# State with reason whether the following functions have inverse

i.$

f:\left\{ {1,2,3,4} \right\} \to \{ 10\} with \\

f = \{ (1,10),(2,10),(3,10),(4,10)\} \\

$

ii.$

g:\left\{ {5,6,7,8} \right\} \to \{ 1,2,3,4\} with \\

g = \{ (5,4),(6,3),(7,4),(8,2)\} \\

$

iii.$

h:\left\{ {2,3,4,5} \right\} \to \{ 7,9,11,13\} with \\

h = \{ (2,7),(3,9),(4,11),(5,13)\} \\

$

Last updated date: 22nd Mar 2023

•

Total views: 308.1k

•

Views today: 3.88k

Answer

Verified

308.1k+ views

If the given functions satisfy both one-one and onto functions then it will have

inverse.

Given,

$

f:\left\{ {1,2,3,4} \right\} \to \{ 10\} with \\

f = \{ (1,10),(2,10),(3,10),(4,10)\} \\

$

Here, the domain of ‘f’ is $\{ 1,2,3,4\} $and co-domain is$\{ 10\} $.

As we know a function is said to be a one-one function if distinct elements of domain

mapped with distinct elements of co-domain.

$f = \{ (1,10),(2,10),(3,10),(4,10)\} $

But, in this case if we see the function ‘f’ each element from the domain is mapped with the

same element from co-domain i.e.., 10.Since, all the elements have the same image 10 which is not satisfying the one-one function condition. Hence, f is not a one-one function.

Therefore, f doesn’t have inverse.

ii.Given,

$

g:\left\{ {5,6,7,8} \right\} \to \{ 1,2,3,4\} with \\

g = \{ (5,4),(6,3),(7,4),(8,2)\} \\

$

Here, the domain of ‘g’ is $\{ 5,6,7,8\} $and co-domain is$\{ 1,2,3,4\} $.

As we know a function is said to be a one-one function if distinct elements of domain mapped with distinct elements of co-domain.

$g = \{ (5,4),(6,3),(7,4),(8,2)\} $

But, in this case if we see the function ‘g’, the elements 5 and 7 from the domain are mapped

with the same element from co-domain i.e... As ‘g’ is not satisfying the one-one function

condition. Hence,’g’ is not a one-one function.

Therefore, g doesn’t have inverse.

iii.Given,

$

h:\left\{ {2,3,4,5} \right\} \to \{ 7,9,11,13\} with \\

h = \{ (2,7),(3,9),(4,11),(5,13)\} \\

$

Here, the domain of ‘h’ is $\{ 2,3,4,5\} $and co-domain is$\{ 7,9,11,13\} $.

As we know a function is said to be a one-one function if distinct elements of domain mapped with distinct elements of co-domain.

$h = \{ (2,7),(3,9),(4,11),(5,13)\} $

Here, each element from the domain is mapped with the different element from the co-domain. Therefore, ‘h’ is a one-one function.

Now, let us check with the onto condition i.e.., each element in the co-domain has a pre-image from the domain.

Here, each element from the co-domain has a pre-image from the domain. Therefore ‘h’ is

an onto function. As, function ‘h’ is both one-one and onto functions.

Hence, the inverse of ‘h’ exists i.e..,

$

h = \{ (2,7),(3,9),(4,11),(5,13)\} \\

{h^{ - 1}} = \{ (7,2),(9,3),(11,4),(13,5)\} \\

$

Hence, among the functions ‘f’, ‘g’, ‘h’ only the function ‘h’ has the inverse.

Note: The alternate method to find whether a function is one-one is by horizontal line test

i.e. ., if a horizontal line intersects the original function in a single region, the function is a

one-to-one function otherwise it is not a one-one function.

inverse.

Given,

$

f:\left\{ {1,2,3,4} \right\} \to \{ 10\} with \\

f = \{ (1,10),(2,10),(3,10),(4,10)\} \\

$

Here, the domain of ‘f’ is $\{ 1,2,3,4\} $and co-domain is$\{ 10\} $.

As we know a function is said to be a one-one function if distinct elements of domain

mapped with distinct elements of co-domain.

$f = \{ (1,10),(2,10),(3,10),(4,10)\} $

But, in this case if we see the function ‘f’ each element from the domain is mapped with the

same element from co-domain i.e.., 10.Since, all the elements have the same image 10 which is not satisfying the one-one function condition. Hence, f is not a one-one function.

Therefore, f doesn’t have inverse.

ii.Given,

$

g:\left\{ {5,6,7,8} \right\} \to \{ 1,2,3,4\} with \\

g = \{ (5,4),(6,3),(7,4),(8,2)\} \\

$

Here, the domain of ‘g’ is $\{ 5,6,7,8\} $and co-domain is$\{ 1,2,3,4\} $.

As we know a function is said to be a one-one function if distinct elements of domain mapped with distinct elements of co-domain.

$g = \{ (5,4),(6,3),(7,4),(8,2)\} $

But, in this case if we see the function ‘g’, the elements 5 and 7 from the domain are mapped

with the same element from co-domain i.e... As ‘g’ is not satisfying the one-one function

condition. Hence,’g’ is not a one-one function.

Therefore, g doesn’t have inverse.

iii.Given,

$

h:\left\{ {2,3,4,5} \right\} \to \{ 7,9,11,13\} with \\

h = \{ (2,7),(3,9),(4,11),(5,13)\} \\

$

Here, the domain of ‘h’ is $\{ 2,3,4,5\} $and co-domain is$\{ 7,9,11,13\} $.

As we know a function is said to be a one-one function if distinct elements of domain mapped with distinct elements of co-domain.

$h = \{ (2,7),(3,9),(4,11),(5,13)\} $

Here, each element from the domain is mapped with the different element from the co-domain. Therefore, ‘h’ is a one-one function.

Now, let us check with the onto condition i.e.., each element in the co-domain has a pre-image from the domain.

Here, each element from the co-domain has a pre-image from the domain. Therefore ‘h’ is

an onto function. As, function ‘h’ is both one-one and onto functions.

Hence, the inverse of ‘h’ exists i.e..,

$

h = \{ (2,7),(3,9),(4,11),(5,13)\} \\

{h^{ - 1}} = \{ (7,2),(9,3),(11,4),(13,5)\} \\

$

Hence, among the functions ‘f’, ‘g’, ‘h’ only the function ‘h’ has the inverse.

Note: The alternate method to find whether a function is one-one is by horizontal line test

i.e. ., if a horizontal line intersects the original function in a single region, the function is a

one-to-one function otherwise it is not a one-one function.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE