Here is how map may be implemented in F Sharp. (At the very least, it is my attempt at implementing it!)
You can run this in the interactive shell.
#light;; let rec my_map fn xs = match xs with | first::rest -> (fn first) :: (my_map fn rest) |  ->  ;; my_map (fun x -> x + 1) [1; 2; 3; 4; 5;];;
val it : int list = [2; 3; 4; 5; 6]