.03 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.04 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.05 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.06 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.07 | Page 18

Find the domain and range of the real valued function:

(vii)  $f\left( x \right) = - \left| x \right|$

Exercise 3.3 | Q 3.08 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.09 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.3 | Q 3.1 | Page 18

Find the domain and range of the real valued function:

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

Exercise 3.4[Page 38]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 3 FunctionsExercise 3.4[Page 38]

Exercise 3.4 | 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

Exercise 3.4 | 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  $g\left( x \right) = \sqrt{x + 1}$

Exercise 3.4 | 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.

Exercise 3.4 | 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).

Exercise 3.4 | 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

Exercise 3.4 | 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 −

Exercise 3.4 | 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

Exercise 3.4 | 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) $\frac{f}{g}$

Exercise 3.4 | 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) $\frac{g}{f}$

Exercise 3.4 | 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)  $2f - \sqrt{5} g$

Exercise 3.4 | 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

Exercise 3.4 | 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) $\frac{5}{8}$

Exercise 3.4 | Q 5.1 | Page 38

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

(i) f + g

Exercise 3.4 | Q 5.2 | Page 38

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

(ii) fg

Exercise 3.4 | Q 5.3 | Page 38

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

(iii) $\frac{f}{g}$

Exercise 3.4 | 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),

$\left( \frac{f}{g} \right) \left( \frac{1}{2} \right), \left( \frac{g}{f} \right) \left( \frac{1}{2} \right)$

Exercise 3.4 | 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).

Exercise 3.4 | 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).

Exercise 3.4 | 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  $\frac{f}{g}$ .

Exercise 3.4 | 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 $\frac{f}{g}$ .

Exercise 3.4 | 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  $\left( \frac{f}{g} \right) \left( x \right)$ .

Exercise 3.5[Pages 41 - 42]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 3 FunctionsExercise 3.5[Pages 41 - 42]

Exercise 3.5 | Q 1 | Page 41

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

Exercise 3.5 | 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).

Exercise 3.5 | Q 3 | Page 41

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

Exercise 3.5 | 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).

Exercise 3.5 | Q 5 | Page 41

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

Exercise 3.5 | Q 6 | Page 41

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

Exercise 3.5 | 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.

Exercise 3.5 | Q 8 | Page 42

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

Exercise 3.5 | Q 9 | Page 42

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

Exercise 3.5 | Q 10 | Page 41

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

Exercise 3.5 | 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)$\left( \sqrt{\frac{\pi}{4}} \right)$ .

Exercise 3.5 | Q 12 | Page 42

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

$f\left( x \right) = \frac{1}{\sqrt{x - \left| x \right|}}$ .

Exercise 3.5 | Q 13 | Page 42

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

Exercise 3.5 | Q 14 | Page 42

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

Exercise 3.5 | 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.

Exercise 3.5 | 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

Exercise 3.5 | 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.

Exercise 3.5 | 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.

[Pages 42 - 45]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 3 Functions[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

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Others

## RD Sharma solutions for Class 11 Mathematics Textbook chapter 3 - Functions

RD Sharma solutions for Class 11 Mathematics Textbook 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 Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics Textbook chapter 3 Functions are Cartesian Product of Sets, Brief Review of Cartesian System of Rectanglar Co-ordinates, Relation, Concept of 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.

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