Vector-Outputing Function

Context:
- range: a Vector Set (possibly from a Vector Space).
- It can range from being a Continuous Vector-Valued Function to being a Discontinuous Vector-Valued Function.
- It can range from being a Vector-Input Vector-Output Function, to being a Tuple-Input Vector-Output Function, to being a Numeric-Input Vector-Output Function, to being a Set-Input Vector-Output Function, to being a Category-Input Vector Output Function.
- It can range from being a Vector-Valued Real Function ...
Example(s):
- [math]\displaystyle{ f(1.5) \rightarrow (3.1, 2.3) }[/math].
- [math]\displaystyle{ f(1.5, 7.1) \rightarrow (0.491, 9.223) }[/math].
- [math]\displaystyle{ f(7, \text{Red}, e) \rightarrow (3.4, 1.1, π) }[/math].
- a Vector Field Function, for some vector field.
- a Radial Basis Function.
- a Infinite-Dimensional-Vector-Valued Function.
- …
Counter-Example(s):
- a Scalar-Valued Function, such as [math]\displaystyle{ f }[/math](1.1,Yellow,3.9) ⇒ π.
- a Category-Valued Function, such as [math]\displaystyle{ f }[/math](1.1,2.3,3.9) ⇒ Red,
- a Tuple-Output Function.
See: Vector-Input Function, Parametric Equation, Scalar-Output Function, Euclidean Vector.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/vector-valued_function Retrieved:2015-2-5.
- A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector. The dimension of the domain is not defined by the dimension of the range.

http://ltcconline.net/greenl/courses/202/vectorFunctions/vecfun.htm
- A vector valued function is a function where the domain is a subset of the real numbers and the range is a vector.
- In two dimensions: r(t) = x(t)i + y(t)j
- In three dimensions: r(t) = x(t)i + y(t)j + z(t)k
- You will notice the strong resemblance to parametric equations. In fact there is an equivalence between vector valued functions and parametric equations.

http://ltcconline.net/greenl/courses/202/vectorIntegration/vectorFields.htm#fields
- We have now seen many types of functions. They are characterized by the domain and the range.
- Below is a list of some of the functions that we have encountered so far.

(de Boor et al., 2007) ⇒ Carl de Boor, Allan Pinkus, and Vilmos Totik. (2007). “Concepts of Approximation Theory.” In: Surveys in Approximation Theory Web Site, July 21, 2007.
- QUOTE: radial basis function is a function in [math]\displaystyle{ \R^d }[/math] of the form [math]\displaystyle{ x \longmapsto g(\parallel \mathbf{x} − \mathbf{a} \parallel) }[/math] where [math]\displaystyle{ g }[/math] is a univariate function, [math]\displaystyle{ \mathbf{a} }[/math] is a point in [math]\displaystyle{ \R^d }[/math], and [math]\displaystyle{ \parallel·\parallel }[/math] denotes the Euclidean norm in [math]\displaystyle{ \R^d }[/math].