Recall the definition of domain and range for the one variable function *y*
= *f*(*x*):

**Domain (of a one variable function)**: The set of all values
of the independent variable for which the function is defined.

**Range(of a one variable function)**: The set of all possible
values which the dependent variable can take, based on the domain.

Similar definitions hold for functions of many variables. However, instead
of the domain consisting of a portion of the real number line, we must now
consider what portion of *n*-dimensional space the domain of a function of
*n* variables inhabits. Points in *n*-dimensional space are called ordered
*n*-tuples because they must be written down in a specific order. You are
already familiar with this concept. To graph a point in three-space, you
need to specify its coordinates as an ordered triple-(*x*,*y*,*z*). Thus, the
following definitions hold:

**Domain**: The domain of a function of *n* variables is the
set of all *n*-tuples in *n*-dimensional space (also written as ) for which the function is defined.

**Range**: The range of a function of *n* variables is the set
of all possible values of the dependent variable.

For example, the function *z* = *f*(*x*,*y*) = *x ^{2}* +

As another example, consider the function . This
function is only defined if the quantity . Thus, either
or . The domain is then all (*x*,*y*) such that either
or . Since the square root is positive, the range is
again .

What about the function *z* = *e*^{-(x2 + y2)}? Clearly the domain is all
ordered pairs in but what about the range? To determine
this, let's rewrite the function as *z* = 1/*e*^{x2 + y2}. Since and *e* raised to any power is positive, we see that *z* > 0.
Further, *z* = 0 only in the limit as *x* and *y* go to infinity. Thus, *z*
> 0 is part of the range. Is there an upper bound to *z*? The function
obviously decreases in value as *x* and *y* increase, so the maximum value
must occur at *x* = *y* = 0. Here, *z* = 1. The range is then .