Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Please Solve RD Sharma Class 12 Chapter Function Exercise 2.4 Question 12 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter Function Exercise 2.4 Question 12 Maths Textbook Solution.

Answers (1)

Answer: 

                \left ( gof \right )^{-1}=f^{-1}og^{-1}

Given:

                f:Q\rightarrow Q,g:Q\rightarrow Q defined by f\left ( x \right )=2x,g\left ( x \right )=x+2.

Hint:

                Bijective function should be fulfil the injective and surjective function.

Solution:

x,y be two elements of domain\left ( Q \right ), such

                \! \! \! \! \! \! \! \! \! f\left (x \right )=f\left ( y \right )\\\\2x=2y\\\\x=y

                f is one-one.

Let y be in the co-domain\left ( Q \right ), such that

                f\left ( x \right )=y

                2x =y

                x=\frac{y}{2}\epsilon \: Q.

f is onto.

So, f is a bijection and hence, it’s invertible
Let          f^{-1}\left ( x \right )=y                                                                                                                                           …(i)                                                                       

       \\\\x=f\left ( y \right )\\\\y=\frac{x}{2}\\\\f^{-1}\left ( x \right )=\frac{x}{2}                                                                                                                                                                                                             Injectivity of g

Let  x,y be two elements of domain\left ( Q \right ).

                \! \! \! \! \! \! \! \! g\left ( x \right )=g\left ( y \right )\\\\x+2=y+2\\\\x=y

           So, g is one-one.

Subjectivity of g:

Let y be in the co-domain\left ( Q \right ), such that g\left ( x \right )=y.

                x+2=y

                x=2-y\: \epsilon \: (domain)

g is onto.

So, g is a bijection, hence it’s invertible.

Let          g^{-1}\left ( x \right )=y

                x=g\left ( y \right )                                                                                                                                                         …(ii)

                x=y+2

                y=x-2

So,          g^{-1}\left ( x \right )=x-2                                                                                                                                   [from (ii)]

\left ( gof \right )^{-1}=f^{-1}og^{-1}.

f\left ( x \right )=2x,g\left ( x \right )=x+2,f^{-1}\left ( x \right )=\frac{x}{2},g^{-1}\left ( x \right )=x-2

\left ( f^{-1} og^{-1}\right )\left ( x \right )=f^{-1}\left ( x-2 \right )                                                   \left [ \because \left ( f^{-1} og^{-1}\right )\left ( x \right )=f^{-1}\left ( g^{-1}\left ( x \right ) \right ) \right ]

\left ( f^{-1}og^{-1} \right )\left ( x \right )=f^{-1}\left ( \frac{x-2}{2} \right )                                                                                                              …(iii)

\left ( gof \right )\left ( x \right )=g\left ( f\left ( x \right ) \right )

                   =g\left (2 x \right )

                   =2x+1

Let          \left ( gof \right )\left ( x \right )=y                                                                                                                                 …(vi)

                x=\left ( gof \right )\left ( y \right )

                x=2y+2

                y=\frac{x-2}{2}

                \left ( gof \right )^{-1}=f^{-1}og^{-1}.

Posted by

infoexpert27

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Single Girl Child Scholarship 2024: Registration extended till January 10; eligibility criteria, amount
anam-khan

CBSE wants international boards reined in; letter to education ministry seeks directions for AIU
anam-khan

PM Modi's Pariksha Pe Charcha 2025 to be held next month, registrations open till January 14

Latest Question