This paper reports on a study conducted with Grade 11 learners as participants, in four public high schools in a rural district of the Eastern Cape Province of South Africa. The central objective was to investigate the mathematical discourses of Grade 11 learners related to the word, algebraic and graphic representations of asymptotes of the hyperbola and exponential functions. Commognition and social learning theories were referred to, with particular emphasis on the characteristics of the mathematical discourse, that is, word use, visual mediators, endorsed narratives and routines. The study adopted a qualitative research design, with exploratory, descriptive and interpretive elements complementing both its data collection and analysis processes. A purposive sampling strategy was opted for in the study. Data was collected by means of a test administered to a total of 112 Grade 11 participants from the selected four high schools. Data was thematically analysed by means of a developed Discourse Profile of the Hyperbola and Exponential Function adapted from the Arithmetic Discourse Profile. The study's findings revealed that, while there was evidence of learning functions, for these leareners, the process of de-ritualisation was not yet complete. There were also challenges in linking different representations of a function. Learners could work efficiently on procedure tasks, but struggled on action-oriented tasks.