Density function of generation function $M_X(t)=\frac12(1+e^t)$

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Given the moment generation function $M_X(t)=(1+e^t)/2$. I try to find the density function, I used the formula $M_X(t)=E(e^{tX})$ but after that I have no idea of what I should do.

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A Bernoulli Distribution with parameter $0<p<1$ has moment generating function $$m(t)=(1-p)+pe^t$$ So if $p=0.5$, then...