Mathematics>Differential Calculus>Functions>Types of Functions

EASY

Earn 100

Draw the graph of signum function and state the discontinuous point of signum function.

Important Questions on Functions

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

For a real number

r

we denote by

[r]

the largest integer less than or equal to

r

. If

x, y

are real numbers with

x, y \geq 1

then which of the following statements is always true?

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

On the real line

R

, we define two functions

f

and

g

as follows:

f (x) = m i n {x - [x], 1 - x + [x]}

g (x) = m a x \{x - [x], 1 - x + [x]\}

Where

[x]

denotes the largest integer not exceeding

x

. The positive integer

n

for which

\int_{0}^{n} (g (x) - f (x)) d x = 100

is

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

The minimum value of

f (x) = \max \{x, 1 + x, 2 - x\}

is

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

Let

f : R \to R

be defined by

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

then

f

is

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

The function

f : N \to I

defined by

f (x) = x - 5 [\frac{x}{5}]

, where

N

is the set of natural numbers and

[x]

denotes the greatest integer less than or equal to

x

, is:

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

For a real number

r

let

[r]

denote the largest integer less than or equal to

r .

Let

a > 1

a real number which is not an integer and

k

be the smallest positive integer such that

[a^{k}] > {[a]}^{k}

Then which of the following statement is always true?

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

The function

f : R \to [- \frac{1}{2}, \frac{1}{2}]

defined as

f (x) = \frac{x}{1 + x^{2}},

is:

EASY

Mathematics>Differential Calculus>Functions>Types of Functions

Let

S

be the set of all functions

f : [0, 1] \to R

which are continuous on

[0, 1]

and differentiable on

(0, 1)

. Then for every

f

in

S

, there exists

c \in (0, 1)

depending on

f

, such that.

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

Let

X

be a set with exactly

5

elements and

Y

be a set with exactly

7

elements. If

α

is the number of one-one functions from

X \to Y

and

β

is the number of onto functions from

Y \to X

, then the value of

\frac{1}{5!} (β - α)

is _____-.

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

If

a \in R

and the equation

- 3 (x - [x])^{2} + 2 (x - [x]) + a^{2} = 0

(where

[x]

denotes the greatest integer

\leq

x

) has no integral solution, then all possible values of

a

lie in the interval

EASY

Mathematics>Differential Calculus>Functions>Types of Functions

Let

f : R \to R

be defined as

f (x) = 3 x .

Then

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

Let

f

and

g

be differentiable functions on

R

such that

f o g

is the identity function. If for some

a, b \in R, g^{'} (a) = 5

and

g (a) = b,

then

f^{'} (b)

is equal to:

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

If the function

f : R - \{1, - 1\} \to A

defined by

f (x) = \frac{x^{2}}{1 - x^{2}},

is surjective, then

A

is equal to

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

Let

f (x)

be a non-constant polynomial with real coefficients such that

f (\frac{1}{2}) = 100 & f (x) \leq 100

for all real

x .

Which of the following statements is NOT necessarily true?

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

If

[x]

denotes the greatest integer

\leq x,

then the system of linear equations

[s i n θ] x + [- c o s θ] y = 0

,

[c o t θ] x + y = 0

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

If

A = {x | x \in N, x \leq 5}, B = \{x | x \in Z, x^{2} - 5 x + 6 = 0\},

then the number of onto functions from

A

to

B

is

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

Let

[t]

denote the greatest integer

\leq t

and

\lim_{x \to 0} x [\frac{4}{x}] = A .

Then the function,

f (x) = [x^{2}] \sin (π x)

is discontinuous, when

x

is equal to:

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

Let $R$ be the set of real numbers and $f : R \to R$ be defined by $f (x) = \frac{\{x\}}{1 + {[x]}^{2}},$ where $[x]$ is the greatest integer less than or equal to $x,$ and $\{x\} = x - [x] .$ Which of the following statement are true?

I. The range of $f$ is closed interval
II. $f$ is continuous on $R$
III. $f$ is one-one on $R$

EASY

Mathematics>Differential Calculus>Functions>Types of Functions

Let

A =

{

x \in R : x

is not a positive integer}

.

Define a function

f : A \to R

as

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

, then

f

is:

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

For

x \in R,

Let

[x]

denotes the greatest integer

\leq x,

then the sum of the series

[- \frac{1}{3}] + [- \frac{1}{3} - \frac{1}{100}] + [- \frac{1}{3} - \frac{2}{100}] + . . . . . + [- \frac{1}{3} - \frac{99}{100}]

is