The aim of this study is to determine students' learning difficulties and misconceptions related to the "inverse function". The study group was composed of 137 first-grade students enrolled in the elementary mathematics teaching program of an Eastern Anatolia University in Turkey during the fall term of the academic year 2010-2011. The case study method was employed in the study. Three different question forms with one open-ended question are used as data collection instruments. Also, students were interviewed to conduct a more detailed analysis of their responses to the open-ended questions, and semi-structured interviews were used as a data source in this study. As a result, it was concluded that the students experienced difficulties in demonstrating surjection of a function when finding the inverse of a given function and had misconceptions like "If a function's inverse is not available, the inverse image of any element in the range is not either" and "The expressions of inverse function, inverse of a function, and inverse relation all have the same meaning." Some specific teaching strategies are recommended based on the conclusions to prevent misconceptions and address learning difficulties specific to the topic of inverse functions.