Moment generating functions and exponential problem

Question

Suppose $X,Y$ are random variables corresponding to mgfs $M_X(t)$ and $M_Y(t)$. Is there a function of X and Y such that its mgf is $M_X(t)+M_Y(t)$?

I think the question eventually becomes if I can find a function $f(X,Y)$, say, such that $e^{tX}+e^{tY}=e^t{f(X,Y)}$. Using some simple tricks I will have $f(X,Y)=\frac{ln(e^{tX}+e^{tY})}{t}.$ This is not at all helpful though, since it also relies on $t$. So I have no idea if there is something else I could do about it?

Answer 1

Sum of two moment generating functions can never be a moment generating function. This is because any moment generating function had the value $1$ at $0$.

Answer 2

Without any further rearrangement, if you plug in $t=0$ at the left hand side of your first equation you will see that such a function cannot exist.