For any real number \(x\), \([x]\) denotes the greatest integer not exceeding \(x\); e.g. \([3.6] = 3\), \([2] = 2\), \([-1.4] = -2\), etc. Functions \(f\) and \(g\) are defined on the domain of all real numbers as follows: \[\begin{equation*} f \colon x \to [x]; \quad g \colon x \to x - [x]. \end{equation*}\]

Find the ranges of \(f\) and \(g\), and sketch the graph of \(g\).

Determine the solution sets of the equations

  1. \(f(x) = g(x),\qquad\) (ii) \(fg(x) = g\,f(x)\).