## Functions

### Mathematics

#### (Past Years Questions)

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The range of the function, $$f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \le... JEE Main 2021 (Online) 1st September Evening Shift Let f : N$$\to$$N be a function such that f(m + n) = f(m) + f(n) for every m, n$$\in$$N. If f(6) = 18, then f(2) . f(3... JEE Main 2021 (Online) 31st August Evening Shift The domain of the function$$f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - ...
JEE Main 2021 (Online) 31st August Evening Shift
Which of the following is not correct for relation R on the set of real numbers ?
JEE Main 2021 (Online) 31st August Morning Shift
The domain of the function $${{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)$$ is :
JEE Main 2021 (Online) 26th August Evening Shift
Let [t] denote the greatest integer less than or equal to t. Let f(x) = x $$-$$ [x], g(x) = 1 $$-$$ x + [x], and h(x) = ...
JEE Main 2021 (Online) 26th August Evening Shift
Out of all patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infe...
JEE Main 2021 (Online) 26th August Morning Shift
Consider function f : A $$\to$$ B and g : B $$\to$$ C (A, B, C $$\subseteq$$ R) such that (gof)$$-$$1 exists, then :...
JEE Main 2021 (Online) 25th July Evening Shift
If [x] be the greatest integer less than or equal to x, then $$\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over... JEE Main 2021 (Online) 25th July Evening Shift Let N be the set of natural numbers and a relation R on N be defined by$$R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - ...
JEE Main 2021 (Online) 27th July Evening Shift
Let f : R $$\to$$ R be defined as $$f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1$$. Then, the va...
JEE Main 2021 (Online) 27th July Evening Shift
Let g : N $$\to$$ N be defined asg(3n + 1) = 3n + 2,g(3n + 2) = 3n + 3,g(3n + 3) = 3n + 1, for all n $$\ge$$ 0. Then whi...
JEE Main 2021 (Online) 25th July Morning Shift
If the domain of the function $$f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{... JEE Main 2021 (Online) 22th July Evening Shift The number of solutions of sin7x + cos7x = 1, x$$\in$$[0, 4$$\pi$$] is equal to JEE Main 2021 (Online) 22th July Evening Shift Let [x] denote the greatest integer less than or equal to x. Then, the values of x$$\in$$R satisfying the equation$${[{...
JEE Main 2021 (Online) 22th July Evening Shift
Let $$f:R - \left\{ {{\alpha \over 6}} \right\} \to R$$ be defined by $$f(x) = {{5x + 3} \over {6x - \alpha }}$$. Then ...
JEE Main 2021 (Online) 20th July Evening Shift
Let [ x ] denote the greatest integer $$\le$$ x, where x $$\in$$ R. If the domain of the real valued function $$f(x) = \... JEE Main 2021 (Online) 20th July Morning Shift Let f : R$$-$${3}$$ \to $$R$$-$${1} be defined by f(x) =$${{x - 2} \over {x - 3}}$$.Let g : R$$ \to $$R be give... JEE Main 2021 (Online) 18th March Evening Shift If the functions are defined as$$f(x) = \sqrt x $$and$$g(x) = \sqrt {1 - x} $$, then what is the common domain of the... JEE Main 2021 (Online) 18th March Morning Shift The real valued function$$f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}$$, where [x] denotes the greatest integ... JEE Main 2021 (Online) 18th March Morning Shift Consider the function f : R$$ \to $$R defined by$$f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \r...
JEE Main 2021 (Online) 17th March Evening Shift
In a school, there are three types of games to be played. Some of the students play two types of games, but none play al...
JEE Main 2021 (Online) 17th March Morning Shift
The inverse of $$y = {5^{\log x}}$$ is :
JEE Main 2021 (Online) 17th March Morning Shift
Let A = {2, 3, 4, 5, ....., 30} and '$$\simeq$$' be an equivalence relation on A $$\times$$ A, defined by (a, b) $$\s... JEE Main 2021 (Online) 16th March Evening Shift Let f be a real valued function, defined on R$$-$${$$-$$1, 1} and given by f(x) = 3 loge$$\left| {{{x - 1} \over {x +...
