If $ f(x)=\frac{x}{x-1} $, then $ \frac{f(a)}{f(a+1)}= $

$ f\left(\frac{-a}{a-1}\right) $

If $f\left( x \right) = \frac{1}{{\sqrt { - x} }}$, then domain of $f\;of$ is

$\left( { - \infty ,0} \right)$

Domain of the function $ f(x)=\sin ^{-1}\left(1+3 x+2 x^{2}\right) $ is

$ \left(-\infty, \frac{-1}{2}\right) \cup(2, \infty) $

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JEE • MATHEMATICS • FUNCTIONS

Functions practice · Easy

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 31-60 of 478 questions

31 Easy

31 of 478

Application +4 | -1

Let

f(x)=\sin \sqrt{[a]} \times

(where [] denotes the greatest integer function). If

\mathrm{f}

is periodic with fundamental period

\pi

, then a belongs to

View answer and explanation

Correct option: D

[4,5)

Option D ([4,5)) is correct because the period of the sine function is related to the square root of the greatest integer of a, and the given period is used to find the range of a.

32 Easy

32 of 478

Understanding +4 | -1

Let

x

be a non-zero rational number and

y

be an irrational number. Then

x y

View answer and explanation

Correct option: B

Irrational

Option B (Irrational) is correct because the product of a non-zero rational number and an irrational number is always irrational.

33 Easy

33 of 478

Understanding +4 | -1

Let

A = \left\{ {x \in R:x \ne 0, - 4 \le x \le 4} \right\}\;and\;f:A \in R

be defined by

f\left( x \right) = \frac{{\left| x \right|}}{x}\;for\,x \in A.

. Then, the range of

f

View answer and explanation

Correct option: A

\left\{ {1, - 1} \right\}

Option A (

\{1, - 1\}

) is correct because the function simplifies to either 1 or -1 depending on the sign of x, given the absolute value in the numerator.

34 Easy

34 of 478

Application +4 | -1

f(x)=\frac{x}{x-1}

, then

\frac{f(a)}{f(a+1)}=

View answer and explanation

Correct option: C

f\left(a^{2}\right)

Option C (

f(a^{2})

) is correct because when you simplify

\frac{{f(a)}}{{f(a+1)}}

, it results in

\frac{{a^2}}{{a^2-1}}

, which is equivalent to

f(a^2)

35 Easy

35 of 478

Analyzing +4 | -1

Which one of the following is a bijective function on the set of real numbers

View answer and explanation

Correct option: A

2 x-5

Option A (

2x-5

) is correct because it is a linear function, which are always bijective (both one-to-one and onto) on the set of real numbers.

36 Easy

36 of 478

Analyzing +4 | -1

Let

g(x)=1+x-[x]

and

f(x)=\operatorname{sgn} x

. Then for all

x, f \circ g(x)

is equal to

View answer and explanation

Correct option: B

Option B (1) is correct because

g(x)

is always greater than 0, making

f(g(x))

equal to 1 since

f(x) = sgn(x)

37 Easy

37 of 478

Understanding +4 | -1

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

, then domain of

f\;of

View answer and explanation

Correct option: D

\{ \}

Option D (

\{\}

) is correct because the composite function f(f(x)) results in an expression with nested square roots of negative values, making the domain an empty set.

38 Easy

38 of 478

Application +4 | -1

f:R \to R\;and\;g:R \to R

are defined by

f\left( x \right) = x - 3\;and\;g\left( x \right) = {x^2} + 1

then the values of x for which

g\left\{ {f\left( x \right)} \right\} = 10

are

View answer and explanation

Correct option: D

0,6

Option D (0, 6) is correct because these values satisfy the equation g{f(x)} = 10.

39 Easy

39 of 478

Analyzing +4 | -1

(\sin \alpha) x^2-2 x+b-2 \geq 0 \forall x \in(-\infty, 1]

and

\alpha \in\left(0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right)

, then the possible real value of

b

View answer and explanation

Correct option: D

Option D (4) is correct because the minimum value of the quadratic must be greater than or equal to zero, leading to b >= 4.

