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Given $R = \{ (x,y):x,y \in W,{x^2} + {y^2} = 25\} $ , where $W$ is the set of all whole numbers. Find the domain and range of $R$

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Last updated date: 03rd Mar 2024
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Hint: The domain of a function is a set of all possible inputs for a function under certain restrictions. In this case, the given restriction is that both $x$ and $y$ belong to a set of whole numbers. These whole numbers can only be a pair of digits whose squares add up to give $25$ . This provides us with the scope for another restriction: that both digits have to be less than or equal to 5 for the given restriction to be feasible.
The range of a function is the set of all output values of a function.

Complete step by step solution:To find the domain and range of such a function, we have to find the list of ordered pairs $(x,y)$ that satisfy the given equation. For the sum to be $25$ there are only two possibly pairings whose squares can add up to give that;
1.$(5,0)$ and $(0,5)$ is the first pairing that satisfies the given equation;
$ \Rightarrow {x^2} + {y^2} = 25$
$ \Rightarrow {5^2} + {0^2} = 25$
And
$ \Rightarrow {0^2} + {5^2} = 25$
So in this case the two values for domain are $\{ 0,5\} $ and the values for range are also $\{ 0,5\} $

2.$(3,4)$ and $(4,3)$ is the second pairing that satisfies the given equation. This is also the known as the Pythagorean triplet $\{ 3,4,5\} $ ;
$ \Rightarrow {x^2} + {y^2} = 25$
$ \Rightarrow {3^2} + {4^2} = 25$
$ \Rightarrow 9 + 16 = 25$
And
$ \Rightarrow {4^2} + {3^2} = 25$
$ \Rightarrow 16 + 9 = 25$
So in this case the two values for domain are $\{ 3,4\} $ and the values for range are also $\{ 3,4\} $

Therefore the values that satisfy the function of $x$ are $\{ 0,3,4,5\} $ . These are the values of the domain.
The values that are the set of outputs of the function or $y$ are $\{ 0,3,4,5\} $ . These are the values of the range.

Note:
In these questions, we need to keep the restrictions in mind before we decide which numbers to use for domain and range. Remember that very few pairs are possible with a square this small so the domain and range are not that big.
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