JEE Main 2021 (Online) 16th March Evening Shift
The range of a$$\in$$R for which the function f(x) = (4a $$-$$ 3)(x + loge 5) + 2(a $$-$$ 7) cot$$\left( {{x \over 2}} \... JEE Main 2021 (Online) 16th March Morning Shift Let [ x ] denote greatest integer less than or equal to x. If for n$$\in$$N,$${(1 - x + {x^3})^n} = \sum\limits_{j = 0}...
JEE Main 2021 (Online) 16th March Morning Shift
The number of elements in the set {x $$\in$$ R : (|x| $$-$$ 3) |x + 4| = 6} is equal to :
JEE Main 2021 (Online) 16th March Morning Shift
Let $$A = \{ 1,2,3,....,10\}$$ and $$f:A \to A$$ be defined as$$f(k) = \left\{ {\matrix{ {k + 1} &amp; {if\,k\,is\,o... JEE Main 2021 (Online) 26th February Evening Shift Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1,$$-...
JEE Main 2021 (Online) 26th February Morning Shift
Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote ...
JEE Main 2021 (Online) 25th February Evening Shift
A function f(x) is given by $$f(x) = {{{5^x}} \over {{5^x} + 5}}$$, then the sum of the series $$f\left( {{1 \over {20}}... JEE Main 2021 (Online) 25th February Evening Shift Let f, g : N$$ \to $$N such that f(n + 1) = f(n) + f(1)$$\forall $$n$$\in$$N and g be any arbitrary function. Which ... JEE Main 2021 (Online) 25th February Morning Slot Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) =$${{x - {1 \over 2}} \over {x - 1}}$... JEE Main 2021 (Online) 24th February Morning Slot For a suitably chosen real constant a, let a function, $$f:R - \left\{ { - a} \right\} \to R$$ be defined by $$f(x) = {... JEE Main 2020 (Online) 6th September Evening Slot If f(x + y) = f(x)f(y) and$$\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$$, x, y$$ \in $$N, where N is the s... JEE Main 2020 (Online) 6th September Morning Slot If the minimum and the maximum values of the function$$f:\left[ {{\pi \over 4},{\pi \over 2}} \right] \to R$$, define... JEE Main 2020 (Online) 5th September Morning Slot A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percenta... JEE Main 2020 (Online) 5th September Morning Slot Let$$\mathop \cup \limits_{i = 1}^{50} {X_i} = \mathop \cup \limits_{i = 1}^n {Y_i} = T$$where each Xi contains 10 ... JEE Main 2020 (Online) 4th September Evening Slot A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read ... JEE Main 2020 (Online) 4th September Morning Slot Let R1 and R2 be two relation defined as follows : R1 = {(a, b)$$ \in $$R2 : a2 + b2$$ \in $$Q} and R2 = {(a, ... JEE Main 2020 (Online) 3rd September Evening Slot Consider the two sets : A = {m$$ \in $$R : both the roots of x2 – (m + 1)x + m + 4 = 0 are real} and B = [–3, 5). Wh... JEE Main 2020 (Online) 3rd September Morning Slot Let f : R$$ \to $$R be a function which satisfies f(x + y) = f(x) + f(y)$$\forall $$x, y$$ \in $$R. If f(1) = 2 an... JEE Main 2020 (Online) 2nd September Evening Slot The domain of the function f(x) =$${\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)$$is (–$$... JEE Main 2020 (Online) 2nd September Morning Slot If R = {(x, y) : x, y $$\in$$ Z, x2 + 3y2 $$\le$$ 8} is a relation on the set of integers Z, then the domain of R–1 ... JEE Main 2020 (Online) 2nd September Morning Slot Let a – 