Understanding of a function

1. A function : A relation between the number of one set and the other set.Or changing a number into another number. 

 

Which i drew for helping myself understand the notation of a function. In the photo, i drew a set named X, and the other set named Y. X has two elements to be mapped to two different elements in Y respecticely. So I can say that for any element you give me in the first set(X), I'll give you a member of y associated with that members of x.

 

2. Domain, Codomain : Domain is a set that you're mapping from. Codomain is a set that you're mapping to. With reference to the above photo, X can be a domain and Y can be a codomain. 

 

3. The difference between codomain and range : Codomain is, as i said, a set that you're mapping to. And also it means that codomain is a set that can be possibly mapped to. You don't need to map to every point in codomain Y. Range means a subset of the codomain that a function actually maps.

 

4. Scalar valued function, Vectors valued function : Depending on which dimention you are mapping to, they can be distinguished. If you map to a one-dimentional value(R), it's called a Scalar valued function. If a set is mapping to two or three-dimentional(or more) values, then it's called a Vector valued function.