Let $$f: \mathbb{R} \to \mathbb{R}$$ and $$g: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} \log_e x, & x \gt 0 \\ e^{-x}, & x \leq 0 \end{cases}$$ and $$g(x) = \begin{cases} x, & x \geq 0 \\ e^x, & x \lt 0 \end{cases}$$. Then, $$g \circ f: \mathbb{R} \to \mathbb{R}$$ is:

A

one-one but not onto

B

neither one-one nor onto

C

onto but not one-one

D

both one-one and onto