Wow, that is quite an interesting vertical motion function. Too bad Humpty-Dumpty didn't live in the world where this physical phenomenon exists. For your function, your egg will never hit the ground. That's because as soon as you drop your egg the force of gravity in your world that is opposite to that experienced where the rest of us live will cause the egg to go upward.
Now if you use the vertical motion function in effect on Planet Earth where I live, that is a different story.
Presumably the ground is at height zero, so:
My calculator said it, I believe it, that settles it