It was established in [1] that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, particularly ensuring the existence of critical points. In this talk, we establish a determination result for merely bounded from below functions in complete metric spaces, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if a function is inf-compact. Additionally, our result is valid for a wide range of descent moduli including the (De Giorgi) local slope, global slope and average descent operators. [1] A. Daniilidis and D. Salas, A determination theorem in terms of the metric slope, Proc. Amer. Math. Soc. 150 (2022), 4325–4333. [2] A. Daniilidis, T. M. Le and D. Salas, Metric compatibility and determination in complete metric spaces, arXiv:2308.14877.