I urge you to try to find ways that this doesn't work.

That was the easy obvious one which I hoped the hint would be sufficient for the original poster to recognize.

Are there more? I think I can see at least one additional class, and perhaps more which neither of these are sufficient to deal with.

Being more or less amenable to symbolic analysis by Mathematica was well beyond what I guessed the original poster, or most, might be able to address.

First of all thank you all for replying to my question in such a short time.
I guess I wasn't clear enough. My goal is to define this function in wolframalpha.com site and not in Mathematica.
How do I type this formula in the "calculate" field. I just need to see its graph. 

Its graph looks like the constant 0 function, or 1 function, depending on how you look at it. It could look like both superimposed, if you look at it like that. Both sets are dense in [0,1], so there aren't any gaps - just both the lines.

1. In Mathematica is there an example where Head === Rational and Element[x, Rationals] differ?

We might be able to guess the implementation of the first, but the second might be more open to question.
Are they actually identical or not?

2. I tried an assortment of ways to get WolframAlpha to plot or even define your function and failed repeatedly.

In my experience, one of the differences between MMA and WA is that MMA allows function definitions and WA doesn't,
but you can sometimes just insert the "body" of the function into your WA query and get around this.

If there is a recipe for how to efficiently discover the syntax to understand a particular WA query I'd love to learn that.

Note: I did accidentally stumble onto appending "last 10 years" onto the end of a WA query and found it is possible
to get the history of some statistic instead of just the latest value, I had long since given up and assumed that
this just could not be done.  I still have not found a way to get an inflation adjusted history though.