Tuesday, October 6, 2015

Why aren't exponents and roots always inverses?

We've been studying functions in my Algebra 2 class. I'm taking an entirely new approach this year. It's going well, but the jury's still out. So far, one thing I'm really happy about is their excellent grasp of inverse functions as a process of 'undoing' whatever the original function did. Today in class we were organizing our thoughts about the different function families. We'd drawn out some nice examples of expoential growth and decay and I asked if we could figure out the inverse functions. I EXPECTED them to see that they could not write a function (we haven't done logs yet), but that they could use the tables and graphs that they just made to draw an inverse graph. Our function was a simple "2 to the power of x."

But they surprised me. "The inverse would be the xth root of 2," someone said.

Several agreed. I was dumbfounded. Why not?

After all, isn't it TRUE that:

for all real values of n?

That is, I know there are some issues with this as a blanket statement. For example, even values of n only work if we restrict the function domain. But for the most part, this is entirely true and logical.

So why is it FALSE that:

Maybe you, like me in class today, are scratching your head now and wondering... could that be right? It isn't. We checked. Choosing an input for the original function, applying the function and then applying the inverse function will not return us to our original value. But WHY?

Help me out here. Can anyone provide a purely sensible argument for why this will not work? Not just a demonstration of HOW it doesn't work. I can supply several of these now. I want to know why.


  1. For the sake of being able to write this out in a comment, let's call your inverse function g(x). If they're truly inverses, then g(f(x)) = x. The reason that this doesn't work becomes more obvious when you build the composite statement. Fleshing out the composite function, we get that (n^x)√(n) = x (if it isn't obvious, the n to the x-eth root of n equals x). Looking at it this way, it's clearly a very rare scenario where this would work. I think the more interesting question given those to functions is to ask when, if ever, g(f(x)) = x.

    1. Thanks Mike. I like how you ask another question that's also interesting. I HAVE done the compositions: f(g(x)) and also g(f(x)) and see of course that they are not x (although, you are right, they CAN be, which IS interesting). My students have only a rudimentary understanding of inverses and compositions at this point, so it would be useless to use this method as a rationale for them. They were satisfied with simply finding a counterexample: g(f(3)) is not equal to 3, so this cannot be the inverse. But to them, the xeth root of n seemed like such the logical choice for the inverse. After all, x^n does indeed yield nth root of x as its inverse. How can I explain this quirk to them so that they feel secure in their understanding of inverses overall (which, BTW, i think is excellent)?

  2. Thank you, Christopher Danielson, for pointing me to your post about logs and non-commutative binary operations: https://christopherdanielson.wordpress.com/2011/12/07/enough-with-the-logs-already/

    I think this is getting me somewhere. Somehow, the fact that exponents are not commutative will throw off the ease with which inverse functions are grasped. For example, it is true that f(x) = x+3 and f(x) = 3+x will both yeild the same inverse function f-1(x) = x-3. What casues problems is this: if g(x) = x - 3, then g-1(x) = x+3... BUT if g(x) = 3-x, then g-1(x) is NOT 3 + x. Since the original g function is not commutative, we cannot use the same 'thought process' to invert them when the arguments are reversed. I like.