I was thinking about inverse function notation, and that got me thinking about function notation, and that got me thinking about operations and how meh our notation for mathematical operations is.
So, let’s start fresh. We’ll pretend we don’t have any operators, just a bunch of numbers and an equal sign. We need to make up operators.
We’ll need something to indicate multiplication. How about ⇧? So 4⇧5 = 20, 3⇧8 = 24, 1.5⇧3.2 = 4.8, and so on. Okay, great.
What if we want to undo multiplication? ⇩ seems like an obvious choice for that. Since 4⇧5 = 20, it must be that 20⇩5 = 4. Hopefully we’ll eventually notice that 4⇧5⇩5 = 4. “Undoing” multiplication means taking its inverse. (I know we have another word for that, but I’m not going to use it right now.)
We can also “undo” without having “done” in the first place: 10⇩5 = 2 because 2⇧5 = 10. Those arrows remind us that what we’re doing is mapping between number lines. The ⇳ means we’re mapping based on a multiplication-based step.
⇩5 is the inverse of ⇧5.
Let’s say we want to multiply several times: 4⇧5⇧5 = 100. 4⇧5⇧5⇧5 = 500. Maybe we want shorthand for that: We’ll put a tiny number. 4⇧²5 = 4⇧5⇧5. 4⇧³5 = 4⇧5⇧5⇧5.
So 4⇧⁶5 says to start with 4 and apply the “multiply by 5” action 6 times. 4⇧⁶5 = 62500.
Maybe we get the idea that this means we don’t need ⇩ after all; we could use ⇧ and put the symbol on the bottom instead to say we’re “undoing” “multiply by 5”. 4⇧₁5 = 4⇩5 = 0.8. The ⇧ tells us we’re moving between number mappings based on “multiply by 5”, the little number tells us how many times we’re moving, and the position tells us whether we’re moving up or down.
To generalize: x⇧nk means “multiply some number by k n times” and x⇧nk means “unmultiply some number by k n times”.
Let’s generalize a little more. We want to have notation that says, “Do something to x a bunch of times.” I’ll use the symbol ⁑ to mean “Do something”. It could be anything, really: Add some stuff, then multiply some stuff. Take the sine of what you get. Have fun. We don’t care.
So x⁑ means “take some number and do those steps to it.” This is a mapping, too. Assuming the result is the same sort of thing (we’ll assume a number), we can do it again. x⁑⁑. x⁑⁑⁑. x⁑⁑⁑⁑.
More generalization: x⁑n means “do that stuff n times”. x⁑n means “undo that stuff n times”.
Perhaps on another planet in this universe, this is sort of what they do. I’m not saying we should do this, I’m not that arrogant yet. I’m saying that thinking about this helped me realize something about what we DO do.
Instead of x⁑, we write f(x). On those occasions where we want to do the same steps twice, we write f(f(x)) but write it as f²(x) if we feel like it. What if we want to undo?
I’m going to sidebar for a second. I’m really not a fan of how we write numbers on the other side of 0. We pick on them. We call them “negative” and “less than zero”. We act like absolute values and positive numbers are the same thing. We don’t even have unique symbols for them: We just take the absolute value version and put a negative sign in front of them.
If we want to write clearly that we’re squaring a negative number, this is how much work we need to do: (-1)². All those symbols! We even give more respect to the square root of negative one (which does get its own symbol) than we do to negative one itself.
If we wanted to stick with the f²(x)-style notation but indicate that we’re going to undo once, we could have done this: f1(x). But we decided to use subscripts for other things, and besides, we already have a way to write negative numbers: f-1(x). That means “find the number that, if you did the stuff that the f action does, would give you x”.
I didn’t know this until a few days ago. I’ve lived half a century. I have a BS in Mathematics. I teach high school math.
I mean, I knew that f-1(x) referred to the inverse function. What I didn’t know was why, and so I never really understood what f-1(f-1(x)) referred to. Because I had so completely ingrained that f-1(x) is the “inverse function,” I thought that f-1(f-1(x)) should be the inverse of the inverse, that is, the function itself.
Now I understand that f-1(f-1(x)) = f-2(x) = x⁑2, that is, find the number such that, if you did the actions of the function TWICE, you’d get x.
Okay, so, fine, it all makes sense now. But it’s lousy notation. And it’s lousy on multiple levels.
First of all, we also use superscripts to mean “create clones of myself and multiply them together”. That is to say, 4⁵ means “take 5 copies of 4 and multiply them all together”. f⁵(x) means “do the action of f five times, starting with x”. In one sense, 4⁵ is really just shorthand for f⁵(1) where f(x) = 4x. That is, 4⁵ = f(f(f(f(f(1))))).
Nobody ever told me that. And it’s clumsy to write, because our notation is clumsy.
At some point, we sneak in that ⇧n = ⇩n. By which I mean, 1/xn=x-n. Going up in the elevator a negative number of floors is the same as going down a positive number of floors.
So it’s understandable that students get confused at f-1(x) and want it to be 1/(f(x)). So much of our notation is based on treating multiplication and division as the basis on which everything else hinges, we often don’t bother stressing that multiplication is just one of a bunch of stuff we could do to numbers.
Keep in mind: Multiplication is SO BASIC that we don’t even have to write the symbol. Put two variables next to each other, and multiplication is inferred.
We do that so adeptly that we internalize it. But if we don’t point it out to students, they might not notice it on their own, and decide that because 2-1=1/2, then f-1(x) = 1/(f(x)). But 2-1=1/2 only because 2-1 is shorthand for f-1(1) where f(x) = 2x. By my earlier notation, 2-1 = 1⇧12 = 1⇩2.
Even worse is what we do to trigonometric functions. We write sin-1x when we mean the inverse sine, but sin2x when we mean sine x times sine x. This is likely because we want sin x • sin x more often than we want sin(sin(x)) and we have a special name for 1/sin x (even though we use it less with each passing year), but that doesn’t change the fact that the notation is inconsistent and confuses students.
Also, to make matters worse for the modern generation: It used to be the fashion to use x, y, z for unknowns and a, b, c for parameters and k, l, m, n for integer parameters and f, g, h for function names. So when f, g, and h were introduced to students as function names, they hadn’t been used yet.
Now, though, we just use whatever of the 26 letters we want for whatever variable we want. The natural base isn’t even immune: e might appear on an Algebra I worksheet as an unknown.
So f(x), in Algebra I class, is a perfectly valid way of writing “the variable f times the variable x”. And it’s not a stretch to interpret f-1(x) as “the variable x divided by the variable f”.
This is a notation problem. f-1(x) is a culmination of that problem, a perfect storm of lousy notation mixed with reckless application. In the historical context in which it was created, f-1(x) made perfect sense, but the sands of time and mathematics have shifted out from underneath it.
So what to do? I don’t know. I am but a meager high school mathematics teacher, incapable of fixing this on my own. The best I can do is share in the misery of batch after batch of confused students, whittling away at my own misunderstanding (as I did throughout this meandering post).