Inverse, composition, and identity: The case of function and linear transformation

• We interview students about composing functions and transformations with inverses.
• All ten students made incorrect predictions about the result of composition.
• Six students were able to resolve their incorrect prediction.
• Students who resolved all used what we call “do-nothing function” reasoning.
• The efficacy of this reasoning may be linked to process-object conceptions of function.

In this report we examine linear algebra students’ reasoning about composing a function or linear transformation with its inverse. In the course of analyzing data from semi-structured clinical interviews with 10 undergraduate students in a linear algebra class, we were surprised to find that all the students said the result of composition of a function and its inverse should be 1. We examined how students attempted to reconcile their initial incorrect predictions, and found that students who succeeded in this reconciliation used what we refer to as “do-nothing function” and “net do-nothing function” reasoning. We provide examples of these patterns of reasoning, and propose explanations for why this reasoning was helpful. We also discuss possible sources for this incorrect prediction, and provide implications for classroom practice.