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In multivariable calculus, surfaces are two-dimensional shapes that reside in a three-dimensional space. A function of two variables, such as \( G(x, y) \), defines a surface by assigning a single z-value to every point on the xy-plane. In this exercise, the function \( G(x, y) = -\sqrt{1 + x^2 + y^2} \) describes a surface. Here, the presence of the square root indicates a relationship between x, y, and the z-value. Understanding what a surface is, paves the way for tackling problems involving functions of more than one variable. Surfaces can be complex, but they serve as a fundamental stepping-stone in understanding multidimensional geometry.

The domain and range are vital concepts in understanding functions, especially in multiple dimensions. For a function \( G(x, y) = -\sqrt{1 + x^2 + y^2} \), the domain refers to all possible pairs of \((x, y)\) values that we can input into the function. Here, the domain is all real values of x and y, because the expression \(1 + x^2 + y^2 \geq 0\) is satisfied for all real numbers. This occurs as any real number, when squared, is non-negative. On the other hand, the range of the function is the set of output values, \( G(x, y) \), that result from plugging in all numbers from the domain. Since the function involves a negative square root, all output values are negative or zero, hence the range is \((-\infty, 0] \). This indicates that at no point is \( G(x, y) \) positive.

Sketching graphs in multivariable calculus involves visualizing the surface defined by a function. For the function \( G(x, y) = -\sqrt{1 + x^2 + y^2} \), the graph depicts a three-dimensional surface. To sketch it, we need to understand key features. At the origin, \((0,0)\), the graph has its peak because \( G(x, y) = 0 \) there. This is where \( x^2 + y^2 = 0 \), matching the condition when the expression inside the square root equals one. After determining critical points, we note that because \( G(x, y) \) is a quadratic form inside the square root, the surface opens downwards as x and y increase. The resulting sketch would illustrate a sombrero-shaped surface tilted downwards, centered at the origin in three-dimensional space.

The graph of \( G(x, y) = -\sqrt{1 + x^2 + y^2} \) represents a specific type of surface called a paraboloid. In this case, it's a downward-opening paraboloid. Paraboloids can be thought of as three-dimensional parabolas, resembling a bowl or dish shape either pointing upwards or downwards. This function forms a downward paraboloid because all values of \( G(x, y) \) are non-positive. The center of this paraboloid is at the origin \((0, 0, 0)\), where it achieves its maximum possible value, which is zero. As x and y move away from the center, the value of \( G(x, y) \) decreases, creating the downward shape. Understanding paraboloids is crucial for visualizing and solving problems in multivariable calculus, as they frequently appear in contexts involving quadratic surfaces.