Functions — JEE Maths Practice Questions & Solutions

mediumPYQ · JEE Main 2026

Let

f(x)+3f\!\left(\dfrac{\pi}{2}-x\right)=\sin x

,

x\in\mathbb{R}

. Let

\alpha

be the maximum value of

f

. If the area bounded by

g(x)=x^2

and

h(x)=\beta x^3

,

\beta>0

, equals

\alpha^2

, then

30\beta^3

equals

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mediumPYQ · JEE Main 2026

Let

A=\{1,2,3,4,5,6\}

. The number of one-one functions

f:A\to A

such that

f(1)\ge3

,

f(3)\le4

and

f(2)+f(3)=5

, is

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mediumPYQ · JEE Main 2026

The number of functions

f:\{1,2,3,4\}\to\{a,b,c\}

, which are not onto, is

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mediumPYQ · JEE Main 2026

For the function

f:[1,\infty)\to[1,\infty)

defined by

f(x)=(x-1)^4+1

, consider the two statements: (I) The set

S=\{x\in[1,\infty):f(x)=f^{-1}(x)\}

contains exactly two elements, and (II) The set

S=\{x\in[1,\infty):f(x)=f^{-1}(x+1)\}

is an empty set.

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mediumPYQ · JEE Main 2026

Let for some

\alpha\in\mathbb{R}

,

f:\mathbb{R}\to\mathbb{R}

satisfy

f(x+y)=f(x)+2y^2+y+\alpha xy

for all

x,y\in\mathbb{R}

. If

f(0)=-1

and

f(1)=2

, then the value of

\displaystyle\sum_{n=1}^{5}\big(\alpha+f(n)\big)

is

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mediumPYQ · JEE Main 2026

If the domain of the function

f(x)=\sqrt{\log_{(0.6)}\left(\left|\dfrac{2x-5}{x^2-4}\right|\right)}

is

(-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty)

, then the value of

a+b+c+d+e

is

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mediumPYQ · JEE Main 2026

Let

[\cdot]

denote the greatest integer function. If the domain of the function

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

is

[\alpha,\beta)

, then

\alpha^2+\beta^2

is equal to

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easyPYQ · JEE Main 2026

Let

[\cdot]

denote the greatest integer function. If the domain of

f(x)=\cos^{-1}\left(\dfrac{4x+2[x]}{3}\right)

is

[\alpha,\beta]

, then

12(\alpha+\beta)

is equal to

View solution →

expert

Let

[k]

and

\{k\}

denote the greatest integer and fractional part of

k

. A real-valued function

f

is defined for all real

x \neq -1

by

f(x) = \left(\left[\log_e(1 + \{x\})\right]\int_{0}^{x^2+1}\sin(e^t)\,dt + \frac{2026 - x^5}{x^5 + 1}\right)^{1/5}.

Find the value of

\frac{f^{-1}(200) - f(200) + f(f(2026))}{f(f(2))}.

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expert

f:\left[0,1\right]\to\mathbb R

,

f(x) = 4x\left(1-x\right)

,

f_n(x) = f\left(f_{n-1}(x)\right)

\forall\,n \ge 1

and

f_0(x) = x

. Let

\left\{b_k(n)\right\}

represent the sequence formed by the solutions of the equation

f_n(x) = f_0(x)

and

B(n)

represent the sum of all terms of the sequence

\left\{b_k(n)\right\}

.

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hard

If the domain of the function

f(x)

which is defined as

f(x) = \frac1{\sqrt{\sin\left(\cos x\right)}}+\sin^{-1}\left(\frac{2x}\pi\right)+\frac1{\left\{-x\right\}} +\frac1{\ln\left(1-\left[\tan\dfrac x2\right]-\left[-\tan\dfrac x2\right]\right)}

is

x \in \left(a,b\right)-\left\{c,d,e\right\}

, then the value of

\left(b+c+d+e\right)-a

is equal to (Where

[\,.\,]

represents the greatest integer function and

\left\{x\right\} = x-[x]

)

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hard

If

f:(0,\pi/n)\to\mathbb{R}

defined by

f(x)=\displaystyle\sum_{k=1}^{n}[1+\sin kx]

, where

[x]

denotes the integral part of

x

, then range of

f(x)

is:

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hard

If

f(x)+f(y) = f\!\left(\dfrac{x+y}{1-xy}\right)

for all

x, y \in \mathbb{R}

with

xy \ne 1

, and

\displaystyle\lim_{x\to 0}\dfrac{f(x)}{x} = 2

, then the value of

\dfrac{15\,f(\sqrt{3})}{\pi\,f'(-2)}

is.

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hard

If the domain of

f(x) = \dfrac{\sin^{-1}(\sin x)}{\sqrt{-\log_{\frac{x+4}{2}}\log_2\!\left(\frac{2x-1}{3+x}\right)}}

is

(a,b)\cup(c,\infty)

, then the value of

a+b+3c

.

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hard

Let

f(x) = [\sec\{x\}]

where

[x]

and

\{x\}

denote greatest integer and fractional parts of

x

respectively, and

g(x) = 2x^2 - 3x(k+1) + k(3k+1)

. Find the number of integral values of

k

such that

g(f(x)) < 0

for all

x \in \mathbb{R}

.

