The Inverse Function Fallacy: On Sign Determination and Forgotten Fundamentals

TL;DR

The paper investigates why students misinterpret inverse functions, particularly assuming a positive root regardless of domain. It introduces the Inverse Function Point method, which uses a domain-restricted input, reflection across $y=x$, and algebraic verification to determine the correct sign of the inverse without graphing. In an initial study with $n=69$ STEM students at RPI, only $13$ solved correctly on the first try and $19\%$ accuracy; after a brief explanation and four practice problems, $32$ of $40$ retested participants solved them correctly ($80\%$). The approach offers a simple, intuitive way to address sign ambiguity, reinforce core principles of inversion, and potentially improve conceptual understanding of inverse functions at scale.

Abstract

Although inverse functions are introduced early in algebra, many students remain unaware that an inverse expression may legitimately involve a negative root. Instead, they default to assuming a positive root, overlooking the role of domain restrictions in determining the correct solution. This paper identifies this misconception as the "inverse function fallacy" and introduces a systematic approach, the Inverse Function Point method, that establishes sign determination to a single domain-based reference point. In a study of 69 STEM students at Rensselaer Polytechnic Institute, only 19% solved a sign-determination problem correctly on their first attempt. Those students were not re-tested. Of the remaining 56 who answered incorrectly, I was able to re-test 40 after teaching the proposed method. In this subgroup, accuracy rose to 80%. These results highlight both the fragility of assumed mathematical knowledge and the potential of simple, intuitive procedures to reinforce conceptual understanding.