Function notation
We have already used the notation that R stands for the real numbers.
(Especially on the blackboard, we often use the notation 
.) Similarly, R2 is a
two-dimensional vector, and R3 is a three-dimensional vector.
Scalar-valued functions
In one-variable calculus, you worked a lot with one-variable functions, i.e., functions from R onto R. If f(x) is such a one-variable functions, we can write f : R → R as a shorthand way of expressing that f is a function from R onto R.
A function like f(x,y) = x + y is a function of two variables. It takes an element
of R2, like (2, 1), and gives a value that is a real number (i.e., an element of R),
like f(2, 1) = 3. Since f maps R2 to R, we write f : R2 → R. We can also use this
“mapping” notation to define the actual function. We could define the above
f(x,y) by writing f : (x,y)
x + y.
To contrast a simple real number with a vector, we refer to the real number as a scalar. Hence, we can refer to f : R2 → R as a scalar-valued function of two variables or even just say it is a real-valued function of two variables.
Everything works the same for scalar valued functions of three or more variables. For example, f(x,y,z), which we can write f : R3 → R, is a scalar-valued function of three variables.
Vector-valued functions
In contrast, a vector-valued function takes on values that are vectors. First, let’s talk about vector-valued functions of a single variable.
A vector-valued function in two dimensions can be written f : R → R2. An example is f(t) = (3t,-t). For a given real number, which we’ll denote by ♣ for fun, f(♣) is the two-dimensional vector (3♣,-♣). Similarly, a vector-valued function in three dimensions can be written f : R → R3. For example, if f(s) = (1 - s,s3, cos s), then f(0) = (1, 0, 1). We sometimes write vector-valued functions using the standard unit vector i, j, and k, as in f(s) = (1 - s)i + s3j + (cos s)k.
Lastly, we can have vector-valued functions of multiple variables. For example, a function could take values in R3, say (x,y,z), and map them to R2, such as f(x,y,z) = (x - y,x22∕z). We can write a function from R3 to R2 as f : R3 → R2. You get the idea.
The domain of a function
The function f(t) = (t,t2) is defined over all real numbers R, i.e., the domain of the function is R. Sometimes a function of one variable may be defined over a subset of real numbers, say some set U ⊂ R; in this case, the domain of the function is U. (Note, the symbol “⊂” just means “is subset of”.) In three dimensions, for example, we can specify the domain by writing f : U ⊂ R → R3, or simply f : U → R3.
Example: since log t isn’t a real number for t ≤ 0, the domain of f(t) = (log t)i + tj, is the set D = (0,∞). We could write this f : (0,∞) → R2. What would the domain be if we replaced log t with log(t - 3) or log(2 - t)? You have to think where log(t - 3) or log(2 - t) is a real number, i.e., where t - 3 > 0 or where 2 - t > 0.
We use the same notation for functions of multiple variables. If we wrote f : U ⊂ R2 → R3, we would mean a function maps values in a subset U of R2 to values in R3,