2b + c = 1. If $$f(x)=\left| {\matrix{ {x + a} &amp; {x + 2} &amp; {x + 1} \cr {x + b} &amp; {x + 3} &am... JEE Main 2020 (Online) 9th January Evening Slot If A = {x$$ \in $$R : |x| &lt; 2} and B = {x$$ \in $$R : |x – 2|$$ \ge $$3}; then : JEE Main 2020 (Online) 9th January Evening Slot Let ƒ : (1, 3)$$ \to $$R be a function defined by$$f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}$$, where [x] denot... JEE Main 2020 (Online) 8th January Evening Slot Let ƒ(x) = xcos–1(–sin|x|),$$x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$$, then which of the following is ... JEE Main 2020 (Online) 8th January Morning Slot The inverse function of f(x) =$${{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}$$, x$$ \in $$(-1, 1), is :... JEE Main 2020 (Online) 8th January Morning Slot If g(x) = x2 + x - 1 and (goƒ) (x) = 4x2 - 10x + 5, then ƒ$$\left( {{5 \over 4}} \right)$$is equal to:... JEE Main 2020 (Online) 7th January Morning Slot Let A, B and C be sets such that$$\phi  \ne $$A$$ \cap $$B$$ \subseteq $$C. Then which of the following stat... JEE Main 2019 (Online) 12th April Evening Slot For x$$ \in $$(0, 3/2), let f(x) =$$\sqrt x $$, g(x) = tan x and h(x) =$${{1 - {x^2}} \over {1 + {x^2}}}$$. If$$\p... JEE Main 2019 (Online) 12th April Morning Slot Let f(x) = loge(sin x), (0 &lt; x &lt; $$\pi$$) and g(x) = sin–1 (e–x ), (x $$\ge$$ 0). If $$\alpha$$ is a positive ... JEE Main 2019 (Online) 10th April Evening Slot Let f(x) = ex – x and g(x) = x2 – x, $$\forall$$ x $$\in$$ R. Then the set of all x $$\in$$ R, where the function h... JEE Main 2019 (Online) 10th April Morning Slot Let f(x) = x2 , x $$\in$$ R. For any A $$\subseteq$$ R, define g (A) = { x $$\in$$ R : f(x) $$\in$$ A}. If S = [... JEE Main 2019 (Online) 10th April Morning Slot The domain of the definition of the function $$f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)$$ is JEE Main 2019 (Online) 9th April Evening Slot If the function ƒ : R – {1, –1} $$\to$$ A defined by ƒ(x) = $${{{x^2}} \over {1 - {x^2}}}$$ , is surjective, then A i... JEE Main 2019 (Online) 9th April Morning Slot Let $$\sum\limits_{k = 1}^{10} {f(a + k) = 16\left( {{2^{10}} - 1} \right)}$$ where the function ƒ satisfies ƒ(x + y) ... JEE Main 2019 (Online) 9th April Morning Slot Let $$f(x) = \int\limits_0^x {g(t)dt}$$ where g is a non-zero even function. If ƒ(x + 5) = g(x), then $$\int\limits_0... JEE Main 2019 (Online) 8th April Evening Slot Let ƒ(x) = ax (a &gt; 0) be written as ƒ(x) = ƒ1 (x) + ƒ2 (x), where ƒ1 (x) is an even function of ƒ2 (x) is an odd fun... JEE Main 2019 (Online) 8th April Evening Slot If$$f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$$,$$\left| x \right| &lt; 1$$then$$f\left( {{{2x} \over... JEE Main 2019 (Online) 8th April Morning Slot Let Z be the set of integers. If A = {x $$\in$$ Z : 2(x + 2) (x2 $$-$$ 5x + 6) = 1} and B = {x $$\in$$ Z : $$-$$... JEE Main 2019 (Online) 12th January Evening Slot If the function f given by f(x) = x3 – 3(a – 2)x2 + 3ax + 7, for some a$$\in$$R is increasing in (0, 1] and decreasin... JEE Main 2019 (Online) 12th January Evening Slot Let S = {1, 2, 3, … , 100}. The number of non-empty subsets A of