F as described in the question is indeed a function, it implies the statement is true.
Now, when a rule is defined in such a way that
(1) every element in the domain is mapped to a "unique" {one and only one value or item} image in the co-domain
(2) Two elements in the domain may have the same image in the co-domain
(3) No element in the domain can have more than one image in the co-domain (which is a mere consequence of item 1 above)
(4) Not all the images n the co-domain may be mapped to an element in the domain;
holds, then such a rule is called a "function"
Hence, we can say a function is a rule which assigns each member in the domain, a unique image in the co-domain.