Image and preimage of a recursive function on a recursive set

Question

I am learning about recursive functions and recursive sets, but I cannot find an answer to my question in my textbook. I think it is somehow a general stuff.

Let $R$ be a recursive subset of $\Bbb N$ and $f$ be a recursive function $\Bbb N→\Bbb N$.

Now I wonder if we can prove that :

(i) Must the set $f[R]$ be recursive, recursively enumerable, or neither?

(i) Must the set $f^{−1}(R)$ be recursive, recursively enumerable, or neither?

Thanks for any help and reference.

Answer 1

Both are recursively enumerable (why?) and the inverse image is in fact recursive. This is because we can decide whether $f(n)\in R$ or not within finite time.

However the image is not recursive in general, even if $R=\mathbb{N}$; for example, consider the function which enumerates the halting set. Its image is the halting set, which is not recursive.