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Question: Given \(R = \\{ (x,y):x,y \in W,{x^2} + {y^2} = 25\\} \) , where \(W\) is the set of all whole numbe...

Given R=(x,y):x,y∈W,x2+y2=25R = \\{ (x,y):x,y \in W,{x^2} + {y^2} = 25\\} , where WW is the set of all whole numbers. Find the domain and range of RR

Explanation

Solution

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 xx and yy belong to a set of whole numbers. These whole numbers can only be a pair of digits whose squares add up to give 2525 . 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)(x,y) that satisfy the given equation. For the sum to be 2525 there are only two possibly pairings whose squares can add up to give that;
1.(5,0)(5,0) and (0,5)(0,5) is the first pairing that satisfies the given equation;
⇒x2+y2=25\Rightarrow {x^2} + {y^2} = 25
⇒52+02=25\Rightarrow {5^2} + {0^2} = 25
And
⇒02+52=25\Rightarrow {0^2} + {5^2} = 25
So in this case the two values for domain are 0,5\\{ 0,5\\} and the values for range are also 0,5\\{ 0,5\\}

2.(3,4)(3,4) and (4,3)(4,3) is the second pairing that satisfies the given equation. This is also the known as the Pythagorean triplet 3,4,5\\{ 3,4,5\\} ;
⇒x2+y2=25\Rightarrow {x^2} + {y^2} = 25
⇒32+42=25\Rightarrow {3^2} + {4^2} = 25
⇒9+16=25\Rightarrow 9 + 16 = 25
And
⇒42+32=25\Rightarrow {4^2} + {3^2} = 25
⇒16+9=25\Rightarrow 16 + 9 = 25
So in this case the two values for domain are 3,4\\{ 3,4\\} and the values for range are also 3,4\\{ 3,4\\}

Therefore the values that satisfy the function of xx are 0,3,4,5\\{ 0,3,4,5\\} . These are the values of the domain.
The values that are the set of outputs of the function or yy are 0,3,4,5\\{ 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.