Question
The domain and range of the relation $$R$$ given by $$R = \left\{ {\left( {x,\,y} \right):y = x + \frac{6}{x};{\text{ where }}x,\,y\, \in \,N{\text{ and }}x < 6} \right\}$$ is :
A.
$$\left\{ {1,\,2,\,3} \right\},\,\left\{ {7,\,5} \right\}$$
B.
$$\left\{ {1,\,2} \right\},\,\left\{ {7,\,5} \right\}$$
C.
$$\left\{ {2,\,3} \right\},\,\left\{ {5} \right\}$$
D.
None of these
Answer :
$$\left\{ {1,\,2,\,3} \right\},\,\left\{ {7,\,5} \right\}$$
Solution :
$$\eqalign{
& {\text{When }}x = 1,\,y = 7\, \in \,N,\,{\text{so }}\left( {1,\,7} \right) \in \,R \cr
& {\text{When }}x = 2,\,y = 2 + 3 = 5\, \in \,N,\,{\text{so }}\left( {2,\,5} \right) \in \,R \cr
& {\text{Again for }}x = 3,\,y = 3 + 2 = 5\, \in \,N,\,{\text{so }}\left( {3,\,5} \right) \in \,R \cr
& {\text{Similarly for }}x = 4,\,y = 4 + \frac{6}{4}\, \notin \,N \cr
& {\text{and for }}x = 5,\,y = 5 + \frac{6}{5}\, \notin \,N \cr
& {\text{Thus, }}R = \left\{ {\left( {1,\,7} \right),\,\left( {2,\,5} \right),\,\left( {3,\,5} \right)} \right\} \cr
& \therefore {\text{ Domain of }}R = \left\{ {1,\,2,\,3} \right\}{\text{ and Range of }}R = \left\{ {7,\,5} \right\} \cr} $$