After attending the middle school data analysis meeting, I decided to do some analysis of my own. I started with the inverse function problem on the PreCalculus midterm since it was the only topic students did well on during the semester but struggled with on the exam.

What types of mistakes do you think students would make?

Have a hypothesis? Okay, now you can view the exact mistakes my students made at the Math Mistakes blog.

Back? Great! So here’s **what I noticed:**

Many students made small errors in basic algebra that may have been due to fatigue (this was question 23 of 25 and most students had to stay past the scheduled 90 minutes to finish the exam).

It was a “check if these are inverses” question but many students attempted to check by finding the inverse themselves, rather than composing the functions. When their solution didn’t look exactly like the given inverse they said “not inverses” and moved on. In fact, the inverse many students found was equivalent to the one given (they found ).

**Next time I teach/assess this: **

In class/homework I need to give more examples of equivalent functions that don’t look equivalent so students are compelled to check more carefully.

I wonder if giving the functions in reverse order would change the results for the better (so students who found the inverse themselves would start with and work from there). I’m inclined to think I would see more algebra mistakes but fewer students who found the inverse correctly and didn’t see that their solution was equivalent to the given one.

My co-worker and I found it difficult to come up with a problem that requires students to solve the inverse since many of our students could immediately “see” the inverse without having to write anything down. I will need to spend more time finding good problems with interesting inverses that aren’t too complex.

**Requests for feedback:**

Did the mistakes my students made match the ones you expected to see? What would you do to remediate this year or prevent such mistakes next year?

Interesting post. Think about the intent of your question, and have that guide you. Are you interested in knowing if students understand the composition definition of inverses, or do you want them to be able to find the inverse of a function? To be honest, I would probably do what your students did: taking f(x) and finding its inverse rather than composing the functions. The composition looks nasty, so why deal with it if I don’t have to? Switching the order around is a good approach, though it is also a bit more challenging algebraically.

OR, inisist that students use the definition of inverse functions in order to verify that these are inverses.

I would really like to see students use composition of functions, but I completely agree that it’s not a friendly problem. What I really need is a pair of functions that don’t look like inverses that are composition friendly.

I agree with Bob, but I think inverting g is significantly more difficult than inverting f – the key step of factoring out a y does not come naturally to most students.

I am not surprised by the mistakes, but I am surprised that you did not see many of these types of mistakes when you were doing the unit. Do you think these students have poor understanding of fractions but were able to rely on shorter term memorization during the unit?

The students do struggle with fractions as much as any class does. During the unit I gave more root/exponent problems so these issues didn’t surface as frequently. We wanted to give a problem that students couldn’t do in their heads on the midterm, but ended up losing them to other issues in the process.

If you want to assess their understanding of the definition of inverses as composition, perhaps using a quadratic / sqaure root function pair would work better. Let f(x) = x^2 + 4, and g(x) = rt (x+2). Have them prove or disporve their inverse-ness.