I think some of the issue stems from the incompleteness of the description. We don't say functions are into or onto without describing the set as well. y = x^2 is onto the non-negative real numbers, but y = x^2 is not onto the set of real numbers.
Most of the time we take the set for granted because we think we're implying the set of real numbers, but for the sake of completeness the set must be included in our wording and explanation.
Steve Goldman Half Hollow Hills HS East Dix Hills, NY