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A function f: D → ℝ maps points (x, y) or (x, y, z) in domain D to real numbers. The domain is the set of inputs where f is defined; the range is the set of outputs. Domains in ℝ² are regions (open, closed, or neither).
From your introduction to multivariable functions, you know that a function of several variables takes a point in the plane (or space) as input and returns a real number. Now the question is: for which points is the function actually defined? The domain is the set of all input points where the formula makes sense — no division by zero, no square roots of negative numbers, no logarithms of non-positive numbers. For a function of one variable, the domain is usually an interval on the number line. For a function of two variables, the domain is typically a region in the plane — a two-dimensional subset of ℝ².
Finding the domain of f(x, y) means identifying every constraint the formula imposes and intersecting the sets where each constraint is satisfied. For f(x, y) = √(1 − x² − y²), the square root requires 1 − x² − y² ≥ 0, which means x² + y² ≤ 1 — the closed unit disk. For f(x, y) = 1/(x − y), the denominator requires x ≠ y — the plane minus the diagonal line. The range is then the set of output values f actually achieves on its domain, which can require more work to determine: for the disk example, z = √(1 − x² − y²) ranges from 0 (on the boundary circle) to 1 (at the origin), so the range is [0, 1].
The geometric distinction between open and closed domains matters for limits and continuity. A domain is open if every point in it has a small disk entirely contained in the domain — there are no boundary points included. It is closed if it contains all its boundary points. It can be neither (like a half-open interval in one variable, extended to two dimensions). This matters because functions on closed, bounded (compact) domains are guaranteed to attain their maximum and minimum values — the two-dimensional version of the Extreme Value Theorem you know from single-variable calculus.
Understanding domain and range sets you up for studying level curves and contour maps — the graphs of the equation f(x, y) = c for different constants c. A level curve lives entirely within the domain (assuming c is in the range), and reading a set of level curves gives geometric insight into how f varies over its domain. You will also need precise domain language when you define limits and continuity in two variables, because the notion of "approaching a point" depends on the topology of the domain — you can approach along many different paths, not just from left or right as in one dimension.