Let f: R → R and g: R → R be two functions defined by f(x)= log_e (x^2+1)- e^-x+1 and g(x)=(1-2 e^2 x)/(e^x). Then, for which of the following range…

Let $\displaystyle f: \mathbf{R} \rightarrow \mathbf{R}$ and $\displaystyle g: \mathbf{R} \rightarrow \mathbf{R}$ be two functions defined by $\displaystyle f(x)=\log _{\mathrm{e}}\left(x^2+1\right)-\mathrm{e}^{-x}+1$ and $\displaystyle g(x)=\frac{1-2 \mathrm{e}^{2 x}}{\mathrm{e}^x}$. Then, for which of the following range of $\displaystyle \alpha$, the inequality $\displaystyle f\left(g\left(\frac{(\alpha-1)^2}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)$ holds ?