Summary: Given a sequence of i.i.d. random functions \(\Psi_n :\mathbb{R}\to \mathbb{R}, n\in \mathbb{N}\), we consider the iterated function system and Markov chain, which is recursively defined by \(X_0^x :=x\) and \(X_n^x :=\Psi_{n-1}(X_{n-1}^x)\) for \(x\in \mathbb{R}\) and \(n\in \mathbb{N}\). Under the two basic assumptions that the \(\Psi_n\) are a.s. continuous at any point in \(\mathbb{R}\) and asymptotically linear at the “endpoints” \(\pm \infty\), we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie’s implicit renewal theory [C. M. Goldie, Ann. Appl. Probab. 1, No. 1, 126–166 (1991; Zbl 0724.60076)] and can also be viewed as an adaptation of Kesten’s work on products of random matrices [H. Kesten, Acta Math. 131, 207–248 (1973; Zbl 0291.60029)] to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, for example, \(\mathsf{ARCH}\) models and random logistic transforms.