## IM Commentary

The purpose of this task is to construct and use inverse functions to model a a real-life context. Students choose a linear function to model the given data, and then use the inverse function to interpolate a data point. We caution that given the open-ended nature of the modelling in this problem (where by choosing different data points they end up with different linear functions), the task is not intended to illustrate any of the statistics content concerning finding lines of best fit. In particular, we caution that reasoning with inverses of best-fit lines involves some subtlety: The least squares regression equation to predict $x$ from $y$ is not necessarily the inverse of the regression equation used to predict $y$ from $x.$ This task instead focuses on the interplay between modeling, predictions, and inverse functions, rather than the statistical process of finding a line of best fit.

Note: While students were not required to use $f^{-1}$ notation for F.BF.4a, in F.BF. 4b, c, and, d, students begin to use this notation. A common student error is to mistake $f^{-1}$ for $\displaystyle \frac {1}{f}$, and students can be asked to notice that in this problem the expression for $h^{-1}(x)$ is not equivalent to the expression for $\displaystyle \frac{1}{h(x)}$.

Adapted from a problem by Hilton Russell.