S such that the product of elements in A is even is : JEE Main 2019 (Online) 12th January Morning Slot Let a function f : (0, $$\infty$$) $$\to$$ (0, $$\infty$$) be defined by f(x) = $$\left| {1 - {1 \over x}} \right|$$... JEE Main 2019 (Online) 11th January Evening Slot Let fk(x) = $${1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$ for k = 1, 2, 3, ... Then for all x $$\in$$ R,... JEE Main 2019 (Online) 11th January Morning Slot Let f : R $$\to$$ R be defined by f(x) = $${x \over {1 + {x^2}}},x \in R$$.&nbsp;&nbsp; Then the range of f is : JEE Main 2019 (Online) 11th January Morning Slot Let N be the set of natural numbers and two functions f and g be defined as f, g : N $$\to$$ N such that f(n) = $$\lef... JEE Main 2019 (Online) 10th January Evening Slot Let A = {x$$ \in $$R : x is not a positive integer}. Define a function$$f$$: A$$ \to $$&nbsp;&nbsp;R &nbsp;&nbsp;as&n... JEE Main 2019 (Online) 9th January Evening Slot For$$x \in R - \left\{ {0,1} \right\}$$, Let f1(x) =$$1\over x$$, f2 (x) = 1 – x and f3 (x) =$$1 \over {1 - x}$$be... JEE Main 2019 (Online) 9th January Morning Slot Let N denote the set of all natural numbers. Define two binary relations on N as R = {(x, y)$$ \in $$N$$ \times $$N ... JEE Main 2018 (Online) 16th April Morning Slot Let f : A$$ \to $$B be a function defined as f(x) =$${{x - 1} \over {x - 2}},$$Where A = R$$-$${2} and B = R$$-$...
JEE Main 2018 (Online) 15th April Evening Slot
Consider the following two binary relations on the set A = {a, b, c} : R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a...
JEE Main 2018 (Online) 15th April Morning Slot
Two sets A and B are as under : A = {($$a$$, b) $$\in$$ R $$\times$$ R : |$$a$$ - 5| &lt; 1 and |b - 5| &lt; 1}; B =...
JEE Main 2018 (Offline)
The function f : N $$\to$$ N defined by f (x) = x $$-$$ 5 $$\left[ {{x \over 5}} \right],$$ Where N is the set of natu...
JEE Main 2017 (Online) 9th April Morning Slot
Let f(x) = 210.x + 1 and g(x)=310.x $$-$$ 1. If (fog) (x) = x, then x is equal to :
JEE Main 2017 (Online) 8th April Morning Slot
The function $$f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]$$ defined as $$f\left( x \right) = {x \over {1 + {x^2... JEE Main 2017 (Offline) Let P = {$$\theta $$: sin$$\theta -$$cos$$\theta $$=$$\sqrt 2 \,\cos \theta $$} and Q = {$$\theta $$: sin$$\t...
JEE Main 2016 (Online) 10th April Morning Slot
For x $$\in$$ R, x $$\ne$$ 0, Let f0(x) = $${1 \over {1 - x}}$$ and fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . . Th...
JEE Main 2016 (Online) 9th April Morning Slot
Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y $$\subseteq$$ X, ...
AIEEE 2012
The domain of the function f(x) = $${1 \over {\sqrt {\left| x \right| - x} }}$$ is
AIEEE 2011
For real x, let f(x) = x3 + 5x + 1, then
AIEEE 2009
Let $$f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1$$ Statement - 1 : The set $$\left\{ {x:f\left( x \ri... AIEEE 2009 Let$$f:N \to Y$$be a function defined as f(x) = 4x + 3 where Y = { y$$ \in $$N, y = 4x + 3 for some x$$ \in $$N }... AIEEE 2008 The largest interval lying in$$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$for which the function$$f\left( x \...
