Function notation

We have already used the notation that R stands for the real numbers. (Especially on the blackboard, we often use the notation $\mathds{R}$.) 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$\to$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$\to$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) $\mapsto$ 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$\to$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$\to$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$\to$R². An example is f(t) = (3t, - t). For a given real number, which we'll denote by $\clubsuit$ for fun, f($\clubsuit$) is the two-dimensional vector (3$\clubsuit$, - $\clubsuit$). Similarly, a vector-valued function in three dimensions can be written f : R$\to$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$\to$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 $\subset$ R; in this case, the domain of the function is U. (Note, the symbol "$\subset$" just means "is subset of".) In three dimensions, for example, we can specify the domain by writing f : U $\subset$ R$\to$R³, or simply f : U$\to$R³.

Example: since log t isn't a real number for t$\le$ 0, the domain of f(t) = (log t)i + tj, is the set D = (0,$\infty$). We could write this f : (0,$\infty$)$\to$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 $\subset$ R²$\to$R³, we would mean a function maps values in a subset U of R² to values in R³,