The focus of this dissertation is on students' thinking regarding the function concept in the context of abstract algebra, focusing on the properties of well- and everywhere-definedness. This dissertation follows a three-paper format, where each paper has a different yet related focus.

In the first paper, I analyze (non-)examples found in textbooks and those used during instruction related to function in abstract algebra to gain insight into how students are expected to reason about functions in this context. I investigate experts' thinking about functions by conducting a textbook analysis and semi-structured clinical interviews with mathematicians. I then elaborate the essential properties of well- and everywhere-definedness into four categories of non-examples that students are expected to successfully reason about in an introductory abstract algebra course.

In the second paper, I explore students' reasoning with examples and non-examples of function related to the four categories by conducting task-based clinical interviews. I provide characteristics of a coordinated way of understanding functions in abstract algebra and illustrate how such a coordinated way of understanding functions enables students to reason productively with function tasks and the properties of well- and everywhere-definedness. This paper addresses what is entailed in reasoning productively about well-definedness and everywhere-definedness.

In the last paper, I present a functions activity focused on recovering an example from a non-example of a function by prompting students to modify the domain, the codomain, and/or the rule. I argue that such an activity can help instructors have a clearer image of how they might support their students in developing a coordinated view of function and is a tool that abstract algebra instructors can use to help students attend to well-definedness and everywhere-definedness.