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hard

If

f(x) = \dfrac{2010x+163}{165x-2010}

,

x > 0

,

x \neq \dfrac{2010}{165}

, then the least value of

f(f(x)) + f\!\left(f\!\left(\dfrac{4}{x}\right)\right)

is.

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hard

Let

f(n)

denote the square of the sum of the digits of natural number

n

, where

f^2(n) = f(f(n))

,

f^3(n) = f(f(f(n)))

, and so on. Then the value of

\dfrac{f^{2011}(2011)-f^{2010}(2011)}{f^{2013}(2011)-f^{2012}(2011)}

.

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hard

If the range of

f(x) = \sqrt{24\{x\}-5-16\{x\}^2}

, where

\{\cdot\}

denotes fractional part, is

[a,b]

, then the value of

b^2-a

.

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hard

Let

\left[x\right]

= greatest integer less than or equal to

x

. If all the values of

x

such that the product

\left[x-\dfrac12\right]\left[x+\dfrac12\right]

is prime, belongs to the set

\left[x_1,x_2\right)\cup\left[x_3,x_4\right)

, find the value of

x_1^2+x_2^2+x_3^2+x_4^2

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hard

If

x+\left[y\right]+\left\{z\right\} = 1.1 ,\qquad \left[x\right]+\left\{y\right\}+z = 2.2 ,\qquad \left\{x\right\}+y+\left[z\right] = 3.3 ,

then (

\left[\ \cdot\ \right]

is G.I.F and

\left\{\ \cdot\ \right\}

is fractional part)

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hard

f\left(x\right)

is a continuous increasing function with

f\left(x\right) \ge x

and

h\left(x\right) = \left(1-x\right)^{-1}

. Then the number of integral values in real numbers not belonging to the domain of

\phi\left(x\right) = \left(f^{2022}\left(x\right)-h^{2025}\left(x\right)\right)^{1/2024}

is/are. (where

f^{\,n}\left(x\right) = f\circ f\circ\cdots\circ f

,

n

times)

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hard

Let

f

be a continuous function on

\mathbb R

satisfying the relation

f\left(f\left(f\left(x\right)\right)\right) = x\ \forall\,x \in \mathbb R

. Then

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hard

Let

\alpha

,

\beta

,

\gamma

be the roots of the equation

f(x) = 3x^3-13x^2+14x-2 = 0

. Let

\left[\alpha\right]

,

\left[\beta\right]

,

\left[\gamma\right]

be the roots of the cubic polynomial equation

g(x) = 0

; then for

h(x) = \dfrac{f(x)+2}{g(x)}

(where

[\,.\,]

represents the greatest integer function)

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hard

Let

\alpha

,

\beta

,

\gamma

be the roots of the equation

f(x) = 3x^3-13x^2+14x-2 = 0

. Let

\left[\alpha\right]

,

\left[\beta\right]

,

\left[\gamma\right]

be the roots of the cubic polynomial equation

g(x) = 0

; then for

h(x) = \dfrac{f(x)+2}{g(x)}

(where

[\,.\,]

represents the greatest integer function)

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medium

A = \{1,2,3\}

,

B = \{1,3,5,7,9\}

. Then

\dfrac{\text{No. of one-one functions}}{\text{No. of strictly monotonic functions}}

.

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medium

The domain of the function

\log_{e}\!\left(\text{sgn}(9-x^{2})\right)+\sqrt{[x]^{3}-4[x]}

(where sgn is signum function and

[\cdot]

is step function) is:

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medium

Domain of the definition

f(x)=\dfrac{\log_{4}(5-[x-1]-[x]^{2})}{x^{2}+x-2}

(where

[\cdot]

denotes greatest integer function) is:

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medium

The range of the function

f(x)=4\cos^{3}x-8\cos^{2}x+1

is:

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medium

If

[x]

denotes the integral part of

x

, then the domain of

f(x)=\cos^{-1}(x+[x])

is:

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medium

Range of

f(x)={}^{16-x}C_{2x-1}+{}^{20-3x}C_{4x-5}

is:

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medium

The domain of the function

y=\sqrt\dfrac{2x-1}{2x^{3}+3x^{2}+x}+\sqrt{\sin^{-1}(\log_{2}x)}

is:

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medium

The domain of the function

y=\sqrt{\sin x+\cos x}+\sqrt{7x-x^{2}-6}

is:

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medium

Let

f(x) = \dfrac{x+1}{x-1}

for all

x \neq 1

. Let

f^{-1}(x) = f(x)

,

f^2(x) = f(f(x))

, and

f^n(x) = f(f^{n-1}(x))

for

n > 1

. Let

P = f^1(2)\cdot f^2(3)\cdot f^3(4)\cdot f^4(5)

. The number of divisors of

P

.

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medium

If

f(x) + f\!\left(1-\dfrac{1}{x}\right) = 1+x

for all

x \in \mathbb{R} - \{0,1\}

,The value of

4f(2)

.

