Let f : R ⟶ R be two differentiable functions satisfying f x - f y = x 3 - y 3 g x - y ∀ x , y ∈ R (where f , g are not identically zero), then which of the following statement about f & g is correct:?

f is one-one onto and g is many one into

f is many one onto and g is one-one onto

f is one-one into and g is many one into

f is many one into and g is many one into

Let f : R ⟶ R be two differentiable functions satisfying f x - f y = x 3 - y 3 g x - y ∀ x , y ∈ R (where f , g are not identically zero), then which of the following statement about f & g is correct:?

f is one-one onto and g is many one into

If f x + 2 f 1 x = 3 x , x ≠ 0 , and S = x ∈ R : f x = f - x , then S

Contains more than two elements

HARD

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Let $f : R ⟶ R$ be two differentiable functions satisfying $f (x) - f (y) = (x^{3} - y^{3}) g (x - y) \forall x, y \in R$ (where $f, g$ are not identically zero), then which of the following statement about $f & g$ is correct:?

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Important Questions on Functions

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Number of functions

f : [0, 1] \to [0, 1]

satisfying

|f (x) - f (y)| = |x - y|

for all

x, y

[0, 1]

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let,

f : R \to R

be a function such that

f (x) = x^{3} + x^{2} f^{'} (1) + x f^{"} (2) + f^{'''} (3), \forall x \in R .

Then

f (2)

equals

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f (x) = \{\begin{array}{l} x \sin (\frac{1}{x}), when x \neq 0 \\ 1, when x = 0 \end{array}\}

and

A = \{x \in R : f (x) = 1\}

. Then,

A

has

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

A function $f (x)$ is given by $f (x) = \frac{5^{x}}{5^{x} + 5}$ , then the sum of the series $f (\frac{1}{20}) + f (\frac{2}{20}) + f (\frac{3}{20}) + \dots + f (\frac{39}{20})$ is equal to:

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

f (x) + 2 f (\frac{1}{x}) = 3 x, x \neq 0,

and

S = \{x \in R : f (x) = f (- x)\}

, then

S

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

A

denote the set of all real numbers

x

such that

x^{3} - [x]^{3} = (x - [x])^{3},

where

[x]

is the greatest integer less than or equal to

x .

Then

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let the function $f : [0, 1] \to ℝ$ be defined by $f (x) = \frac{4^{x}}{4^{x} + 2}$ . Then the value of $f (\frac{1}{40}) + f (\frac{2}{40}) + f (\frac{3}{40}) + \dots + f (\frac{39}{40}) - f (\frac{1}{2})$ is_________

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : R \to R

be defined as

f (x + y) + f (x - y) = 2 f (x) f (y), f (\frac{1}{2}) = - 1 .

Then the value of

\sum_{k = 1}^{20} \frac{1}{\sin (k) \sin (k + f (k))}

is equal to :

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

S

be the set of points where the function

, f (x) = |2 - | x - 3 ||, x \in R,

is not differentiable. Then

\sum_{x \in S} f (f (x))

is equal to

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

\sum_{k = 1}^{10} f (a + k) = 16 (2^{10} - 1),

where the function

f

satisfies

f (x + y) = f (x) f (y)

for all natural numbers

x, y

and

f (1) = 2 .

Then the natural number

a

is:

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

a + α = 1, b + β = 2

and

a f (x) + α f (\frac{1}{x}) = b x + \frac{β}{x}, x \neq 0,

then the value of the expression

\frac{f (x) + f (\frac{1}{x})}{x + \frac{1}{x}}

is ___________.

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let $R$ be the set of all real numbers and let $f$ be a function $R$ to $R$ such that $f (x) + (x + \frac{1}{2}) f (1 - x) = 1$ , for all $x \in R$ . Then $2 f (0) + 3 f (1)$ is equal to.

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

f (x)

is an even function and satisfies the relation

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

where

g (x)

is an odd function, then

f (5)

equals

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

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

in terms of

f (x)

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

f (x + y) = f (x) + f (y) - x y - 1 \forall x, y \in R

and

f (1) = 1

, then the number of solutions of

f (n) = n, n \in N

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

f (x) + 2 f (1 - x) = x^{2} + 1, \forall x \in R

, then the range of

f

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

S = \{1, 2, 3, 4, 5, 6, 7\} .

Then the number of possible functions

f : S \to S

such that

f (m \cdot n) = f (m) \cdot f (n)

for every

m, n \in S

and

m \cdot n \in S

, is equal to _____.

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : R \to R

be a function which satisfies

f (x + y) = f (x) + f (y), \forall x, y \in R

. If

f (1) = 2

and

g (n) = \sum_{k = 1}^{(n - 1)} f (k),

n \in N

then the value of

n

, for which

g (n) = 20

, is

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : x \to y

be such that

f (1) = 2

and

f (x + y) = f (x) f (y)

for all natural numbers

x

and

y

.

\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1)

then

a

is equal to

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

a, b, c \in R

. If

f (x) = a x^{2} + b x + c

is such that

a + b + c = 3

and

f (x + y) = f (x) + f (y) + x y, \forall x, y \in R

, then

\sum_{n = 1}^{10} f (n)

is equal to:

Let f:R⟶R be two differentiable functions satisfying fx-fy=x3-y3gx-y ∀x, y∈R (where f, g are not identically zero), then which of the following statement about f & g is correct:?

Important Questions on Functions

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Let $f : R ⟶ R$ be two differentiable functions satisfying $f (x) - f (y) = (x^{3} - y^{3}) g (x - y) \forall x, y \in R$ (where $f, g$ are not identically zero), then which of the following statement about $f & g$ is correct:?