Density-based clustering aims to find groups of similar objects (i.e., clusters) in a given dataset. Applications include, e.g., process mining and anomaly detection. It comes with two user parameters (ε, MinPts) that determine the clustering result, but are typically unknown in advance. Thus, users need to interactively test various settings until satisfying clusterings are found. However, existing solutions suffer from the following limitations: (a) Ineffective pruning of expensive neighborhood computations. (b) Approximate clustering, where objects are falsely labeled noise. (c) Restricted parameter tuning that is limited to ε whereas MinPts is constant, which reduces the explorable clusterings. (d) Inflexibility in terms of applicable data types and distance functions. We propose FINEX, a linear-space index that overcomes these limitations. Our index provides exact clusterings and can be queried with either of the two parameters. FINEX avoids neighborhood computations where possible and reduces the complexities of the remaining computations by leveraging fundamental properties of density-based clusters. Hence, our solution is effcient and flexible regarding data types and distance functions. Moreover, FINEX respects the original and straightforward notion of density-based clustering. In our experiments on 12 large real-world datasets from various domains, FINEX frequently outperforms state-of-the-art techniques for exact clustering by orders of magnitude.