Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ f(x, y)=x^{2}+y^{2} $$

Step-by-Step Solution

Answer

The region \(R\) in the \(x y\)-plane that corresponds to the domain of the function \(f(x, y) = x^{2} + y^{2}\) is the entire \(x y\)-plane, as the function accepts all real numbers for \(x\) and \(y\).

Step 1: Identify the function

The given function is \(f(x, y) = x^{2} + y^{2}\). This is a 2-variable function where both \(x\) and \(y\) are squared and added together.

Step 2: Understand properties of squared numbers

When any real number is squared, the result is always a non-negative value. It is either zero (when the original number is zero) or positive (when the original number is not zero).

Step 3: Analyze the domain of the given function

Given that the function squares both \(x\) and \(y\), and any real number can be squared, it follows that the domain of \(f(x, y)\) must be all real numbers for both \(x\) and \(y\). This includes both positive and negative numbers, as well as zero.

Step 4: Describe the region \(R\) in the \(x y\)-plane

Region \(R\), which describes the domain of \(f(x, y)\), is the entire \(x y\)-plane. This is because any point with coordinates \((x, y)\) in the \(x y\)-plane, regardless of where it is located, would produce a valid output when plugged into the function \(f(x, y) = x^{2} + y^{2}\).