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…

f(3)=(3-5)^2=(-2)^2=4

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?

Posting Learning Targets yay or nay

Thanks to @JustinAion,  I got thinking…

It depends… on my class and the students and the activity…to determine if I actually post it.

However, when I do, I refer to it at the beginning, throughout the task – to remind students of the end goal, and again as a wrap up – whether reflection, exit ticket of discussion/summary to end class.  And I like to refer to it the following day as we begin the next lesson, just as a quick review.

I, personally, would prefer to have an overarching Essential Question for each lesson to use rather than a specifically stated target.  However, I sometimes struggle a lot with Writing EQs, would love a colleague to collaborate on these.

Here’s a section of the unit organizers I’ve used this past year (thanks @lisabej_manitou). 

And a link to this file.
Unit Organizer
Functions Overview

I give them to students toward beginning of unit, we complete the words worth knowing for vocabulary (thanks @mathequalslove). Then read through actual targets.  When quizzes are given back or practice problems checked, students have a place to reflect/record thwir level of learning as well.  Because students have this in their INBs, I can quickly refer to them if not posted on the board on any given day. 

Inequalities in 2 Triangles

After some discussion about the hingle theorem… the question was …since 40° is twice the size of 20°, won’t side e be twice as long as side d?

Well, we found some counterexamples to prove it wouldnt always be true.

But what about if the triangles are Isosceles?  The picture is actually marked wrong…all of those sides should be marked congruent. 

Two students said it would always work…they believe if  the triangles are isosceles, the third side will be proportional to the angles. 
My projector bulb is shot.  So I could not pull up geogebra to play around in class.  I told them we’d explore tonight and begin there tomorrow.