Question

Describe the domain and range of the function. $$ f(x, y)=x^{2}+y^{2} $$

Short answer

The domain of the function \(f(x, y)=x^{2}+y^{2}\) is all real numbers (\(x,y∈R\)), and the range is all non-negative real numbers (\(f(x, y)≥0\)).

Step-by-step solution

Determine the Domain

The domain of a function encompasses all possible input values. The function \( f(x, y)=x^{2}+y^{2} \) will be defined for all real numbers for \(x\) and \(y\) because squaring a real number, either it's positive, negative or zero, always gives a real number. Thus, the domain of the function is all real numbers: \(x,y∈R\).

Determine the Range

The range of a function represents all possible output values. In this case, because the function \( f(x, y)=x^{2}+y^{2} \) squares the input values, the minimum possible value of the function is 0 (when \(x = 0\) and \(y = 0\)), and the maximum possible value is infinity. Moreover, as the square of a real number is always non-negative, there will be no negative number in the range. Thus, the range of the function is all non-negative real numbers, or \(f(x, y)≥0\).