The function f x = log x + x 2 + 1 is

Neither an even nor an odd function

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

neither an odd nor an even function

If [ x ] denotes the greatest integer ≤ 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

have infinitely many solutions if θ ∈ π 2 , 2 π 3 and has a unique solution if θ ∈ π , 7 π 6

has a unique solution if θ ∈ π 2 , 2 π 3 ∪ π , 7 π 6

have infinitely many solution if θ ∈ π 2 , 2 π 3 ∪ π , 7 π 6

has a unique if θ ∈ π 2 , 2 π 3 and have infinitely many solutions if θ ∈ π , 7 π 6

Let t denote the greatest integer ≤ t . Then the equation in x , x 2 + 2 x + 2 - 7 = 0 has :

exactly four integral solutions

Show that the function f : R → R defined by f x = x x 2 + 1 , ∀ x ∈ ℝ is neither one-one nor onto. Also, if g : R → R is defined as g ( x ) = 2 x - 1 , find fog ( x ) .

Given, y = x x 2 + 1 ⇒ y x 2 − x + y = 0 Here a = y , b = - 1 and c = y ∴ x = − ( − 1 ) ± 1 − 4 y 2 2 y Clearly, for every value of y , x will have two different values . So the function is many-one not one - one. Since 1 − 4 y 2 ≥ 0 ⇒ ( 1 + 2 y ) ( 1 − 2 y ) ≥ 0 ⇒ − 1 2 ≤ y ≤ 1 2 That means no matter what is x , y belongs to the interval − 1 2 , 1 2 . Hence, the function is not onto. Now, f o g x = f g x = f 2 x - 1 = 2 x - 1 2 x - 1 2 + 1 = 2 x + 1 4 x 2 - 4 x + 1 + 1 = 2 x + 1 2 2 x 2 - 2 x + 1

EASY

Earn 100

Statement-1: If $y = f (x)$ is an even differentiable function, then $f^{'} (0) = 0$ .

Statement-2 : If $f (x)$ is even then $f^{'} (x)$ is odd.

33.33% studentsanswered this correctly

Important Questions on Functions

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd Functions

The function

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

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

Which of the following functions is neither even nor odd?

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

The even function of the following is

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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>Even and Odd Functions

The function

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

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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?

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd Functions

The real valued function

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

defined on

R \ {0}

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

The value of

\lim_{x \to 0^{+}} \frac{x}{p} [\frac{q}{x}]

is (where

[\cdot]

represents greatest integer function)

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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:

HARD

Mathematics>Differential Calculus>Functions>Even and Odd Functions

[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

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

Which of the following function from

z

into

z

is bijection? (

z

is set of integers)

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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}]

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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:

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd Functions

Let

[t]

denote the greatest integer

\leq t

. Then the equation in

x, {[x]}^{2} + 2 [x + 2] - 7 = 0

has :

HARD

Mathematics>Differential Calculus>Functions>Even and Odd Functions

Consider the sequence of numbers

[n + \sqrt{2 n} + \frac{1}{2}]

for

n \geq 1,

where

[x]

denotes the greatest integer not exceeding

x

. If the missing integers in the sequence are

n_{1} < n_{2} < n_{3} < \dots . .,

then find

n_{12}

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd Functions

Let $[x]$ be the greatest integer less than or equal to $x$ , for a real number $x$ . Then the following sum

$[\frac{2^{2020} + 1}{2^{2018} + 1}] + [\frac{3^{2020} + 1}{3^{2018} + 1}] + [\frac{4^{2020} + 1}{4^{2018} + 1}] + [\frac{5^{2020} + 1}{5^{2018} + 1}] + [\frac{6^{2020} + 1}{6^{2018} + 1}]$

is :-

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

f : R \to R,

such that

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

then

f

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd Functions

Show that the function

f : R \to R

defined by

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

is neither one-one nor onto. Also, if

g : R \to R

is defined as

g (x) = 2 x - 1

, find

fog (x)

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

Let

f : R \to R

be defined as

f (x) = 3 x .

Then

Statement-1: If y=fx is an even differentiable function, then f'0=0. Statement-2 : If fx is even then f'x is odd.

Important Questions on Functions

Learn and Prepare for any exam you want !

Statement-1: If $y = f (x)$ is an even differentiable function, then $f^{'} (0) = 0$ .

Statement-2 : If $f (x)$ is even then $f^{'} (x)$ is odd.