**1.** Let where , , …, are all independent variables. Then the **“Domain”** of is the set of -tuples . Note that an ordered pair is

(1)

The set of corresponding values is the **“Range”** (or *Codomain*) of the function. So we have

(2)

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

(3)

Notice that for any . So

(4)

This is not the codomain! It’s contained in the codomain, though. The image isalwaysa subset of the codomain.

Example 2.Let . For to be defined, we need

(5)

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

(6)

The domain is the set of such that

(7)

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

(8)

How can we draw this? Well, the first trick is to draw when and :

(9)

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

(10)

Observe then that controls the radius of the circles: for nonzero , we have

(11)

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

(12)

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

(13)

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”.