40 Easy

40 of 478

Application +4 | -1

\mathbf{f}(\mathbf{x})=\log \left(\frac{1+\mathrm{x}}{1-\mathrm{x}}\right), \mathbf{- 1}<\mathbf{x}<\mathbf{1}

, then

\mathbf{f}\left(\frac{3 \mathrm{x}+\mathrm{x}^3}{1+3 \mathrm{x}^2}\right)-\mathrm{f}\left(\frac{2 \mathrm{x}}{1+\mathrm{x}^2}\right)

\mathbf{M}

f(x)

then the value of

M

View answer and explanation

Correct option: A

Option A (1) is correct because simplifying the expression leads to M = 1.

41 Easy

41 of 478

Application +4 | -1

The domain of definition of the function

f\left( x \right) = \sqrt {{{\log }_{10}}\left( {\frac{{5x - {x^2}}}{4}} \right)}

View answer and explanation

Correct option: A

\left[ {1,\mathrm{ }4} \right]

Option A ([1, 4]) is correct because it satisfies the conditions for the logarithm and the square root to be defined.

42 Easy

42 of 478

Application +4 | -1

The polynomial function

\mathrm{f}(\mathrm{x})

satisfies the equation

\mathrm{f}(\mathrm{x})-\mathrm{f}(\mathrm{x}-2)=(2 \mathrm{x}-1)^2

for all

\mathrm{x}

. If

\mathrm{p}

and

\mathrm{q}

are the coefficients of

x^2

and

x

respectively in

f(x)

, then

p+q

is equal to

View answer and explanation

Correct option: B

5 / 6

Option B is correct because the sum of the coefficients p and q, which are the coefficients of x² and x respectively in f(x), is 5/6.

43 Easy

43 of 478

Application +4 | -1

The domain for which the functions

f(x)=2 x^2-1

and

g(x)=1-3 x

are equal, is

View answer and explanation

Correct option: C

\left\{-2, \frac{1}{2}\right\}

Option C (

\left\{-2, \frac{1}{2}\right\}

) is correct because these are the x-values where the functions f(x) and g(x) have the same output.

44 Easy

44 of 478

Understanding +4 | -1

R

denotes the set of all real numbers then the function

f: R \rightarrow R

defined

f(x)=[x]

View answer and explanation

Correct option: D

Neither one-one nor onto

Option D (Neither one-one nor onto) is correct because the greatest integer function is neither one-to-one nor onto when mapping from R to R.

45 Easy

45 of 478

Application +4 | -1

|x-1| > 5

, then

View answer and explanation

Correct option: C

x \in(-\infty,-4) \cup(6, \infty)

Option C (x ∈ (-∞,-4) ∪ (6, ∞)) is correct because the absolute value inequality |x-1| > 5 implies that x-1 < -5 or x-1 > 5.

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46 Easy

46 of 478

Analyzing +4 | -1

Let

f\;:N \to N

be defined by

f\left( x \right) = {x^2} + x + 1,

then f is

View answer and explanation

Correct option: C

One-one but not onto

The correct answer is C (One-one but not onto) because the function is injective (one-to-one) but not surjective (onto) when mapping from N to N.

47 Easy

47 of 478

Understanding +4 | -1

A

contains 10 elements then total number of functions defined from

A

A

is:

View answer and explanation

Correct option: C

10^{10}

Option C (

10^{10}

) is correct because the total number of functions from a set A to itself, where A has 10 elements, is

10^{10}

48 Easy

48 of 478

Analyzing +4 | -1

f: R \rightarrow R, f(x)=\left\{\begin{array}{cc}x|x|-4, & x \in Q \\ x|x|-\sqrt{3} & x \in Q\end{array}\right.

, then

f(x)

View answer and explanation

Correct option: D

many to one and into

Option B (many to one and onto) is incorrect. The correct answer is many to one and into because f(2) = f(3^(1/4)) indicating many-to-one, and f(x) ≠ -√3 for all x ∈ R, indicating into.

49 Easy

49 of 478

Application +4 | -1

Let

f:\left( { - 1,\;1} \right) \to B

, be a function defined by

f\left( x \right) = {\tan ^{ - 1}}\frac{{2x}}{{1 - {x^2}}}

, then

f

is both one-one and onto when

B

is the interval

View answer and explanation

Correct option: A

\left[ { - \frac{\pi }{2},\;\frac{\pi }{2}} \right]

Option B is correct because the range of the function is (-π/2, π/2) when f is both one-one and onto.

