I've been using a new approach to introduce functions to my Algebra 2 students (who presumably aren't completely new to functions). As much as I can, I'm trying to let need dictate the math. At this point our definition is somewhat incomplete, but working well so far. Today we see a need for more explicit notation, because it's not always clear to us what exactly the function rule is:
I was amazed how quickly they got comfortable with function notation, given this set-up, but I am willing to accept that it may be partly due to previous teachers and previous foundations.
From here we jumped right into the idea of function compositions (but not composition notation, because we don't need it yet):
It was entirely logical to follow these with a discussion of inverses that are a bit more complex:
Here are two samples of their student work at this point. There still seems to be some ambiguity around the word opposite (a great entry point for an upcoming lesson). Beyond that, however, their understanding appears to be rock solid. Everybody did a nice job of explaining the function. Only one got mixed up on defining the inverse. A teacher's dream.