Core Syllabus Module

Functions

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2024

LEVELJEE Advanced

Let $f : R \to R$ and $g : R \to R$ be functions defined by $f (x) = {∣ x ∣ sin (\frac{1}{x}), 0, x \neq = 0 x = 0$ and $g (x) = {1 - 2 x, 0, 0 \leq x \leq \frac{1}{2} otherwise$ . Let $a, b, c, d \in R$ . Define the function $h : R \to R$ by $h (x) = a f (x) + b (g (x) + g (\frac{1}{2} - x)) + c (x - g (x)) + d g (x), x \in R$ . Match each entry in List-I to the correct entries in List-II.

List-I

(P)

a = 0, b = 1, c = 0

and

d = 0

, then

(Q)

a = 1, b = 0, c = 0

and

d = 0

, then

(R)

a = 0, b = 0, c = 1

and

d = 0

, then

(S)

a = 0, b = 0, c = 0

and

d = 1

, then

List-II

(1)

h

is one-one.

(2)

h

is onto.

(3)

h

is differentiable on

R

(4)

the range of

h

[0, 1]

(5)

the range of

h

{0, 1}

JEE Main 2024 (04 Apr Shift 1)

LEVELJEE Advanced

If the domain of the function $sin^{- 1} (\frac{3 x - 22}{2 x - 19}) + lo g_{e} (\frac{3 x ^{2} - 8 x + 5}{x ^{2} - 3 x - 10})$ is $(α, β]$ , then $3 α + 10 β$ is equal to:

If the function $f : R \to R$ is defined by $f (x) = ∣ x ∣ (x - sin x)$ , then which of the following statements is TRUE?

(A)

f

is one-one, but NOT onto

(B)

f

is onto, but NOT one-one

(C)

f

is BOTH one-one and onto

(D)

f

is NEITHER one-one NOR onto

JEE Advanced 1996

LEVELJEE Advanced

A function $f : I R \to I R$ , where $I R$ is the set of real numbers, is defined by $f (x) = \frac{α x ^{2} + 6 x - 8}{α + 6 x - 8 x ^{2}}$ . Find the interval of values of $α$ for which $f$ is onto. Is the function one-to-one for $α = 3$ ? Justify your answer.

JEE Main 2022 (24 June Shift 1)

LEVELJEE Advanced

The number of one-one function $f : {a, b, c, d} \to {0, 1, 2, \dots, 10}$ such that $2 f (a) - f (b) + 3 f (c) + f (d) = 0$ is ______.

JEE Main 2023 (08 April Shift 2)

LEVELJEE Advanced

If domain of the function $lo g_{e} (\frac{6 x ^{2} + 5 x + 1}{2 x - 1}) + cos^{- 1} (\frac{2 x ^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ)$ , then $18 (α^{2} + β^{2} + γ^{2} + δ^{2})$ is equal to

JEE Advanced 2014

LEVELJEE Advanced

Let $f_{1} : R \to R, f_{2} : [0, \infty) \to R, f_{3} : R \to R$ and $f_{4} : R \to [0, \infty)$ be defined by $f_{1} (x) = {∣ x ∣ e^{x} if x < 0, if x \geq 0;$ $f_{2} (x) = x^{2}$ ; $f_{3} (x) = {sin x x if x < 0, if x \geq 0;$ and $f_{4} (x) = {f_{2} (f_{1} (x)) f_{2} (f_{1} (x)) - 1 if x < 0, if x \geq 0.$

List-I

(P)

f_{4}

(Q)

f_{3}

(R)

f_{2} \circ f_{1}

(S)

f_{2}

List-II

(1)

Onto but not one-one

(2)

Neither continuous nor one-one

(3)

Differentiable but not one-one

(4)

Continuous and one-one

JEE Advanced 2015

LEVELJEE Main

Let $f (x) = sin (\frac{π}{6} sin (\frac{π}{2} sin x))$ for all $x \in R$ and $g (x) = \frac{π}{2} sin x$ for all $x \in R$ . Let $(f \circ g) (x)$ denote $f (g (x))$ and $(g \circ f) (x)$ denote $g (f (x))$ . Then which of the following is (are) true?

