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.