Prove that the function f : R → R , given by f x = 2 x is one-one and onto.

Given f : R → R and f x = 2 x We know that A function f : X → Y is one-one, for every x , y ∈ X , if f x = f y then x = y Take any two real numbers x 1 and x 2 f x 1 = f x 2 ⇒ 2 x 1 = 2 x 2 ⇒ x 1 = x 2 ∴ f is one-one We know that A function f : X → Y , for every y ∈ Y there exists an element x ∈ X such that f ( x ) = y Take y = f x ⇒ y = 2 x ⇒ x = y 2 Here for every real number y , there exists a real number x ∴ f is an onto function ∴ The given function is one-one and onto Hence Proved

The function f : R → - 1 2 , 1 2 defined as f x = x 1 + x 2 , is:

Neither injective nor surjective

Let A = { x ∈ R : x is not a positive integer} . Define a function f : A → R as f x = 2 x x - 1 , then f is:

Neither injective nor surjective

Let S , T , U be three non-void sets and f : S → T , g : T → U be so that g o f : S → U is surjective. Then,

g is surjective, f may not be so

f is surjective, g may not be so

f and g both may not be 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

Let f : R → R be defined by f ( x ) = x 2 - x 2 1 + x 2 for all x ∈ R . Then,

f is one-one but not onto mapping

f is onto but not one-one mapping

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

EASY

Earn 100

Define into function and write one example.

Important Questions on Relations and Functions

MEDIUM

Mathematics>Algebra>Relations and 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

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

Prove that the function

f : R \to R

, given by

f (x) = 2 x

is one-one and onto.

MEDIUM

Mathematics>Algebra>Relations and 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>Algebra>Relations and Functions>Types of Functions

A = {1, 2, 3, 4},

then the number of functions on the set

A,

which are not one - one, is

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Functions

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>Algebra>Relations and Functions>Types of Functions

Which of the following function from

z

into

z

is bijection? (

z

is set of integers)

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

A =

{

x \in R : x

is not a positive integer}

.

Define a function

f : A \to R

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

, then

f

is:

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

S, T, U

be three non-void sets and

f : S \to T, g : T \to U

be so that

g o f : S \to U

is surjective. Then,

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

If the function

f : R \to R

is defined by

f (x) = |x| (x - \sin x)

, then which of the following statements is TRUE?

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

A = [- 1, 1]

and

f : A \to A

be defined as

f (x) = x |x|

for all

x \in A

, then

f (x)

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Functions

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

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : (- \frac{π}{2}, \frac{π}{2}) \to R

be given by

f (x) = {(\log (\sec x + \tan x))}^{3} .

Then

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

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>Algebra>Relations and Functions>Types of Functions

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>Algebra>Relations and Functions>Types of Functions

Let

f : R \to R

be defined by

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

for all

x \in R .

Then,

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

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

and

g : [- 1, 1] \to [0, 2]

be two functions such that

g

is injective and

g o f : [0, 1] \to [0, 2]

is surjective. Then,

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

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

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

A

and

B

be finite sets and

P_{A}

and

P_{B}

, respectively denote their power sets. If

P_{B}

has

112

elements more than those in

P_{A}

, then the number of functions from

A

B

which are injective is

EASY

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : R \to R

be defined as

f (x) = 3 x .

Then

HARD

Mathematics>Algebra>Relations and Functions>Types of Functions

Let

f : R \to R

be defined by

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

then

f

Define into function and write one example.

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !