What are the multiplicative functions?

Ala Paul Solmon’s problem, what if we threw this in front of kids?

Our mystery function has the following property: f(a*b) = f(a)*f(b)

Find 3 functions that couldn’t possibly be our function. Find 3 functions that might be our function.My thought behind this problem is that (a) this problem would be fun for a bit, if difficult and (b) could serve as a good introduction to multiplying/simplifying radicals, drawing out the fact that it’s actually pretty unusual that radicals have this property.

But the problem is too open-ended. Anyone have an idea of how to make this a little bit more user-friendly?

Update: Might this also be the beginning of a problem?

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