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medium

If

f(x) = \cos\!\left(2010\{x^3\}(2011^{[x^2]}+2012x)\right)

,

x \in \mathbb{R}

, where

\{\cdot\}

denotes fractional part and

[\cdot]

denotes greatest integer function, then the value of

f_{\max}

.

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medium

Let

f

be defined on the natural numbers as follows:

f(1) = 1

and for

n > 1

,

f(n) = f(f(n-1)) + f(n - f(n-1))

. then the value of

\dfrac{1}{30}\displaystyle\sum_{r=1}^{20} f(r)

.

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medium

Let

f: \mathbb{R} \to \mathbb{R}

be defined as

f(x) = x^3 + x + 1

,

1 \leq x \leq 2

. The graph of

y = g(x)

is the reflection of the graph of

y = f(x)

through the line

y = x

. If the domain of

g(x)

is

[a, b]

, then

|a-b|

is.

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medium

Number of solutions of the equation

f(x-1)+f(x+1) = \sin\alpha

,

0 < \alpha < \dfrac{\pi}{2}

, where

f(x) = \begin{cases} 1-|x|; & |x| \leq 1 \\ 0; & |x| > 1 \end{cases}

is.

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medium

If

0\le x\le\dfrac{\pi}{3}

, then range of

f(x)=\sec\left(\dfrac{\pi}{6}-x\right)+\sec\left(\dfrac{\pi}{6}+x\right)

is:

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medium

The number of linear functions satisfying

f[x+f(x)] = x+f(x)

for all

x \in \mathbb{R}

is.

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medium

f(x) = \dfrac{ax+b}{cx+d}

for all

x \in \mathbb{R}-\left\{-\dfrac{d}{c}\right\}

. If

f(5) = 5

,

f(13) = 13

, and

f(f(x)) = x

for all

x

, then the range of

f(x) = \mathbb{R}-\{x\}

. Then

x

.

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medium

If

n(A) = 4

,

n(B) = 5

and the number of functions from

A

to

B

such that range contains exactly 3 elements is

k

, then

k/60

is.

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medium

If

a

and

b

are constants,

f(x) = a\sin x + bx\cos x + 2x^2

. If

f(2) = 15

, then

f(-2)

is.

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medium

If

f(x) = 3[2x]+1000\displaystyle\sum_{r=1}^{2008}\dfrac{x+r-[x+r]}{2008}

, then

f(3)

(where

[\cdot]

is GIF) is.

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medium

The number of solutions of

[\sin x]+\left[\dfrac{x}{2\pi}\right]+\left[\dfrac{2x}{5\pi}\right] = \dfrac{9x}{10\pi}

in the interval

(30,40)

(where

[\cdot]

is GIF).

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medium

If

f(x) = \dfrac{\alpha x+1}{x-\beta}

\forall x \in \mathbb{R}-\{\beta\}

, with

\alpha\beta \ne 1

and

\alpha \ne 1

, is a self-inverse function such that

\frac{f(4)}{4} = \frac{f(12)}{12} = f\left(\frac{1+\beta}{1-\alpha}\right),

then "

\beta

" is equal to

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medium

Consider

f:\mathbb{R}^{+}\to\mathbb{R}

such that

f\left(1\right) = 1

and

f\left(x\right)f\left(y\right)+f\left(\frac3x\right)f\left(\frac3y\right) = 2f\left(xy\right) \qquad\forall\,x,y \in \mathbb{R}^{+}.

Find

f\left(99\right)

.

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medium

The graphs of

y = \sqrt{x+3}

and

y = \sqrt{1-x}+f(k)

intersect, where

k

is a real parameter.

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medium

Let

f:\mathbb Z \to \mathbb R

be defined as

f\left(x+2\right) = f(x)+2\left(x+1\right)

if

x

is even and

f\left(x+2\right) = f(x)+1

if

x

is odd, and

f(1) = 1

,

f(2) = 5

, then

View solution →

medium

If

f(x)

is a real valued function such that

f\left(x+1\right) = 1+\left[2-3f(x)+3f^2(x)-f^3(x)\right]^{1/3} ,

where

f(x) \ne 0

and

f\left(x+2022\right) = \dfrac{2023}{\lambda}f(x)

, then

\lambda

is equal to

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medium

Suppose

f(x) = \begin{cases}\left|\log_3 x-1\right|, & 0<x \le 9 \\ 4-\sqrt x, & x>9\end{cases}

defined on the positive set of real numbers; provided that

a

,

b

,

c \in \mathbb{R}

which are not equal to each other satisfying

f\left(a\right) = f\left(b\right) = f\left(c\right)

, then

abc

can be

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easy

The range of the function

y=[x^{2}]-[x]^{2}

,

x\in[0,2]

(where

[\cdot]

denotes the integral part) is:

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easy

If

[x]

denotes the integral part of

x

, then domain of the function

f(x)=\dfrac{\sqrt{3-x}}{(x-1)(x-2)(x-3)}+\sin^{-1}\!\left[\dfrac{3x-2}{2}\right]

is:

View solution →

easy

The number of integers in the domain of

f(x) = \dfrac{1}{\sqrt{\ln\cos^{-1}x}}

.

View solution →

Functions — JEE Maths Practice Questions & Solutions

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