The composite functional equation \[ f(xG(f(x))) = f(x)G(f(x)), \quad x \in \mathbb R_+,\tag{1} \] is considered. \textit{P. Kahlig}, \textit{A. Matkowska} and \textit{J. Matkowski} [Aequationes Math. 52, 260--283 (1996; Zbl 0861.39013)] dealt with the special case: \(G(u) = u^p\). Here the continuous solutions \(f: \mathbb R_+ \to \mathbb R_+\) of (1) are determined provided that \(G\) is continuous and strictly increasing, \(G(\gamma) = 1\) for a \(\gamma > 0\). The graph of any solution is a broken line possibly with a piece of a constant function (this is necessarily the case when \(a = 0\) or \(a = \infty\)). The authors remark that if \(f\) is invertible then its inverse function satisfies a linear equation for which the theory can be found, e.g., in the book by \textit{M. Kuczma}, \textit{B. Choczewski} and \textit{R. Ger} [Iterative functional equations (1990; Zbl 0703.39005)] works.