* Multiple Correct Options

(A)

Range of

f

[- 1/2, 1/2]

(B)

Range of

f \circ g

[- 1/2, 1/2]

(C)

lim_{x \to 0} \frac{f ( x )}{g ( x )} = \frac{π}{6}

(D)

There is an

x \in R

such that

(g \circ f) (x) = 1

JEE Main 2023 (11 Apr Shift 2)

LEVELJEE Main

Let $f$ and $g$ be two functions defined by $f (x) = {x + 1, ∣ x - 1∣, x < 0 x \geq 0$ and $g (x) = {x + 1, 1, x < 0 x \geq 0$ . Then $(g \circ f) (x)$ is

(A)

Continuous everywhere but not differentiable exactly at one point

(B)

Continuous everywhere but not differentiable at

x = 1

(C)

Differentiable everywhere

(D)

Not continuous at

x = 1

JEE Main 2024 (01 Feb Shift 1)

LEVELJEE Main

Let $f : R \to R$ and $g : R \to R$ be defined as $f (x) = {lo g_{e} x e^{- x}, x > 0, x \leq 0$ and $g (x) = {x e^{x}, x \geq 0, x < 0$ . Then, $g \circ f : R \to R$ is :

(A)

one-one but not onto

(B)

neither one-one nor onto

(C)

onto but not one-one

(D)

both one-one and onto

JEE Main 2024 (05 Apr Shift 2)

LEVELJEE Main

Let $f, g : R \to R$ be defined as : $f (x) = ∣ x - 1∣$ and $g (x) = {e^{x}, x + 1, x \geq 0 x \leq 0$ . Then the function $f (g (x))$ is

(A)

neither one-one nor onto.

(B)

one-one but not onto.

(C)

onto but not one-one.

(D)

both one-one and onto.

JEE Main 2024 (05 Apr Shift 1)

LEVELJEE Main

If $S = {a \in R : ∣2 a - 1∣ = 3 [a] + 2 {a}}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ and ${t}$ represents the fractional part of $t$ , then $72 \sum_{a \in S} a$ is equal to ______

JEE Main 2024 (04 Apr Shift 2)

LEVELJEE Main

Consider the function $f : R \to R$ defined by $f (x) = \frac{2 x}{1 + 9 x ^{2}}$ . If the composition of $f, 10 times (f \circ f \circ f \circ \dots \circ f) (x) = \frac{2 ^{10} x}{1 + 9 α x ^{2}}$ , then the value of $3 α + 1$ is equal to ______

JEE Main 2023 (31 January Shift 2)

LEVELJEE Main

The absolute minimum value of the function $f (x) = ∣ x^{2} - x + 1∣ + [x^{2} - x + 1]$ , where $[t]$ denotes the greatest integer function, in the interval $[- 1, 2]$ , is :

JEE Main 2023 (31 January Shift 1)

LEVELJEE Main

If the domain of the function $f (x) = \frac{[ x ]}{1 + x ^{2}}$ , where $[x]$ is greatest integer $\leq x$ , is $(2, 6)$ , then its range is

(A)

(\frac{5}{26}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}

(B)

(\frac{5}{26}, \frac{2}{5}]

(C)

(\frac{5}{37}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}

(D)

(\frac{5}{37}, \frac{2}{5}]

JEE Main 2023 (25 January Shift 2)

LEVELJEE Main

Let $f : R \to R$ be a function defined by $f (x) = lo g_{m} {2 (sin x - cos x) + m - 2}$ , for some $m$ , such that the range of $f$ is $[0, 2]$ . Then the value of $m$ is

JEE Main 2023 (25 January Shift 1)

LEVELJEE Main

For some $a, b, c \in N$ , let $f (x) = a x^{b} + 3$ and $g (x) = x^{b} + c, x \in R$ . If $(f \circ g)^{- 1} (x) = (\frac{x - 7}{2})^{1/3}$ , then $(f \circ g) (a c) + (g \circ f) (b)$ is equal to

JEE Advanced 2014

LEVELJEE Main

Let $f : (- π /2, π /2) \to R$ be given by $f (x) = (lo g (sec x + tan x))^{3}$ . Then

* Multiple Correct Options

(A)

f (x)

is an odd function

(B)

f (x)

is one-one function

(C)

f (x)

is an onto function

(D)

f (x)

is an even function

JEE Main 2026 (22 January Shift 2)

LEVELJEE Main

Let the domain of the function $f (x) = lo g_{3} lo g_{5} (7 - lo g_{2} (x^{2} - 10 x + 85)) + sin^{- 1} (\frac{3 x - 7}{17 - x})$ be $(α, β]$ . Then $α + β$ is equal to :

JEE Main 2023 (06 April Shift 2)

LEVELJEE Main

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f (x) = \frac{1}{⌈ x ⌉ - x}$ , where $⌈ x ⌉$ denotes the smallest integer greater than or equal to $x$ . Then among the statements (S1): $A \cap B = (1, \infty) - N$ and (S2): $A \cup B = (1, \infty)$

Neither (S1) nor (S2) is true

(D)

