A mathematical (Vector Calculus) term.
A vector-valued function (also called a vector function) is simply any function that has a vector as its output (range). That is (limiting ourself to the Real Numbers) a function that maps Rn -> Rm, where n > 0, m > 1. This is opposed to your normal scalar function that outputs a single value (m = 1, for instance f(x) = x + 2)
There are two main notations for saying a function is vector-valued. First is to draw some form of an arrow on top of the function's name (think f with a -> on top of it), useful for doing work on paper. Second is to write the name of the function in bold (f), useful for functions written using a keyboard.
An example
f(x) = (x + 1, x + 2)
f(2) = (3, 4)
Some other notational notes:
Wntrmute says just a couple of other notations for a vector valued function : in my degree course we tend to denote a vector-valued function by simple underlining of the function symbol, as it's easier than drawing an arrow! This may be a european trait, or just a UK thing (lecturer was austrian). Also, it's quite common to denote a vvf with a capital letter (usually F) then the scalar components relative to the coordinate system in lower case with a subscript (e.g. F(x,y,z)= f1(x,y,z)i + f2(x,y,z)j + f3(x,y,z)k in 3-space; thus dispensing with bold or underlining entirely if you know the coordinate system.
(FYI the i, j, and k in 3d vectors represent three special unit vectors. Particularly i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1). Also, capital letters are commonly used to denote vectors.)
Note on the arrow notation: At least in my calc classes we've shorthanded the arrow by making it look like this __\ (the line and top of the arrow).
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