Map (higher-order function)
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In many programming languages, map is the name of a higher-order function that applies a given function to a sequence of elements (such as a list) and returns a sequence of results. They are examples of both catamorphisms and anamorphisms.
For example, if we define a function square as follows:
square x = x * x
Then calling (map square [1,2,3,4,5]) will return [1,4,9,16,25].
Map itself may be defined recursively as:
map f [] = [] map f (x:xs) = f x : map f xs
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[edit] Language comparison
The map function is especially common in functional programming languages, but some high-level procedural languages support it as well, and in others it may be defined. Common Lisp provides a whole family of map-like functions. The one that corresponds to the behavior described here is called mapcar. In C++'s Standard Template Library, the map function is called transform.
[edit] Optimizations
The mathematical basis of maps allow for a number of optimizations. If one has map f . map g ('.' is function composition) then it is the same as the simpler map (f . g) xs; that is, 
. This particular optimization eliminates an expensive second map by fusing it with the first map; thus it is a "map fusion"[1].
Map functions can and often are defined in terms of a fold such as foldr, which means one can do a "map-fold fusion": f z . map g is equivalent to foldr (f . g) z.
[edit] Haskell's Functor class
In the Haskell programming language, map is generalized to a polymorphic function called fmap using the Functor type class. For every Functor instance, fmap must be defined such that it obeys the Functor laws:
fmap id = id (identity)fmap (f . g) = fmap f . fmap g (composition)
