JEE Math Practice Question

Let the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined by \[ f(x)=e^{x-1}-e^{-|x-1|} \quad \text { and } \quad g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right) \] Then the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is

1. A.$(2-\sqrt{3})+\frac{1}{2}\left(e-e^{-1}\right)$
2. B.$(2+\sqrt{3})+\frac{1}{2}\left(e-e^{-1}\right)$
3. C.$(2-\sqrt{3})+\frac{1}{2}\left(e+e^{-1}\right)$
4. D.$(2+\sqrt{3})+\frac{1}{2}\left(e+e^{-1}\right)$