Answer

Verified

447.6k+ 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

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

How many crores make 10 million class 7 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE