Question

What is the domain of the multivariable function f(x, y) = √(x² + y²)?

a) All real numbers
b) x ≥ 0, y ≥ 0
c) x ≠ 0, y ≠ 0
d) x > 0, y > 0

Answer 1

The domain of the function f(x, y) = √(x² + y²) includes all real numbers for both x and y since the expression under the square root is always non-negative, therefore the correct option is (a) All real numbers.

Step-by-step explanation:

The domain of a multivariable function is the set of all possible input values (x, y) that the function can accept. For the function f(x, y) = √(x² + y²), the expression under the square root, x² + y², is always non-negative because the square of any real number, whether it's positive or negative, is non-negative. Therefore, there is no restriction on the values of x and y based on the square root operation.

Square roots are defined for non-negative numbers, which means that for any real values of x and y, the expression inside the square root will be non-negative, and thus the square root will be defined. The correct answer is that the domain of f(x, y) is all real numbers for both x and y, as there are no restrictions on their values to avoid taking square roots of negative numbers. Hence, the correct option is (a) All real numbers.