The definition of “into” will call back to my A to Z piece on “bijections”. It particularly call on what mathematicians mean by a function. When a mathematician talks about a functions she means a combination three components. The first is a set called the domain. The second is a set called the range. The last is a rule that matches up things in the domain to things in the range.
We said the function was “onto” if absolutely everything which was in the range got used. That is, if everything in the range has at least one thing in the domain that the rule matches to it. The function that has domain of -3 to 3, and range of -27 to 27, and the rule that matches a number x in the domain to the number x3 in the range is “onto”.