Calculus How To

Function of Two Variables

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Types of Functions >

A function of two variables

The formal definition of a function of two variables is similar to the definition for single variable functions.

A function of two variables f(x, y) has a unique value for f for every element (x, y) in the domain D.

has two inputs (independent variables).

map of functions

A two variable function (bottom) maps a coordinate point (x, y) to a single value (z). The one variable function (top) maps one input (x) to one output (y).

Many common functions have two inputs, including:

  • Area of a rectangle with two sides l and w: A = lw
  • Work done by force F with displacement d: W = Fd,
  • Volume of a right circular cylinder with radius r and height h (V = πr2h). Note that π is a constant and doesn’t count here as a variable.

While a single variable function maps the value of one variable to another, a function of two variables maps ordered pairs (x, y) to another variable.


Notation for a function of two variables is very similar to the notation for functions of one variable. For example:

How to Find the Domain of a Function of Two Variables

The domain is the set of points where the function is defined. Some authors will specify the domain; Many times (especially in real life) you’ll have to figure out what makes sense.

A good starting point is to assume that the domain is all real numbers (from -∞ to ∞), then look for areas where the function doesn’t work. This is where a good knowledge of algebra will come in handy, but if you’re rusty— here are a couple of basic steps (which will catch most of the undefined areas).

Step 1: Ask yourself: Where does this make sense? If a formula is given, like A = lw, then
assume that the domain is defined for points that make sense, unless a specific domain is given. What “makes sense” will depend on the specific situation, but here are a couple of examples:

  • Area only makes sense for values of length and width greater than 0.
  • Volume only makes sense for positive valued inputs.

Step 2: Look for division by zero.
If a formula has x in a denominator, like this one:
function of two variables

Then figure out what would make the denominator zero. The function will be undefined at those points. For this particular formula, the domain is all real numbers (-&infin, ∞) except for x = ±2

Step 2: Look for values that will make quantities under a square root negative. For example:
f(x, y) = √(9 – x2 – y2)
If (x2 – y2) equals anything less than 9, then the value under the square root becomes negative. Therefore, the domain here is all reals except where (x2 – y2) ≤ 9. In more formal notation, you can write that as:
The domain of f(x, y) = {(x, y) ∈ ℝ | x2 – y2 ≤ 9}

  • ℝ (doublestruck R) = the set of all real numbers,
  • ∈ = “is in the set of”.


Larson, R. & Edwards, B. Multivariable Calculus. Houghton Mifflin.

Stephanie Glen. "Function of Two Variables" From Calculus for the rest of us!

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