I have a problem and I think I know how to solve it so here it is: Determine $F(x)$ if, for all real $x$ and $y$, $F(x)F(y)-F(xy) = x+y$.
I tried a couple of cases and these were my results:
When $x=0$ and $y=0$ then $F(0) = 1$. When $y=0$ and $x$ is anything real, then $F(x) = 1 + x$ (or 1 + y if we did with $x=0$). So it seems like $F(x) = x+1$ works but I don't know if that is enough. Any hints or help would be greatly appreciated!
As David notices, the $x=y=0$ case can also be satisfied by $F(0)=0$, and you need to show that this is not in fact possible.
Except for this gap, what you have is enough to prove that $F(x)=x+1$ is the only possible function that can satisfy the equation.
However you also need to show that $F(x)=x+1$ is in fact a solution, by substituting it into the equation and noticing that you get an identity after simplifying.