# Give examples of two one-one functions \[{{f}_{1}}\] and \[{{f}_{2}}\] from R to R such that \[{{f}_{1}}+{{f}_{2}}:R\to R\] defined by \[\left( {{f}_{1}}+{{f}_{2}} \right)\left( x \right)={{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] is not one-one.

Last updated date: 18th Mar 2023

•

Total views: 305.7k

•

Views today: 3.88k

Answer

Verified

305.7k+ views

Hint: For this question make \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=\text{constant}\] as y = k is the easiest function which is not one-one. Take \[{{f}_{1}}\left( x \right)\] and \[{{f}_{2}}\left( x \right)\] as linear function in x such that \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] is constant.

Here we have to find two one-one functions \[{{f}_{1}}\] and \[{{f}_{2}}\] R to R such that \[\left( {{f}_{1}}+{{f}_{2}} \right)x={{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] is not one-one.

We know that one-one function is a function that maps distinct elements of its domain to distinct elements of its co-domain that is for a particular value of x, there is a particular value of y and that value of y should not repeat for any other value of x.

Now, we have to make \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] such that it is not one-one.

We know that \[f\left( x \right)=\text{constant}\] is the easiest function which is not one-one because its value of y keeps getting repeated for all values of x.

Therefore, we will choose \[{{f}_{1}}\left( x \right)\] and \[{{f}_{2}}\left( x \right)\] such that

\[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=k\]

Now, we are given that \[{{f}_{1}}\left( x \right)\] and \[{{f}_{2}}\left( x \right)\] must be one-one.

We know that the easiest one-one function is \[y=ax+b:R\to R\] where a and b are constants as it gives different values of y for different values of x.

Therefore, we take \[{{f}_{1}}\left( x \right)=9x+5\].

Now to make \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] constant, 9x must disappear.

Therefore, we take \[{{f}_{2}}\left( x \right)=-9x+8\].

Therefore, we get \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=\left( 9x+5 \right)+\left( -9x+8 \right)\].

\[\Rightarrow {{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=13\]

Therefore, finally we get

\[{{f}_{1}}\left( x \right)=9x+5\]

\[{{f}_{2}}\left( x \right)=-9x+8\]

which are one-one functions.

Also, we get \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=13\] which is not a one-one function.

Note: Students could also check if a function is one-one or not by making the line on the graph of the function which is parallel to the x axis. If this line cuts the graph just 1 time then, it is one-one function, otherwise it is not one-one.

Here we have to find two one-one functions \[{{f}_{1}}\] and \[{{f}_{2}}\] R to R such that \[\left( {{f}_{1}}+{{f}_{2}} \right)x={{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] is not one-one.

We know that one-one function is a function that maps distinct elements of its domain to distinct elements of its co-domain that is for a particular value of x, there is a particular value of y and that value of y should not repeat for any other value of x.

Now, we have to make \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] such that it is not one-one.

We know that \[f\left( x \right)=\text{constant}\] is the easiest function which is not one-one because its value of y keeps getting repeated for all values of x.

Therefore, we will choose \[{{f}_{1}}\left( x \right)\] and \[{{f}_{2}}\left( x \right)\] such that

\[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=k\]

Now, we are given that \[{{f}_{1}}\left( x \right)\] and \[{{f}_{2}}\left( x \right)\] must be one-one.

We know that the easiest one-one function is \[y=ax+b:R\to R\] where a and b are constants as it gives different values of y for different values of x.

Therefore, we take \[{{f}_{1}}\left( x \right)=9x+5\].

Now to make \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)\] constant, 9x must disappear.

Therefore, we take \[{{f}_{2}}\left( x \right)=-9x+8\].

Therefore, we get \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=\left( 9x+5 \right)+\left( -9x+8 \right)\].

\[\Rightarrow {{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=13\]

Therefore, finally we get

\[{{f}_{1}}\left( x \right)=9x+5\]

\[{{f}_{2}}\left( x \right)=-9x+8\]

which are one-one functions.

Also, we get \[{{f}_{1}}\left( x \right)+{{f}_{2}}\left( x \right)=13\] which is not a one-one function.

Note: Students could also check if a function is one-one or not by making the line on the graph of the function which is parallel to the x axis. If this line cuts the graph just 1 time then, it is one-one function, otherwise it is not one-one.

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