Answer
451.8k+ views
Hint: For solving this problem, we should be aware about the basic properties of functions and relations. In this case, we should be aware about how domains are associated as we carry out the function manipulations. Further, we should know that (fof)(x)=f(f(x)). Thus, this means that we can put the value of f(x) inside the input of f(x) to find the value.
Complete step-by-step answer:
To proceed further, we have,
f(x) = \[\left\{ \begin{align}
& x\text{ for x}\in \text{Q} \\
& 1-x\text{ for x}\notin Q \\
\end{align} \right\}\]
Thus, to find (fof)(x)=f(f(x)), we have to solve this question in two cases.
Case 1: f(x) = x for x$\in $Q
fof(x) = f(f(x)) = f(x) = x
Since, f(x) = x in this case.
Thus, we get, fof(x) = x for x$\in $Q
Case 2: f(x) = 1-x for x$\notin $Q
fof(x) = f(f(x)) = f(1-x) = 1-(1-x) =x
Since, f(x) =1-x in this case.
Thus, we get, fof(x) = x for x$\notin $Q.
Thus, we have,
fof(x)= \[\left\{ \begin{align}
& x\text{ for x}\in \text{Q} \\
& x\text{ for x}\notin Q \\
\end{align} \right\}\]
Thus, we have, fof(x) = x for all numbers in set S. Thus, fof(x) = x for the entire range of [0,1].
Hence, the correct answer is (a) [0,1].
Note: While solving the questions involving functions and relations, we should be careful to keep in mind the domain and range at every point of our calculation. For example, although not observed in this problem, if in one of the intermediate steps, we come across a place where we get $\dfrac{1}{1-x}$ term, then we have to remove x=1 from the solution set since, otherwise the denominator would be zero which is not possible.
Complete step-by-step answer:
To proceed further, we have,
f(x) = \[\left\{ \begin{align}
& x\text{ for x}\in \text{Q} \\
& 1-x\text{ for x}\notin Q \\
\end{align} \right\}\]
Thus, to find (fof)(x)=f(f(x)), we have to solve this question in two cases.
Case 1: f(x) = x for x$\in $Q
fof(x) = f(f(x)) = f(x) = x
Since, f(x) = x in this case.
Thus, we get, fof(x) = x for x$\in $Q
Case 2: f(x) = 1-x for x$\notin $Q
fof(x) = f(f(x)) = f(1-x) = 1-(1-x) =x
Since, f(x) =1-x in this case.
Thus, we get, fof(x) = x for x$\notin $Q.
Thus, we have,
fof(x)= \[\left\{ \begin{align}
& x\text{ for x}\in \text{Q} \\
& x\text{ for x}\notin Q \\
\end{align} \right\}\]
Thus, we have, fof(x) = x for all numbers in set S. Thus, fof(x) = x for the entire range of [0,1].
Hence, the correct answer is (a) [0,1].
Note: While solving the questions involving functions and relations, we should be careful to keep in mind the domain and range at every point of our calculation. For example, although not observed in this problem, if in one of the intermediate steps, we come across a place where we get $\dfrac{1}{1-x}$ term, then we have to remove x=1 from the solution set since, otherwise the denominator would be zero which is not possible.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)