**Domain of a function of two or more variables**
#[The

**natural domain** of a function $f$ of two or more variables $(x,y,...)$ is the set of pairs $(x,y,...)$ for which the expression $f(x,y,...)$ makes sense.][El

**dominio natural** de una función $f$ de dos o más variables $(x,y,...)$ es el conjunto de pares $(x,y,...)$ para los cuales la expresión $f(x,y,...)$ tiene sentido.]#

**#[Examples][Ejemplos]# (2 variables)**
**1. ** \t $f(x,y) = \dfrac{1}{x-y}$ makes sense for all $(x,y)$ for which $x-y \ne 0$, so the natural domain of $f$ is the set of all points other than those on the line $x-y = 0$ or $x = y$, shown as the shaded region in the following figure:

**2. ** $f(x,y) = \sqrt{x^2+y^2-1}$ makes sense for all $(x,y)$ for which $x^2+y^2-1 \ge 0$, so the natural domain of $f$ is the set of all points $(x,y)$ with $x^2+y^2-1 \ge 0$, or $x^2+y^2\ge 1$, shown as the shaded region in the following figure:

**#[Real world situations][Situaciones del mundo real]#**
In many applications, the variables $x, y, ...$ represent numbers of items, and so cannot be negative. Further, if the variables represent the numbers of items sold or manufactured, there would also be some kind of upper limit on the total number of items, or the number of each item.

**#[Example][Ejemplo]#**
The cost of preparing $x$ bouquets and $y$ boxes of chocolates in a day is $C(x,y) = 4x+3y+ 100$ dollars, and you can prepare up to 10 bouquets and up to 20 boxes of chocolates in a day, then the domain of $C$ is the set of all pairs $(x,y)$ with $0 \leq x \leq 10$ and $0 \leq y \leq 20$. (By contrast, the natural domain of $C$ would consist of all possible pairs $(x,y)$.)