A function \(f\) is defined on \(\mathbb{R}\) by \[\begin{equation*} f \colon x \to \left| x + [x] \right| \end{equation*}\]

where \([x]\) indicates the greatest integer less than or equal to \(x\), e.g. \([3] = 3\), \([2.4] = 2\), \([-3.6] = -4\). Sketch the graph of the function for \(-3 \le x \le 3\). What is the range of \(f\)? Is the mapping one-one?

The function \(g\) is defined by \(g \colon x \to \left| x + [x] \right|\), \(x \in \mathbb{R}_+\), \(x \notin \mathbb{Z}_+\). Find the rule and domain of the inverse function \(g^{-1}\).