Question

Answer the following question in one word or one sentence or as per exact requirement of the question:
If \[R = \left\{ {\left( {x,y} \right):x,y \in X,{x^2} + {y^2} \leqslant 4} \right\}\], then write the domain and range of R.

Answer 1

Hint: We will first write all the possible elements in our relation R. After that we will look at all the elements component wise. The elements possible to take values in the first place will be the domain and the values on the right is range.

Complete step-by-step answer:
We see that in the ordered pair $(x,y)$ we have integers on both the places. And, in the relation, we are squaring each of the integers.
We see that ${2^2} = 4$ and ${3^2} = 9$. Therefore, the maximum value of x and y can be 2 only.
Similarly, we see that ${\left( { - 2} \right)^2} = 4$ and ${\left( { - 3} \right)^2} = 9$. Therefore, the minimum value of x and y can be -2 only.
Therefore, we have $ - 2 \leqslant x \leqslant 2, - 2 \leqslant y \leqslant 2$.
Hence, the values of x can be taken from the set { -2, -1, 0, 1, 2 } and the values of y can also be taken from the set { -2, -1, 0, 1, 2 }.
Now, let us find the ordered pair. If x = -2, we will get y = 0. Because any other value of y will exceed the inequality in relation. Hence, we have (-2, 0).
Now, if x = -1, y can take the value of 0, 1, -1. Hence, we get (-1, 0), (-1, 1) and (-1, -1).
Now, if x = 0, y can take the value of 0, 1, -1, 2, -2. Hence, we get (0, 0), (0, 1), (0, -1), (0, 2) and (0, -2).
Now, if x = 1, y can take the value of 0, 1, -1. Hence, we get (1, 0), (1, 1) and (1, -1).
If x = 2, we will get y = 0. Because any other value of y will exceed the inequality in relation. Hence, we have (2, 0).
Hence, we have the relation:\[R = \left\{ {\left( {0,0} \right),\left( { - 1,0} \right),\left( {1,0} \right),( - 1,0),(1,0),(0,2),(0, - 2),(2,0),( - 2,0)} \right\}\]
Now, we see there are 9 ordered pairs in the relation.
On the left components, we have 0, -1, 1, 2, -2.
Similarly, the same values occupy the right side.

Hence, Domain of R = Range of R = {-2, -1, 0, 1, 2}.

Note: The students must know the definitions of domain and range. Domain is the set of values we can put in the given function and the values we receive in the output is called range.
A relation is a relationship between 2 sets among which both sets can be the same as well.