AIEEE 2007
A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is i...
AIEEE 2005
A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is given co...
AIEEE 2005
Let $$f:( - 1,1) \to B$$, be a function defined by $$f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}$$, then...
AIEEE 2005
The domain of the function $$f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}$$...
AIEEE 2004
The graph of the function y = f(x) is symmetrical about the line x = 2, then
AIEEE 2004
If $$f:R \to S$$, defined by $$f\left( x \right) = \sin x - \sqrt 3 \cos x + 1$$, is onto, then the interval of $$S$$ is...
AIEEE 2004
The range of the function f(x) = $${}^{7 - x}{C_{x - 3}}$$ is
AIEEE 2004
Domain of definition of the function f(x) = $${3 \over {4 - {x^2}}}$$ + $${\log _{10}}\left( {{x^3} - x} \right)$$, is
AIEEE 2003
If $$f:R \to R$$ satisfies $$f$$(x + y) = $$f$$(x) + $$f$$(y), for all x, y $$\in$$ R and $$f$$(1) = 7, then $$\sum\li... AIEEE 2003 A function$$f$$from the set of natural numbers to integers defined by$$$f\left( n \right) = \left\{ {\matrix{ {{{n... AIEEE 2003 The function $$f\left( x \right)$$ $$= \log \left( {x + \sqrt {{x^2} + 1} } \right)$$, is AIEEE 2003 The domain of $${\sin ^{ - 1}}\left[ {{{\log }_3}\left( {{x \over 3}} \right)} \right]$$ is AIEEE 2002 ## Numerical More The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____________. JEE Main 2021 (Online) 31st August Evening Shift If A = {x $$\in$$ R : |x $$-$$ 2| &gt; 1}, B = {x $$\in$$ R : $$\sqrt {{x^2} - 3}$$ &gt; 1}, C = {x $$\in$$ R : |x $$-... JEE Main 2021 (Online) 27th August Morning Shift Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S$$\to$$S such that f(m . n) = f(m) . f(n) fo... JEE Main 2021 (Online) 27th July Morning Shift Let A = {n$$\in$$N | n2$$\le$$n + 10,000}, B = {3k + 1 | k$$\in$$N} an dC = {2k | k$$\in$$N}, then the sum of all t... JEE Main 2021 (Online) 27th July Evening Shift Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A$$\to$$A such that f(1) + f(2) = 3$$-$$... JEE Main 2021 (Online) 22th July Evening Shift If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + x g(x3) is divisible by x2 + x + 1, then P(... JEE Main 2021 (Online) 18th March Evening Shift If a +$$\alpha$$= 1, b +$$\beta$$= 2 and$$af(x) + af\left( {{1 \over x}} \right) = bx + {\beta \over x},x \ne 0$$,... JEE Main 2021 (Online) 24th February Evening Slot Let &nbsp;A = {n$$ \in $$N: n is a 3-digit number} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;B = {9k + 2: k$$ \in $$N}... JEE Main 2021 (Online) 24th February Morning Slot Suppose that a function f : R$$ \to $$R satisfies f(x + y) = f(x)f(y) for all x, y$$ \in $$R and f(1) = 3. If$$\sum... JEE Main 2020 (Online) 6th September Evening Slot Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of ... JEE Main 2020 (Online) 6th September Morning Slot Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A $$\to$$ B | 2 $$\in$$ f(A)... JEE Main 2020 (Online) 5th September Evening Slot The number of distinct solutions of the equation $${\log _{{1 \over 2}}}\left| {\sin x} \right| = 2 - {\log _{{1 \over ... JEE Main 2020 (Online) 9th January Morning Slot Let X = {n$$ \in $$N : 1$$ \le $$n$$ \le $$50}. If A = {n$$ \in $$X: n is a multiple of 2} and B = {n$$ \in$\$ ...
JEE Main 2020 (Online) 7th January Evening Slot