To find the CDF, we integrate the density function:
$$F(x)=\int_{-\infty}^{x} f(t) dt$$
For $x<0$:
$$F(x)=\int_{-\infty}^{x} e^{2t} dt = \frac{1}{2}e^{2x}$$
For $x>0$:
$$F(x)=\int_{-\infty}^{0} e^{2t} dt + \int_{0}^{x} e^{-2t} dt = \frac{1}{2} -
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