To derive moment generating function(mgf) of $f(x) = 1/2 e^x $ when $x<0$ and $= e^{-2x}$ when $x > 0$,
Since mgf $M(t) = E(e^{tx})$, we get
$E(e^{tx}) = \int_{-\infty}^\infty e^{tx}\cdot f(e^{tx})dx = \int_0^\infty e^{tx}\cdot e^{-2e^{tx}} + \int_{-\infty}^0e^{tx}\cdot {1\over 2}e^{e^{tx}}$
But is there any way to integrte into more simple format?
$$M(t) = E(e^{tx}) = \int\limits_{-\infty}^{0}e^{tx}\cdot 0.5e^{-x}dx + \int\limits_{0}^{\infty}e^{tx}e^{-2x}dx = \frac{0.5}{t-1} + \frac{1}{2-t}$$
Note, that $t>1$ for the first integral to exist and $t<2$ for the second integral to exist. Therefore the mgf exists for $t \in (1,2)$.