Question

The domain and range of a Relation $R = \\{ (x,y):x,y \in N,x + 2y = 5\\}$ is?
A.$\\{ 1,3\\} ,\\{ 2,1\\}$
B.$\\{ 2,1\\} ,\\{ 3,2\\}$
C.$\\{ 1,3\\} ,\\{ 1,1\\}$
D.$\\{ 1,2\\} ,\\{ 1,3\\}$

Answer 1

A domain of a function is a set of all possible values of $x$ for which the function is defined. Range of a function $y$ is a set of all values obtained from the possible values of $x$.

Complete step-by-step answer:
The given relation between two variables $x,y$ is
$x + 2y = 5 \\\ \Rightarrow 2y = 5 - x \\\ \Rightarrow y = \dfrac{{5 - x}}{2} \\\$
It is given that both $x,y \in N$ thus natural numbers $y$ will only exist for the natural number $x$.
Now, $x$ can take values 1, 2, 3, 4, 5…… and so on
For these values of $x$, the values of $y$ will be 2, 1.5, 1, 0.5, 0, -0.5 ….. And so on
But $y$ can have only natural numbers as its value thus only possible values of $y$ are 2, 1 for values of $x$ as 1, 3.
Hence
Domain= {1, 3}
Range = {2, 1}

Note: For finding range of a function, just look for all the possible values of the independent variable $x$ and then find the corresponding values of $y$ which is the range.