I don't understand this statement. f: A -> A doesn't require that A be the image but only the codomain, so for example the function would conform to f: A -> A.
But I also don't see the difficulty in finding a bijective function. Just write down
...
and fill in with values from A such that you use every single element, then it will be bijective. (Sorry the rectangles are a bit uncentered vertically, couldn't find the proper LaTeX.)
The word `into' is sometimes used to mean `injection' (much like `onto'). When there exists an injection, f, from A to B then it means you can find a copy of A in B, and so f maps A into B.
I would also presume, as did undefined, that 1A means the identity function ( or ) as this would be too easy an answer and noone would exclude any other bijection.
Now, bijections between finite sets are just permutations. So, permute your elements. For example,
will be a function which switches 1 and 2. It is thus a permutation, as it permutes these two elements and keeps everything else fixed.