1. Let where , , …, are all independent variables. Then the “Domain” of is the set of -tuples . Note that an ordered pair is
The set of corresponding values is the “Range” (or Codomain) of the function. So we have
Sometimes we write for the domain of , and or for the range (or codomain) of . We DO NOT write for the range, because this is the collection of all points mapped by .
Example 1. Consider defined by
Notice that for any . So
This is not the codomain! It’s contained in the codomain, though. The image is always a subset of the codomain.
Example 2. Let . For to be defined, we need
The boundary of the domain is , which we can doodle:
Since the boundary is not in the domain, then the domain of is open.
Example 3. Consider
The domain is the set of such that
We can doodle this:
Observe that the boundary is the circle; the disc is the circle and everything enclosed in it.
2. So we have just discussed domains and codomains, but we have not discussed the graph of the function . What would this look like? Well, when we plot , it’s on the plane . So plotting would be on the 3-space . Lets start considering examples of what this looks like.
Example 4. Consider the graph given by
How can we draw this? Well, the first trick is to draw when and :
These are parabolas in the and planes, respectively. Now we can start drawing “Level Curves”, i.e., curves where we fix to be some constant.
For example, when , we have a circle
Observe then that controls the radius of the circles: for nonzero , we have
The left hand side must be non-negative, and can be zero only when . So we get a surface that looks like
We call this surface a “Paraboloid” as we have parabolas along the – and -axes.
This is the general scheme for picturing a surface: draw level curves for some constant , which produces the level curve for corresponding to .
Example 5. Consider . What does this surface look like?
Solution: First we observe the curves along the -axis, i.e., when is . Similarly when we have . What sort of curves are these? Well, and are the curves, for .
Also note that describes a circle, whose radius happens to be . This tells us the level curves are simply circles. So we have a cone:
This surface is precisely a “Cone”.
Example 6. What if we deform the previous example, writing
What surface does this describe?
Solution: Well, we see that the radius of the level curves deform . So the resulting surface looks like a cone, but along the -axis we don’t have a straight-line: we have a hyperbola . Similarly along the -axis we have another hyperbola . Thus our surface is doodled as:
This “deformed cone” is called a “One Sheeted Hyperboloid”.
Example 7. Another variation, consider the surface
What does it look like?
Solution: Well, we see that always, whereas only when . So we have two “surfaces” or “Sheets” here. The level curves are again circles, and this enables us to doodle the graph:
Along the -axis and -axis, we have hyperbolas. For this reason, we call the surface a “Two-Sheeted Hyperbaloid”.
Example 8. Suppose we have a surface given by . What does it look like?
Solution: Observe the level curves for gives us hyperbolas, and for we again have hyperbolas (the same as before but reflected about the line ). For we have a cone.
When we take we see for some constant . These curves look like smiles. For , we have for some constant . These curves look like frowns.
This surface is a saddle, and we can doodle it as
Since the curves when we hold constant form hyperbolas, and the curves when we hold or constaant are parabolas, people sometimes call this saddle a “Hyperbolic Paraboloid”.