A Struggle with Function Inverses


So as we were doing this today – simple enough using inverse operations to find the inverse function…

But as we did this one f(x) = (x-5)^2 to this f^-1(x) = sqrt(x) + 5

Student question… because you had x-5 in parenthesis, won’t the +5 also be inside the radical?

I’m just curious how anyone else responds to this.

We picked #’s and evaluated the expression, modeling transformation on a number line…but referring to inverses as a way to return where we started.  

f(8)=(8-5)^2 = 9

Simple enough…working backwards… sqrt(9)=3+5=8…back where we started x=8.

But what happens if x=3…


Working backwards…sqrt(4) = 2+5 = 7…not where we started at x=3.

Is this where you talk about restricting the domain in order for the inverse function to be defined?

Please don’t judge, I’ve taught straight up procedures in the past, even focusing on mostly linear inverses.  But I want my student to understand what they’re doing and why and be able to expand their understanding to more complex scenarios on their own.

Listening to them discuss this initial question, made me realize how Building Functions and Transformations has huge impact on foundational understanding.

The other realization today…very few of my students are comfortable and fluent in equivalent expressions. 

For example…
f(x)= 6-2x some asked if they could rewrite -2x+6? Sure!

But then when discussing their results, we saw several equivalents…but they did not recognize, rather argued others were incorrect.


We picked a value for x, then tested to see if they were equivalent.  A few were a bit perplexed they were the same value. 

This is one of those assumptions I’ve made but realize we need to address/refresh.

Would a matching, always/sometimes/never be sufficient?


3 responses »

  1. I tend to explain in terms of transformations. What affected x in the original function (horizontal translation 5 units to the right) will affect y in the inverse (vertical translation up 5). I think this helps them think about what is happening more and their result should match what they find procedurally.

    • Thanks mary! To be honest, it’s one of those things I always did as a student but never thought about the why. I want them to have that conceptual understanding and the realization of how important transformation was a game changer for me.

  2. Pam, while we do discuss inverses using transformation language, we also teach them to solve algebraically (swap x and y, solve for y). This helps enormously for the more complicated inverses.

    By the way, we start with a patty paper activity – students trace a function on the patty paper, flip it over a certain way (x=y) and examine what happens. Here is the link to the introductory lesson we use: http://www.txar.org/training/materials/Algebra_II/08_MAPInversesStudentLesson03.14.07.pdf

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