$X$ is uniformly distributed over the interval $[-1, 1]$.

Which of the following is the probability density function of $Y = e^X$?

A. $$f_Y(y) = \begin{cases} \frac{1}{2y} & \frac{1}{e} \le y \le e\\ 0 & \text{otherwise.}\\ \end{cases}$$
B. $$f_Y(y) = \begin{cases} 0.0729e^y & \frac{1}{e} \le y \le e \\ 0 & \text{otherwise.}\\ \end{cases}$$
C. $$f_Y(y) = \begin{cases} \frac{1}{2} \ln \, y + \frac{1}{2} & \frac{1}{e} \le y \le e \\ 0 & \text{otherwise.}\\ \end{cases}$$
D. $$f_Y(y) = \begin{cases} \frac{2e^2}{e^4 - 1}y & \frac{1}{e} \le y \le e \\ 0 & \text{otherwise.}\\ \end{cases}$$
E. $$f_Y(y) = \begin{cases} \frac{e}{1 - e^2} & \frac{1}{e} \le y \le e \\ 0 & \text{otherwise.} \\ \end{cases}$$