Is there an equation for a Non-Surjective, Strictly Increasing, and Natural Number function?

Question

My best guess is e^x+1, but I'm getting mixed results whether or not that is correct.

Answer 1

If you are asking for a function from the natural numbers to the natural numbers which is not surjective and strictly increasing, then e^x + 1 does not work, because for example e^1 + 1 is not a natural number. A simpler example would be f(x) = x + 1. This function is increasing (plenty of ways to see this, one of which being the derivative, which is 1, and always positive), not surjective (since x+1 never equals 0, when you only plug in other natural numbers), and a function from the natural numbers to themselves.