What is a transformation? I am not a native English speaker and I have been pointed out that the word "transformation" as a synonym of "function" is grammatically incorrect. However, I even found a wikipedia and a mathworld entries where they use "transformation" as a synonym of function:
http://en.wikipedia.org/wiki/Transformation_(function)
http://mathworld.wolfram.com/Transformation.html
Which one is correct?
 A: Transformation usually refers to an operation that can change the status of an object to another state.
For example, ‘translation’ is a type of transformation that shifts an object from its current position to a new location where the image of the object appears.
However, all these transformational operations can be done via functions.
For example, from the given y = f(x), if we define g(x) = f(x) + k, then g(x) has the power of shifting the graph of y = f(x) k units upward. Thus, g(x) is a function (of the function f(x)) and it can also do the job of translation.
A: All transformations are functions, but it would be wrong to describe most functions as transformations. Typically a transformation operates on a space or other mathematical structure, and preserves at least some of the structure's features. Often (but my no means always) the result of the transformation resides in the original space, or at least in a similar kind of space. Both "transformation" and "function" are nouns, so no grammatical error is committed when substituting one for the other.
(Added) As an example, consider the positive real numbers as a "space", with the logarithmic transformation (any base you like). When a product is transformed, it becomes a sum in the transformed space, which is the whole real line. Structural features of multiplication, such as associativity, commutativity, and continuity, carry over to the corresponding addition. Of course, the logarithm here is a function as well.
