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 , + ∞

The function f ( x ) = sec log x + 1 + x 2 is

neither an odd nor an even function

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

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

MEDIUM

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The integral of an odd function is usually leads to zero. Is it true and false? Explain the following statements with proper example.

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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

P (x)

be a polynomial with real coefficients such that

P ({s i n}^{2} x) = P ({c o s}^{2} x),

for all

x \in [0, \frac{π}{2}] .

Consider the following statements:

I.

P (x)

is an even function.

II.

P (x)

can be expressed as a polynomial in

{(2 x - 1)}^{2}

III.

P (x)

is a polynomial of even degree. Then

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}

MEDIUM

Mathematics>Algebra>Sets, Relations and Functions>Functions

The function

f (x) = \sec [\log (x + \sqrt{1 + x^{2}})]

EASY

Mathematics>Algebra>Sets, Relations and Functions>Functions

f : R \to R,

such that

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

then

f

HARD

Mathematics>Algebra>Sets, Relations and Functions>Functions

Let

f (x) = a^{x} (a > 0)

be written as

f (x) = f_{1} (x) + f_{2} (x),

where

f_{1} (x)

is an even function and

f_{2} (x)

is an odd function. Then

f_{1} (x + y) + f_{1} (x - y)

equals:

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

The integral of an odd function is usually leads to zero. Is it true and false? Explain the following statements with proper example.

Important Questions on Sets, Relations and Functions

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