Recognize a function of two variables and identify its domain and range.
Sketch a graph of a function of two variables.
Sketch several traces or level curves of a function of two variables.
Recognize a function of three or more variables and identify its level surfaces.
Our first step is to explain what a function of more than one variable is, starting with functions of two independent variables. This step includes identifying the domain and range of such functions and learning how to graph them. We also examine ways to relate the graphs of functions in three dimensions to graphs of more familiar planar functions.
Functions of two variables
The definition of a function of two variables is very similar to the definition for a function of one variable. The main difference is that, instead of mapping values of one variable to values of another variable, we map ordered pairs of variables to another variable.
Definition
A
function of two variables
maps each ordered pair
in a subset
of the real plane
to a unique real number
The set
is called the
domain of the function. The
range of
is the set of all real numbers
that has at least one ordered pair
such that
as shown in the following figure.
The domain of a function of two variables consists of ordered pairs
Determining the domain of a function of two variables involves taking into account any domain restrictions that may exist. Let’s take a look.
Domains and ranges for functions of two variables
Find the domain and range of each of the following functions:
This is an example of a linear function in two variables. There are no values or combinations of
and
that cause
to be undefined, so the domain of
is
To determine the range, first pick a value for
We need to find a solution to the equation
or
One such solution can be obtained by first setting
which yields the equation
The solution to this equation is
which gives the ordered pair
as a solution to the equation
for any value of
Therefore, the range of the function is all real numbers, or
For the function
to have a real value, the quantity under the square root must be nonnegative:
This inequality can be written in the form
Therefore, the domain of
is
The graph of this set of points can be described as a disk of radius
centered at the origin. The domain includes the boundary circle as shown in the following graph.
The domain of the function
is a closed disk of radius 3. To determine the range of
we start with a point
on the boundary of the domain, which is defined by the relation
It follows that
and
If
(in other words,
then
This is the maximum value of the function. Given any value
c between
we can find an entire set of points inside the domain of
such that
Since
this describes a circle of radius
centered at the origin. Any point on this circle satisfies the equation
Therefore, the range of this function can be written in interval notation as
