1. Arrow-Diagram Representations
An Introduction
Jamie Nordstrom
West Valley HS - Yakima

2. The function machine milked for all it is worth
Mathematics: Modeling Our World
Mathematics: Modeling Our World (MMOW)
Published by COMAP Inc.
Bedford, MA
Based in function composition
I first saw the arrow-diagram representation as a part of the Mathematics: Modeling Our World curriculum. It was introduced as a way to check on a students’ understanding of the order of operations, as well as to solve simple equations with a single occurance of the independant variable.
After playing around with the idea for a little while, I recognized it as the old function machine, but broken down even farther - really finding its strength in representing function composition. It was interesting, but as I taught primarily upper-level courses I didn’t see a lot of application for myself.
Then I sat in on a lecture at CWU by Aaron Montgomery, who had his own version of the concept - with a more graphical approach that addressed function transformations in his pre-calculus classes. That sent me digging back into the idea - and the more I played with it, the more utility I found.

3. The basics of “arrow-diagrams”
The notation
Order of operations
Solving equations
Building function inverses
Composition of functions
Decomposition of functions
I didn't’ think I could get everything into a single presentation, so I have broken the presentation into 2-parts, I’ll be doing the second one tomorrow, and will focus on transformations, writing equations and the chain rule in calculus.
This presentation will start where I started and hopefully will find a natural progression through the topics.
It is only in the last couple of months that I have started to think about being more deliberate about the notation I use for some operations, but I think in the end I see more value as a conceptual framework than as something overly formal that requires more memorization.

4. The Notation f( ) x f(x) ( ) +3 x x + 3 ( ) x x
The function machine represents the powerful input-output relationship at the heart of the idea of function, but its utility beyond that is suspect.
The arrow shows the direction of the relationship
The descriptions on the arrow are powerful because they are general
No “x”’s should show up on an arrow x
f(x)
( ) +3 x
x + 3 2
( )
The function machine certainly plays an important role in communicating the whole idea that a function is an input-output relationship. To do what we want to do, though, the big box needs to be streamlined.
We don’t necessarily need to to break the inner workings of the function machine up, but it is helpful for what we are trying to accomplish at this point. But we shouldn’t ever forget that any function (even a relation in some instances) can be represented by an arrow.
We leave empty parenthesis without mention of x on the arrow description because we don’t necessarily know who or what is going to show up at the beginning of the arrow, we just want to know what to do with whoever or whatever shows up.
My convention - developed over time - is to draw the arrow from left to right, arching over the top, and with an arrowhead to indicate where the input and output are. x x 2

5. That seems less useful than a function machine!
Linking arrows together to represent an expression
( ) - 4
( )^3
3( )
( )/8
( ) + 1 x
That seemed pretty limiting, but the good stuff starts to happen when we link arrows together - the output of one arrow becoming the input of the next arrow
As a student builds the arrow diagram, you are able to quickly see how well he or she knows the order of operations. If they don’t, then arrows won’t show up in the correct order.
Of course, in this case, the 3rd and 4th arrows could be switched because multiplication and division are really the same thing.

6. That seems less useful than a function machine!
Linking arrows together to represent an expression
( ) - 4
( )^3
3( )
( )/8
( ) + 1 x
x - 4
Sometimes, in order to drive the concept home with the kids, I will have them record where they are in the process after each arrow.
The idea is that you had better be able to see each piece of the expression showing up as it appears in the expression.

7. That seems less useful than a function machine!
Evaluate the following expression for x = 2
( ) - 4
( )^3
3( )
( )/8
( ) + 1 2 -2 -8
-24 -3 -2
While we are here, we might as well use the arrow diagram we have built to evaluate this expression for a particular value of x.
With the arrow diagram in place, if a student arrives at an incorrect final value, you can see exactly why and where it happened

8. What if we know the output, and need the input
= 4
( ) - 4
( )^3
3( )
( )/8
( ) + 1 x 2 8 24 3 4
8( )
( ) - 1
( )/3
( ) + 4
Provided that each function on each of the arrows is invertible, then we can reverse the direction, as long as we create a path to follow. As it exists, there is no path to get from the far right of the diagram, but we can build that path, and teach about inverse operations at the same time.
So, there are some natural questions at this point - but if you don’t ask them, I’ll just let things go.
If you are a properties of real numbers person, you can use this to emphasize additive and multiplicative inverses.
x=6

9. What if we generalize that idea...
f(x) =
( ) - 4
( )^3
3( )
( )/8
( ) + 1 y
8(x-1)
x -1 x
8( )
( ) - 1
( )/3
( ) + 4
By switching the x and the y (that is what inverses are all about, right?), we can follow the impact of the arrow on x as it works it way back to y on the right, and now we have the inverse function when we arrive at the end.
One of the really cool features of this is that the inverse proves itself with the arrow diagram!
This actually alludes to our next topic, but we will pause because I think there might be some reasonable questions at this point.

10. Questions before we transition?
Even though not every function is perfectly invertible, we can accommodate either a domain restriction, or we can be creative in how we structure that arrow coming back the other direction. I use +/- square root with x^2, I use +/-() for absolute value, and for sine, I split into 2 arrows, one with inverse sine, and the other with pi minus the inverse sine.
One place where arrow diagrams find a limitation is with combinations of functions. We will find, however, that unless we are trying to build arrows coming back (ie solving equations or building inverse functions), we wil be able to squeeze plenty of utility.