50 Easy

50 of 478

Application +4 | -1

Let

R

be the set of real numbers and the mapping

f: R \rightarrow R

and

g: R \rightarrow R

be defined by

f(x)=5-x^2

and

g(x)=3 x-4

, then the value of

(f o g)(-1)

View answer and explanation

Correct option: A

-44

Option A (-44) is correct because it results from correctly evaluating the composite function

(f \circ g)(-1)

51 Easy

51 of 478

Application +4 | -1

Let

f:R \to ,\;g:R \to R

be two functions given by

f\left( x \right) = 2\;x - 3,\;g\left( x \right) = {x^3} + 5.

Then,

{\left( {fog} \right)^{ - 1}}\left( x \right)

is equal to

View answer and explanation

Correct option: D

{\left( {\frac{{x - 7}}{2}} \right)^{1/3}}

Option D (

{\left( {\frac{{x - 7}}{2}} \right)^{1/3}}

) is correct because it is the correct inverse function of the composite function

fog(x)

52 Easy

52 of 478

Analyzing +4 | -1

Consider function

f:A \to B

and

g:B \to C(A,\,B,C \subseteq R)

such that

{(gof)^{ - 1}}

exists, then

View answer and explanation

Correct option: C

f

is one-one and

g

is 0 into

Option C (

f

is one-one and

g

is 0 into) is correct because for the inverse of a composite function to exist, the first function must be injective (one-one) and the second function can be into.

53 Easy

53 of 478

Application +4 | -1

Domain of

f(x)=\left(x^{2}-1\right)^{-1 / 2}

View answer and explanation

Correct option: A

(-\infty,-1) \cup(1, \infty)

Option A ((-∞,-1) ∪(1, ∞)) is correct because the domain requires the expression inside the square root to be positive and the denominator to be non-zero.

54 Easy

54 of 478

Analyzing +4 | -1

Let

f:R \to R

be a function defined by

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

Then,

f

View answer and explanation

Correct option: D

Neither one-one nor onto

Option D (Neither one-one nor onto) is correct because the function is neither injective (not one-to-one) nor surjective (not onto).

55 Easy

55 of 478

Application +4 | -1

If the domain of the function

f(x)=\sqrt{3 \cos ^{-1}(4 x)-\pi}

[a, b]

, then the value of

(4 a+64 b)

View answer and explanation

Correct option: A

Option A (9) is correct because using the properties of binomial expansion and the given condition y=12x, we can solve for n.

56 Easy

56 of 478

Application +4 | -1

Domain of the function

f(x)=\sin ^{-1}\left(1+3 x+2 x^{2}\right)

View answer and explanation

Correct option: C

\left[-\frac{3}{2}, 0\right]

Option C ([-3/2, 0]) is correct because the domain requires the argument of the arcsin function to be between -1 and 1.

57 Easy

57 of 478

Understanding +4 | -1

The number of bijective functions from set A to itself when A contains 106 elements is

View answer and explanation

Correct option: C

106\;!

Option C (

106\\;!

) is correct because the number of bijective functions from a set A to itself when A contains n elements is n!.

58 Easy

58 of 478

Understanding +4 | -1

f: R \rightarrow R

, then

f(x)=|x|

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the function is neither one-to-one nor onto, as it is many-to-one and its range is a subset of the codomain.

59 Easy

59 of 478

Application +4 | -1

-3 x+17 < -13

, then

View answer and explanation

Correct option: A

x \in(10, \infty)

Option A (x ∈ (10, ∞)) is correct because solving the inequality -3x + 17 < -13 results in x > 10.

60 Easy

60 of 478

Understanding +4 | -1

Domain and range of

f(x)=\frac{|x-3|}{x-3}

are respectively

View answer and explanation

Correct option: B

R-\{3\},\{1,-1\}

Option B (

R - \\{3\\}, \\{1, -1\\}

) is correct because the domain excludes x=3 where the denominator is zero, and the range consists of 1 and -1 depending on the sign of (x-3).

Revision notes and key takeaways

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