Algebra 2 is All About Exponents

I don't know if this makes function families easier or more complicated, but I realized this week that everything we cover in Secondary Algebra (1 and 2) can be reduced to two basic function families: f(x) = n^x and f(x) = x^n (trig functions excluded, but we don't cover those until precalc at my school). I don't remember this ever getting pointed out to me when I studied functions, and seriously, how many years have I been teaching this? I just find it very interesting.

We made these organizers in class today (and yesterday). We've already got our brains wrapped around transformations and compositions, although we have thus far stealthily avoided talking about operations on functions (adding/ subtracting and multiplying/dividing)... other than to say, "Ugh, two x's, that looks messy." Which of course, it IS, right? And isn't that the point?  At least, I think that's my point this time through. Basic functions that involve these families, simple transformations, and compositions where one step follows another in a specified order... this kind of function is not hard to work with. They are logical and orderly. It's only when we start multiplying and dividing or adding/subtracting functions that stuff gets tricky and we need to start pulling out new tricks like factoring and zero product rule, and complex numbers, and extraneous solutions, and reducing rational expressions, and limits, oh my.

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.