Are these functions secure PRFs?
Informally, the security of a PRF is defined as follows:
The adversary is allowed to ask questions from its oracle.
The oracle choose one of the functions $F$ or $f$ at the beginning, where $f$ is a truly random function with the same domain and image as $F$.
When the adversary $A$ sends a query on $x$, the oracle returns the answer $F(k,x)$ or $f(x)$, depending on which one it has chosen at the beginning.
We say that $F$ is a secure PRF if $A$ can not distinguish which function the oracle has chosen.
Now whit this explanation, for your first function, the adversary sends a query on a nonzero $x^*$, then  a query on zero, when it receives $k$, it compute $F(k,x^*)$ if it was the same as the answer it has received for $x^*$ from the oracle, it distinguishes that the oracle has chosen $F$.
For the second function, the same attack does not work, because the adversary does not know the whole key yet.