JEE Math Practice Question

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

1. A.(2, 3)
2. B.($$-$$2, $$-$$1)
3. C.(1, 2)
4. D.($$-$$1, 1)