Justify that the mapping $f$ establishes a bijection from the interval $\left[ - 1 , + \infty \left[ \right. \right.$ onto the interval $\left[ - e ^ { - 1 } , + \infty [ \right.$, where $f : \mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto x\mathrm{e}^{x}$.