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

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

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).

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

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

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

(a) the image set of the domain of f

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

(b) {x : f(x) = −2}

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

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

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

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

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

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

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

f, g, h are three function defined from R to R as follow:

(i) f(x) = x²

f, g, h are three function defined from R to R as follow:

(ii) g(x) = sin x

Find the range of function.

f, g, h are three function defined from R to R as follow:

(iii) h(x) = x² + 1

Find the range of function.

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)} 

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)}

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.

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.

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

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.

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 \[g\left( x \right) = \begin{cases}x^2 , & 0 \leq x \leq 2 \\ 3x, & 2 \leq x \leq 10\end{cases}\]

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

Express the function f : X → R given by f(x) = x³ + 1 as set of ordered pairs, where X = {−1, 0, 3, 9, 7}

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find the domain and range of the real valued function:

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

Find the domain and range of the real valued function:

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

Find the domain and range of the real valued function:

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

Find the domain and range of the real valued function:

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

Find the domain and range of the real valued function:

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

Find the domain and range of the real valued function:

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

Find the domain and range of the real valued function:

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

Find the domain and range of the following real function:

f(x) = `sqrt(9 - x^2)`

Find the domain and range of the real valued function:

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

Find the domain and range of the real valued function:

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

Find f + g, f − g, cf (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

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

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

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).

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

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 − f 

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

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

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

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)  \[2f - \sqrt{5} g\]

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

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

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

(i) f + g

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

(ii) fg

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

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

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

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

If f, g and h are real functions defined by 

The function f is defined by

Let f, g: R → R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and `f/g`

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

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

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

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

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

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

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.

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

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

If f, g, h are real functions given by f(x) = x², g(x) = tan x and h(x) = log_e x, then write the value of (hogof)\[\left( \sqrt{\frac{\pi}{4}} \right)\] .

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

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

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

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.

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

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

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.

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

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

Which one of the following is not a function?

If f(x) = cos (log x), then the value of f(x²) f(y²) −

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

Let f(x) = |x − 1|. Then,

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

Which of the following are functions?

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

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

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

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 

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

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

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

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

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

If f(x) = cos (log_e 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

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

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

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

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

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

If f : [−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} =

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 =

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,

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,

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

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

If f(x) = sin [π²] x + sin [−π]² x, where [x] denotes the greatest integer less than or equal to x, then

The domain of the function

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

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

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

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

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

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

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

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

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

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

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

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

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

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