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).

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 = πr^{2}h). 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

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

- One variable function: f(x) = x
^{2} - Two variable function: f(x, y) = x
^{2}+ 2y

## 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:

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 – x^{2} – y^{2})

If (x^{2} – y^{2}) equals anything less than 9, then the value under the square root becomes negative. Therefore, the domain here is all reals except where (x^{2} – y^{2}) ≤ 9. In more formal notation, you can write that as:

The domain of f(x, y) = {(x, y) ∈ ℝ | x^{2} – y^{2} ≤ 9}

Where

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

## References

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

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!