Both (S1) and (S2) are true

Micro-Topic Strategy Matrix

Domain and Range of a Function

Classification of Functions

Even and Odd Functions

Periodic Functions

Composite Functions

Inverse of a Function

Chapter Elite-20 Challenge Zone

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the domain of the function sin−1(2x−193x−22​)+loge​(x2−3x−103x2−8x+5​) is (α,β], then 3α+10β is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the function f:R→R is defined by f(x)=∣x∣(x−sinx), then which of the following statements is TRUE?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of one-one function f:{a,b,c,d}→{0,1,2,…,10} such that 2f(a)−f(b)+3f(c)+f(d)=0 is ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If domain of the function loge​(2x−16x2+5x+1​)+cos−1(3x−52x2−3x+4​) is (α,β)∪(γ,δ), then 18(α2+β2+γ2+δ2) is equal to

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f and g be two functions defined by f(x)={x+1,∣x−1∣,​x<0x≥0​ and g(x)={x+1,1,​x<0x≥0​. Then (g∘f)(x) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R and g:R→R be defined as f(x)={loge​xe−x​,x>0,x≤0​ and g(x)={xex​,x≥0,x<0​. Then, g∘f:R→R is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f,g:R→R be defined as : f(x)=∣x−1∣ and g(x)={ex,x+1,​x≥0x≤0​. Then the function f(g(x)) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The absolute minimum value of the function f(x)=∣x2−x+1∣+[x2−x+1], where [t] denotes the greatest integer function, in the interval [−1,2], is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the domain of the function f(x)=1+x2[x]​, where [x] is greatest integer ≤x, is (2,6), then its range is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a function defined by f(x)=logm​​{2​(sinx−cosx)+m−2}, for some m, such that the range of f is [0,2]. Then the value of m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For some a,b,c∈N, let f(x)=axb+3 and g(x)=xb+c,x∈R. If (f∘g)−1(x)=(2x−7​)1/3, then (f∘g)(ac)+(g∘f)(b) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:(−π/2,π/2)→R be given by f(x)=(log(secx+tanx))3. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the domain of the function f(x)=log3​log5​(7−log2​(x2−10x+85))+sin−1(​17−x3x−7​​) be (α,β]. Then α+β is equal to :

If the domain of the function $sin^{- 1} (\frac{3 x - 22}{2 x - 19}) + lo g_{e} (\frac{3 x ^{2} - 8 x + 5}{x ^{2} - 3 x - 10})$ is $(α, β]$ , then $3 α + 10 β$ is equal to:

If the function $f : R \to R$ is defined by $f (x) = ∣ x ∣ (x - sin x)$ , then which of the following statements is TRUE?

The number of one-one function $f : {a, b, c, d} \to {0, 1, 2, \dots, 10}$ such that $2 f (a) - f (b) + 3 f (c) + f (d) = 0$ is ______.

If domain of the function $lo g_{e} (\frac{6 x ^{2} + 5 x + 1}{2 x - 1}) + cos^{- 1} (\frac{2 x ^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ)$ , then $18 (α^{2} + β^{2} + γ^{2} + δ^{2})$ is equal to

Let $f$ and $g$ be two functions defined by $f (x) = {x + 1, ∣ x - 1∣, x < 0 x \geq 0$ and $g (x) = {x + 1, 1, x < 0 x \geq 0$ . Then $(g \circ f) (x)$ is

Let $f : R \to R$ and $g : R \to R$ be defined as $f (x) = {lo g_{e} x e^{- x}, x > 0, x \leq 0$ and $g (x) = {x e^{x}, x \geq 0, x < 0$ . Then, $g \circ f : R \to R$ is :

Let $f, g : R \to R$ be defined as : $f (x) = ∣ x - 1∣$ and $g (x) = {e^{x}, x + 1, x \geq 0 x \leq 0$ . Then the function $f (g (x))$ is

The absolute minimum value of the function $f (x) = ∣ x^{2} - x + 1∣ + [x^{2} - x + 1]$ , where $[t]$ denotes the greatest integer function, in the interval $[- 1, 2]$ , is :

If the domain of the function $f (x) = \frac{[ x ]}{1 + x ^{2}}$ , where $[x]$ is greatest integer $\leq x$ , is $(2, 6)$ , then its range is

Let $f : R \to R$ be a function defined by $f (x) = lo g_{m} {2 (sin x - cos x) + m - 2}$ , for some $m$ , such that the range of $f$ is $[0, 2]$ . Then the value of $m$ is

For some $a, b, c \in N$ , let $f (x) = a x^{b} + 3$ and $g (x) = x^{b} + c, x \in R$ . If $(f \circ g)^{- 1} (x) = (\frac{x - 7}{2})^{1/3}$ , then $(f \circ g) (a c) + (g \circ f) (b)$ is equal to

Let $f : (- π /2, π /2) \to R$ be given by $f (x) = (lo g (sec x + tan x))^{3}$ . Then

Let the domain of the function $f (x) = lo g_{3} lo g_{5} (7 - lo g_{2} (x^{2} - 10 x + 85)) + sin^{- 1} (\frac{3 x - 7}{17 - x})$ be $(α, β]$ . Then $α + β$ is equal to :