Questions: Functions of Several Variables: Definition and Domain

Question 1

What is the domain of f(x, y) = √(9 − x² − y²)?

• A. All (x, y) such that x² + y² ≤ 9 — the closed disk of radius 3
• B. All (x, y) such that x² + y² < 9 — the open disk of radius 3
• C. All (x, y) such that x² + y² > 9 — the exterior of the circle
• D. All (x, y) with x ≥ 0 and y ≥ 0 — the first quadrant only

Answer: (A)

The square root requires its argument to be non-negative: 9 − x² − y² ≥ 0, which rearranges to x² + y² ≤ 9. This is the closed disk of radius 3 centered at the origin, including the boundary circle x² + y² = 9 (where the function equals 0). The domain is now a 2D region, not an interval — the key shift from single-variable to multivariable calculus.

Question 2

A student claims that f(x, y, z) = x² + y² + z² can be visualized as a surface in 3D space, just like f(x, y) = x² + y² is visualized as a paraboloid surface. What is wrong with this claim?

• A. Visualizing f(x, y, z) as a surface would require a 4th dimension for the output value, which cannot be drawn
• B. f(x, y, z) is not a valid function because functions cannot accept more than two inputs
• C. 3D functions produce vector outputs, not scalar values, so surface visualization does not apply
• D. The function f(x, y, z) = x² + y² + z² is not continuous and therefore cannot be graphed

Answer: (A)

f(x, y) can be visualized as a surface because the two inputs provide the horizontal position (x, y) and the output f provides the height z — three dimensions total. For f(x, y, z), the three inputs already fill 3D space, and the output value would require a 4th axis. The graph would live in ℝ⁴, which cannot be drawn. Functions of three variables instead appear as scalar fields — temperature, pressure, or potential — where the output value is associated with each point in space rather than graphed above it.

Question 3

The domain of g(x, y) = ln(x + y) includes the point (−3, 5).

• T. True
• F. False

Answer: True

The natural logarithm requires a positive argument: x + y > 0. At (−3, 5), we have x + y = −3 + 5 = 2 > 0, so the point is in the domain. The domain of g is the half-plane above the line y = −x, i.e., all points where x + y > 0. Notice that the domain condition defines a region in the plane, not an interval on the number line — the key geometric shift when moving to functions of two variables.

Question 4

A function of two variables f(x, y) maps points in ℝ² to points in ℝ², producing a two-component output.

• T. True
• F. False

Answer: False

A function of several variables, as defined in multivariable calculus, maps n-tuples to a single real number: f: D ⊆ ℝⁿ → ℝ. So f(x, y) takes a point in ℝ² and produces one real number — not a pair of numbers. The function's value can be visualized as a 'height' at each point in the plane. A function that maps ℝ² → ℝ² would be a vector-valued function (or vector field), which is a different object entirely.

Question 5

How does finding the domain of a two-variable function f(x, y) differ from finding the domain of a single-variable function f(x)? What geometric form do the domain restrictions typically take?

Model answer: For a single-variable function, domain restrictions (such as no division by zero, no negative square roots) exclude isolated points or intervals from the real line, producing a domain that is a subset of ℝ¹. For f(x, y), the same types of algebraic restrictions — but applied to expressions involving both x and y — exclude entire curves or regions from the plane ℝ². For example, 'x − y ≠ 0' excludes the line y = x; 'x² + y² ≤ 1' defines a disk. The domain is now a 2D region, and describing it requires specifying curves and inequalities rather than isolated points.

This geometric shift is why multivariable calculus requires visualizing domains as regions in the plane rather than intervals on the number line. The boundary of the domain often plays a crucial role in optimization: a function may achieve its maximum or minimum on the boundary of its domain, not in the interior — which is why understanding domain geometry is foundational for partial derivatives and constrained optimization.