## Abstract

Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers’ and students’ inverse function meanings. Responding to this call, we characterize 25 pre-service teachers’ inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers’ activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss our inferences of their meanings based on their activities. We conclude with implications for the teaching and learning of inverse function and areas for future research.

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## Notes

- 1.
Here we use “variable” to refer to a notational letter or symbol.

- 2.
When creating tasks, we attempted to use notation consistent with what we conjectured the students were introduced through schooling. No students questioned or objected to the notation in any prompt. For instance, students could have questioned using

*f*(*x*) (i.e., a function’s output value) to name a function (i.e.,*f*). - 3.
It can be argued all of her techniques consistently involve switching something or performing a transformation. However, we did not infer Alyssa conceived each technique in terms of some underlying invariance.

- 4.
Students used

*f*(*x*) and*y*interchangeably. See Thompson (2013) for more on muddled uses of function notation. - 5.
The other student turned her focus to determining and working with analytically represented functions and did not return to the context.

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## Acknowledgements

This material is based upon work supported by the NSF under Grant No. DRL-1350342. Any opinions, findings, conclusions, or recommendations expressed are those of the authors. We thank Neil Hatfield for directing us to a notation for inverse function and Nick Wasserman for his feedback on an earlier draft.

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Paoletti, T., Stevens, I.E., Hobson, N.L.F. *et al.* Inverse function: Pre-service teachers’ techniques and meanings.
*Educ Stud Math* **97, **93–109 (2018). https://doi.org/10.1007/s10649-017-9787-y

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### Keywords

- Function
- Inverse function
- Pre-service secondary teachers
- Meanings