Consider f : N → Z defined by f ( x ) = x 2 - 1 . Then

f is a function which is one-one, but not onto

f is a function which is neither one-one nor onto

Let A = x ∈ R : - 1 ≤ x ≤ 1 and f : A → A be a mapping defined by f x = x x . Then f is

neither injective nor surjective

If f : R → R be defined by f ( x ) = e x and g : R → R be defined by g ( x ) = x 2 . The mapping gof : R → R be defined by ( g o f ) ( x ) = g [ f ( x ) ] ∀ x ∈ R . Then

gof is injective but g is not bijective

gof is bijective but f is not injective

gof is injective and g is injective

gof is surjective and g is surjective

EASY

Earn 100

Use horizontal test and find whether the function is one-one or not.

$y = \frac{x^{2}}{4} + 4 x + 5$

Important Questions on Function of a Real Variable

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Consider

f : N \to Z

defined by

f (x) = x^{2} - 1 .

Then

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

The mapping

f : N \to N

given by

f (n) = n^{3} + 3,

n ε N

where

N

is the set of natural number, is

EASY

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

The real-valued function

f : [- 2, 2] \to ℝ

defined by

f (x) = \frac{x}{x + 3}

HARD

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Let

f : R \to R

be defined by

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

then

f

EASY

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Let

A = \{x \in R : - 1 \leq x \leq 1\}

and

f : A \to A

be a mapping defined by

f (x) = x |x|

. Then

f

HARD

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

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 $R$

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

A function

f : [- 7, 6) \to R

is defined as

f (x) = \{x^{2} + 2 x + 1; - 7 \leq x < - 5 : x + 5; - 5 \leq x < 2, x - 1; 2 \leq x < 6\}

. Find

\frac{4 f (- 3) + 2 f (4)}{f (- 6) - 3 f (1)}

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

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

B

EASY

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Let

f : R \to R

be a function defined by

f (x) = | x | + 1

. Then which of the following is true?

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

The number of bijective functions from the set

A

to itself, if

A

contains

108

elements is

EASY

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Let

f : R \to R

be defined as

f (x) = 3 x .

Then

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

f : R \to R

be defined by

f (x) = e^{x}

and

g : R \to R

be defined by

g (x) = x^{2} .

The mapping gof :

R \to R

be defined by

(g o f) (x)

= g [f (x)] \forall x \in R .

Then

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

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

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Let

S = \{1, 2, 3, 4\}

. Then the number of elements in the set {

f : S \times S \to S : f

is onto and

f (a, b) = f (b, a)

\geq a \forall (a, b) \in S \times S

} is

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

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:

EASY

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Which of the following function from

z

into

z

is bijection? (

z

is set of integers)

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Let

A = {a, b, c}

and

B = {1, 2, 3, 4} .

Then the number of elements in the set

C = {f : A \to B ∣ 2 \in f (A)

and

f

is not one-one

}

\dots

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Let

[x]

denote the greatest integer less than or equal to

x .

Then, the values of

x \in R

satisfying the equation

{[e^{x}]}^{2} + [e^{x} + 1] - 3 = 0

lie in the interval:

EASY

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

If $f : S \to R,$ where $S$ is the set of all non-singular matrices of order $2$ over $R$ and $f [(\begin{array}{l} a & b \\ c & d \end{array})] = a d - b c,$ then

MEDIUM

Mathematics>Differential Calculus>Function of a Real Variable>Line Test

Number of solution of the equation

| \cos x | = 2 [x]

are (where

| x |, [x]

are modulus and greatest integer function respectively).

Use horizontal test and find whether the function is one-one or not. y=x24+4x+5

Important Questions on Function of a Real Variable

Learn and Prepare for any exam you want !

Use horizontal test and find whether the function is one-one or not.

$y = \frac{x^{2}}{4} + 4 x + 5$