The basic answer to the question is yes, to plus or minus a couple of notes. As the other answer states, even if the range becomes bigger, the outside sounds may not be particularly pleasant, especially dependent on the words sung. I've lost count of the number of times I've had to change the key of songs in different bands - it often makes a heck of a difference for the vocals, but not much for the instrumentalists. Often the 'original key' is just that - the one the song was written in - sometimes because that's what the writer could do, or where the idea came from, sometimes to just suit the tessitura of the person who it was written for. It doesn't make it the 'right' key. The right key for a singer is the one he sings it best in. Some songs have a small range, so it's no problem, but with others, get used to the fact that a change of key may be needed. Your tutor should be able and happy to do this for you.