finding the MGF of $-X$ given the MGF of $X$

Question

Given the following MGF for an rv $X$ on the interval $-1 < t < 1$:

$$M_X(t) = \frac{e^{t+1} - 1}{(t+1)(e-1)}$$

Determine the MGF of $Y = -X$

any help would be appreciated!

Answer 1

$$ M_{-X}(t) = \operatorname{E}(e^{t(-X)}) = \operatorname{E}(e^{(-t)X}) = M_X(-t). $$