11. Composition of functions
Find f(g(x)) for
f( )
g( )
( ) - 2
( )^2
-( )
( )+4
1/( )
3( )
g( )
f( )
-( )
( ) - 2
( )^2
( )+4
1/( )
3( )
The traditional method of proving that functions are inverses of each other it to compose the function with its inverse and to compose the inverse with the function. I have said that arrow diagrams are all about composition, but we haven’t specifically addressed that yet - so here we go.
First, we build an arrow diagram for each function. Notice a couple of important things in these diagrams. First, I have a longer arrow encompassing the two arrows that make up the function f, and one encompassing the 4 arrows that make up function g. I do this so we can see what to do next, and also to point out that a single arrow can certainly represent more sophisticated steps than parent functions and basic transformations.
You will also notice the “change a sign” function on the first arrow of g, and the “flip it over” function on the third arrow of g (in calculus, I would probably write that as ()^-1.
Enough of that, though. To build f(g(x)), we begin by putting x into g, so the g function goes first. And then the output of g becomes the input of f.

12. Composition of functions
Find f(g(x)) for
g( )
f( )
-( )
( ) - 2
( )^2
( )+4
1/( )
3( )
Now we build the function f(g(x) by starting with an x at the far left and populating the parenthesis with the output of the previous arrow.
Just in case any of my college professors ever see this, I make sure to use something other than an equal sign to connect the outputs of each arrow (because the aren’t equal).
There is, of course, the question of simplification, and there are specific situations that I have used arrow diagrams to look at simplifications, but that has pretty much been restricted to transformations of linear functions. Simplification is very important, but is not really a strength of arrow diagrams.

13. Composition of functions
Find g(f(x)) for
g( )
f( )
( ) - 2
( )^2
-( )
( )+4
1/( )
3( )
Well, we should probably do it the other way as well. g(f(x)) just requires that we switch the orders of the sets of arrows.
You may also have noticed that I am no longer putting the results of each arrow directly on the arrow-diagram. It starts to get a little crowded in there and it starts to emphasis what the arrows are really representing. The arrow doesn’t care what the input and output are, it is just there to do its part when it is asked to do so.

14. Looking back at the inverse example
f(x) =
( ) - 4
( )^3
3( )
( )/8
( ) + 1 y
8(x-1)
x -1 x
8( )
( ) - 1
( )/3
( ) + 4
If we look back at the inverse example, we can now see that the arrows across the bottom from right to left that form the inverse become the input for the arrows going back across the top (the original function) and so the complete loop represents a composition of f and f inverse. The order of the composition depends on whether you start at the left side (f first) or the right side (f inverse first).
What if we started somewhere in the middle - that’s an interesting mathematical question.

15. Decomposition of functions
Write the function h(x) as a composition of functions f and g where h(x)=f(g(x))
𝝅( )
sin( )
( )^2
-( )
( )+4
√( )
3( )
Decomposition of functions is the idea of breaking an existing function into separate composed functions. It can be an important part of seeing structure in expressions (catch that CCSSM reference?)
As usual, we start by building the arrow diagram for the function h. Now for the easy part - just pick a spot to break the function.

16. Decomposition of functions
Write the function h(x) as a composition of functions f and g where h(x)=f(g(x))
g()
f()
𝝅( )
sin( )
( )^2
-( )
( )+4
√( )
3( )
If we pick this to be the spot, then we start with an x at the far right and build the function g up to the marked point, then start over with an x at the marked point and build the function f from there.

17. Decomposition of functions
Write the function h(x) as a composition of functions f and g where h(x)=f(g(x))
g()
f()
𝝅( )
sin( )
( )^2
-( )
( )+4
√( )
3( )
Or maybe we prefer to break the function h at this point

18. Decomposition of functions
Write h(x) as a composition of functions f, g and m where h(x)=f(g(m(x)))
m()
g()
f()
𝝅( )
sin( )
( )^2
-( )
( )+4
√( )
3( )
What if we wanted to show this as a composition of more than 2 functions? Then we pick 2 break points.
The left side of the diagram becomes the input for function m, which stops at the red arrow, we then start with a new x at the red arrow and build g until the orange arrow, and finally pick up from there and run to the end to get f.

19. What if a function is formed from combinations?
Find h(x) = f(g(x)) for the given f and g
g()
f()
2()
√()
e^()
()^2+3()
()-5
So, we mentioned that combinations of functions (adding, subtracting, multiplying and dividing) are a weakness with arrow diagrams because of the multiple occurrences of the variable. Fortunately, depending on what we are trying to do, all hope is not lost.
Let’s look at some adjustments we can make for the sake of some function composition. So we adjust by putting extra sets of empty parenthesis on the arrow - this ends up being very valuable in calculus.
You might rightfully ask at this point why we even needed to have 2 arrow for function f, couldn’t we have just included the -5 on the first arrow? And the answer is yes - that is part of the versitility of the,arrow diagrams. I tend to try to make them as simple as possible, but they don’t have to be.

20. What is left to talk about?
Graphing functions
All things related to transformations
The chain rule
Writing equations
I haven’t really ever presented this material outside of my department, so I think I made a clean break on where to cover things. So, what is left for tomorrow?
Plenty - transformations are one of the coolest applications of arrow diagrams, but the chain rule is probably my favorite because it is the first insight that I had beyond what I had seen done in the modeling our world material
If i have time to talk about writing equations, I will, however, it sometimes feels like you are forcing it to do something it doesn’t really want to do, or that something else might be able to accomplish more efficiently.

21. Resources
I have some documents that cover many of the same things I have covered here that I can include in an , and am always willing to answer questions or even better - to hear additional insights.
Youtube: (google “Nordstrom Arrow Diagram”) - you’ll see videos with a blue background - those are all on my channel.
I’ve had to move relatively quickly and haven’t been able to present a lot of examples. I tried to pick examples that showed a lot of things,but probably missed something.
I have made a few videos using my SMART board and have those on a youtube channel
The honest truth is that everytime I start working with them, I gain additional insights - so if you spend some time looking at the, you’ll probably see something I haven’t and I would love to hear about it.