If a core was made of the entire earth from one side to the other, and the density of the material measured, what type of function would it trace? Constant, linear, log?

This is more of a factual question...

You should know for one that denser materials tend to sink in less dense materials.

As such, you'll understand that metals, being the densest materials will be found at the centre of the Earth.

Hm... now that you know this, can you give a guess of how the density will vary as radius increases?