# What are some real world applications of the Inverse Function Theorem?

I am teaching an AP calc AB course and just covered the Inverse Function Theorem. While I know some of the applications in higher math, I could not come up with any real world applications to use for my students. What are some examples I can give to my students?

Edit: I mean the version of the theorem that says $${f^{-1}}'(x) = \frac{1}{f'(f^{-1}(x))}$$.

• Which inverse function theorem are you speaking of? – Qi Zhu Nov 8 at 14:58
• I'd guess it's sort of useful for calculation to know that $(dy/dx)^{-1}=dx/dy$ always makes sense in an appropriate way. Don't know if that's too physics-y for your students. – WoolierThanThou Nov 8 at 15:19
• Are you referring to the version of the inverse function theorem which says that $$(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}?$$ – TM Gallagher Nov 8 at 15:42
• The OP might have meant the multivariate case of this. – J.G. Nov 8 at 21:27
• Edited to clarify. @TMGallagher. – Abe Schulte Nov 10 at 0:14

The things that I have generally presented to my calculus students (besides the fact that it enables you to say some powerful things about the derivative of $$f$$ inverse without actually having a "formula" for $$f$$ inverse which is wonderful to math-lovers in it's own right) are generally focused on implementing the inverse function theorem in situations where the function $$f$$ is not invertible everywhere (for instance, general polynomials of order 2 or higher, trig functions, etc) because they fail the horizontal line test. So while it is very difficult/tedious (or impossible to do by hand) to compute the "general formula" for the inverse of the function in question, a careful application of the inverse function theorem can still tell you the rate at which the inverse is changing at specific points.
One could take any sort of real-world modeling problem which uses a polynomial or rational model (for instance, how inflated your tires are versus how long the tires will last is generally modeled by a quadratic function; the amount of time it takes people to list their favorite vegetables can be modeled by a cubic function; modeling total cost $$C(x)$$ from a spreadsheet of data and then using this to compute average cost $$C(x)/x$$; and many more) and use the inverse function theorem to determine how the "horizontal" variable (typically $$x$$ or $$t$$, though much of this wouldn't make sense to consider for situations where the independent variable is time) changes with respect to changes in the "vertical" variable.
For instance, consider the time it takes people to list their favorite vegetables. Here you could consider the number of vegetables listed in $$t$$ seconds to be $$V(t)$$ and a straightforward question might be something like "how many more vegetables do you estimate a person to be able to list by spending 11 seconds thinking instead of 10 seconds?" Using the inverse function theorem, students can more quickly answer things like "how many more seconds do you estimate it will take a person to list 16 vegetables instead of 15?"
Lastly, you can use the inverse function theorem to make simple observations about relationships between variables--like where one will be increasing with respect to the other or decreasing with respect to the other--based on careful analysis of the sign of $$f'$$ and $$(f^{-1})'$$ at various points. The same analysis can be done just using precalc techniques but it can be much less confusing in many cases to use positivity of the derivative.