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The domain of a function is a crucial concept in mathematics, especially in multivariable calculus. It essentially consists of all possible input values that can be used in a function without causing any undefined situations, such as division by zero.
For multivariable functions like \(V(x, y) = 4x^2 + y^2\), the domain includes all pairs \((x, y)\) that make the expression valid. In this particular exercise, there are no restrictions such as square roots of negative numbers or division by zero. Thus, the domain is all real numbers for both \(x\) and \(y\).
Therefore, we say the domain is the set of all ordered pairs \((x, y)\) where \(x\) and \(y\) are elements of the set of real numbers.

Real numbers are the foundation upon which the concept of the domain is built. They include all numbers that exist on the number line, encompassing both rational numbers, like fractions and whole numbers, and irrational numbers, like the square root of 2.
In calculus, dealing with real numbers is common since they allow us to find precise locations on a line or plane. Functions like \(V(x, y) = 4x^2 + y^2\) utilize real numbers because they provide a broad understanding of continuous change and can apply to real-world scenarios without limitations.

• Rational numbers include integers and fractions.
• Irrational numbers cannot be expressed as fractions, like \(\pi\).

Every input in our function must be real to maintain the integrity of mathematical operations.

Polynomials play key roles in expressions and functions, as they are simple enough to be widely understood yet versatile enough to model complex scenarios. A polynomial is an algebraic expression made up of variables and constants combined using addition, subtraction, multiplication, or non-negative integer exponents.
In the function \(V(x, y) = 4x^2 + y^2\), each term represents a polynomial. This particular function is actually a sum of polynomials. The polynomial nature of \(4x^2\) and \(y^2\) ensures they are always defined, because polynomials do not include operations that create undefined situations such as division by zero. Therefore, they are defined for all real numbers.

• Each term is individually defined everywhere in the real number system.
• Polynomials are smooth and continuous, with no breaks or abrupt changes.

This quality is why the domain of polynomial functions is often easily determined.