For the following questions, choose the correct answers from the codes ( a ) , ( b ) , ( c ) and ( d ) defined as follows: Let f : R → R be a function such that f x = e x - e - x e x + e - x Statement I: f x is into function. Statement II: f x is many-one function, and the many-one function is not onto.

Statement I is true. Statement II is false.

Statement I is true, Statement II is also true; Statement II is the correct explanation of Statement I.

Statement I is true, Statement II is also true; Statement II is not the correct explanation of Statement I.

Statement I is false. Statement II is true.

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If $f (g (x))$ is one-one function, then:

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Important Questions on Functions

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 41

Which of the following functions have their range equal to

R

(the set of real numbers)?

HARD

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 42

Which of the following pairs of functions are identical?

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 43

For the following questions, choose the correct answers from the codes

(a), (b), (c)

and

(d)

defined as follows:
Statement

I

: The function

f (x) = x \sin x

and

f^{'} (x) = x \cos x + \sin x

are both non-periodic.
Statement

II

: The derivative of differentiable function (non-periodic) is non-periodic function.

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 44

Statement

I

: The maximum value of

s i n \sqrt{2} x + s i n a x

cannot be

2

.
(where

a

is positive rational number)
Statement

II

\frac{\sqrt{2}}{a}

is irrational.

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 45

For the following questions, choose the correct answers from the codes

(a), (b), (c)

and

(d)

defined as follows:
Let

f : R \to R

be a function such that

f (x) = \frac{e^{|x|} - e^{- x}}{e^{x} + e^{- x}}

Statement I:

f (x)

is into function.
Statement II:

f (x)

is many-one function, and the many-one function is not onto.

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 46

For the following questions, choose the correct answers from the codes

(a), (b), (c)

and

(d)

defined as follows:
Statement I: The range of

f (x) = s i n (\frac{π}{5} + x) - s i n (\frac{π}{5} - x) - s i n (\frac{2 π}{5} + x) + s i n (\frac{2 π}{5} - x)

[- 1, 1]

.
Statement II:

c o s \frac{π}{5} - c o s \frac{2 π}{5} = \frac{1}{2}

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 47

For the following questions, choose the correct answers from the codes

(a), (b), (c)

and

(d)

defined as follows:
Statement I: The period of

f (x) = 2 c o s \frac{1}{3} (x - π) + 4 s i n \frac{1}{3} (x - π)

3 π

.
Statement II: If

T

is the period of

f (x)

, then the period of

f (a x + b)

\frac{T}{|a|}

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 48

For the following question, choose the correct answers from the codes $(A), (B), (C) and (D),$ defined as follows:
$f$ is a function defined on the interval $[- 1,1]$ such that $f (\sin 2 x) = s i n x + c o s x$ .

Statement I: If $x \in [- \frac{π}{4}, \frac{π}{4}]$ , then $f (\tan^{2} x) = \sec x .$

Statement II: $f (x) = \sqrt{1 + x}, \forall x \in [- 1,1]$ .

If fgx is one-one function, then:

Important Questions on Functions

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If $f (g (x))$ is one-one function, then: