If x ∈ ( a , b ) satisfies the inequality x - 3 x 2 + 3 x + 2 ≥ 1 , then the largest possible value of b - a is

No real valued x satisfies the inequality

The domain of the function f x = 1 log 10 1 - x + x + 2 is

- 3 , - 2 . 5 ∪ - 2 . 5 , - 2

/ f ( x ) = x 3 - 5 x 2 - 8 x 3 is

negative and monotonically increasing for x ∈ - ∞ , 5 - 57 2 and positive and monotonically increasing for x ∈ 5 + 57 2 , + ∞

positive and monotonically increasing for x ∈ - ∞ , 5 - 57 2 and x ∈ 5 + 57 2 , + ∞

negative and monotonically decreasing for x ∈ - ∞ , 5 - 57 2 and x ∈ 5 + 57 2 , + ∞

positive and monotonically increasing for x ∈ - ∞ , 5 - 57 2 and negative and monotonically decreasing for x ∈ 5 + 57 2 , + ∞

Let f x = sin π 6 sin π 2 sin x , for all x ∈ R and g x = π 2 sin x , for all x ∈ R . Let f o g x denote f g x and g o f x denote g f x . Then which of the following is (are) true?

lim x → 0 f ( x ) g ( x ) = π 6

There is an x ∈ R such that g o f x = 1

The domain of definition of f ( x ) = 1 - | x | 2 - | x | is

( - ∞ , - 2 ) ∪ [ - 1 , 1 ] ∪ ( 2 , ∞ )

EASY

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In the representation of a function, in which of the following method a piece of clear information regarding the nature of the function (like increasing or decreasing of the graph, where the rate of change fluctuates more, maxima, minima, etc.) can be obtained?

$A .$ Analytical representation

$B .$ Graphical representation

$C .$ Tabular representation

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Important Questions on Sets, Relations and Functions

HARD

Mathematics>Algebra>Sets, Relations and Functions>Functions

For

a > b > c > 0

, the minimum value of the function

f (x) = | x - a | + | x - b | + | x - c |

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

Domain of

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

MEDIUM

Mathematics>Algebra>Sets, Relations and Functions>Functions

Let

f (x) = {(x + 2)}^{2} - 2, x \geq - 2

. Then

f^{- 1} (x) =

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

Given

f (x) \dots x^{2} + \log_{3} x

and

g (y) \dots 2 y + f (y)

, the value of

g (3)

equals

MEDIUM

Mathematics>Algebra>Sets, Relations and Functions>Functions

The range of the function

f (x) = \frac{2 + x}{2 - x}, x \neq 2

MEDIUM

Mathematics>Algebra>Sets, Relations and Functions>Functions

x \in (a, b)

satisfies the inequality

\frac{x - 3}{x^{2} + 3 x + 2} \geq 1

, then the largest possible value of

b - a

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

Let

f : A \to B, g : B \to A

be defined as

f (x) = x^{2} \forall x \in A

and

g (x) = x^{1 / 2} \forall x \in B

f (x)

and

g (x)

are inverse functions to each other when

HARD

Mathematics>Algebra>Sets, Relations and Functions>Functions

The range of the polynomial

P (x) = 4 x^{3} - 3 x

x

varies over the interval

(- \frac{1}{2}, \frac{1}{2})

MEDIUM

Mathematics>Algebra>Sets, Relations and Functions>Functions

The domain of the function

f (x) = \frac{1}{\log_{10} (1 - x)} + \sqrt{(x + 2)}

HARD

Mathematics>Algebra>Sets, Relations and Functions>Functions

Let

f (x) = \sqrt{[\sin 2 x] - [\cos 2 x]}

(where

[.]

denotes the greatest integer function), then the range of

f (x)

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

y = 4 + 9 \sin 5 x

then which holds good?

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

The domain of the function

f (x) = \sqrt{\frac{1}{|x - 2| - (x - 2)}}

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

The range of the function

f (x) = \frac{1}{2 - \cos 3 x}

HARD

Mathematics>Algebra>Sets, Relations and Functions>Functions

f (x) = \frac{x^{3} - 5 x^{2} - 8 x}{3}

HARD

Mathematics>Algebra>Sets, Relations and Functions>Functions

Let

f

be a continuous function defined on

[0, 1]

such that

\int_{0}^{1} f^{2} (x) d x = {(\int_{0}^{1} f (x) d x)}^{2}

. Then the range of

f

HARD

Mathematics>Algebra>Sets, Relations and Functions>Functions

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 o g) (x)

denote

f (g (x))

and

(g o f) (x)

denote

g (f (x))

. Then which of the following is (are) true?

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

Let

f : (- 1, 1) \to (- 1, 1)

be continuous,

f (x) = f (x^{2})

for all

x \in (- 1, 1)

and

f (0) = \frac{1}{2}

,

then the value of

4 f (\frac{1}{4})

MEDIUM

Mathematics>Algebra>Sets, Relations and Functions>Functions

If the domain of the function

f (x) = \frac{\cos^{- 1} \sqrt{x^{2} - x + 1}}{\sqrt{\sin^{- 1} (\frac{2 x - 1}{2})}}

is the interval

(α, β],

then

α + β

is equal to:

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

The domain of definition of

f (x) = \sqrt{\frac{1 - | x |}{2 - | x |}}

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

Let

f : [2, \infty) \to R

be the function defined by

f (x) = x^{2} - 4 x + 5

, then the range

Important Questions on Sets, Relations and Functions

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