# Those Funky, Funky Exponential Functions!

A neat math discussion came from an unexpected place today when a teacher in my department sought me out with the TI Nspire of one of her students in hand. The student was in a pre-calc class where exponential functions were being examined, and attempted the graph the following:



What should we expect to see?  How does this graph behave?  Here is what my new Nspire app gave me, which matches what the student calculator showed.

There seems to be a little funkiness around the origin which confused the Nspire, but the bigger issue is that the meat and potatoes of the graph is just wrong.  This about these values of this function and you’ll see why:



This function be-bops around in a quite interesting manner, and the TI-84 shows the graph nicely, as individual dots.  After going through some usual diagnostics in my head, and the list of dumb things kids sometimes do to calculators which cause them to act funny, the problem seems to be with Nspire. But this got me thinking about this strange function, and it’s behavior.  What happens if x = 1/2?  2/3?  3/2?  What’s the domain of this function?  And how do some of my other online math tool friends handle this one?

Wolfram|Alpha is our first contestant.  Show me your stuff:

How cool!  And what a neat discussion of complex numbers, and an interesting overlap between real and complex parts.  Wondering if anyone has insight into the domain and range though.  Is it that this function has no domain?  Or is it that  the domain is simply too difficult to express nicely?

Next up is my old friend Desmos.  I know you won’t let me down.  First, entering the function, Desmos does nothing (trust me, no screencap…nothing happens).  But, activate the table and you can plot some points.  I also added a few of my own at the end of the table:

A good effort, but wish there was some indication of the graph’s behavior without the table.

Overall, this is a tricky little function with a lot to talk about.  Put it on the board for your classes and let them think about:

• What rational values cause the function to be undefined over the reals?
• What rational values cause the function to have negative value?  Positive value?

Then, the plot THICKENS!  Later in the day, I was showing of my Nspire app and the goofy function to some math friends at a meeting, when an English teacher collegue joined the fray.  After giving us the obligatory “what a bunch of geeks” staredown, she grabbed my iPad and gave a few finger swipes at the graphs, changing the window values and……

Holy crud!  How cool, yet….pretty much not useful at all!  During the day, I also sent a note out to TI about what I had found, and a response was given later in the day.  Thanks for getting back to me so quickly TI folk!