The meaning of f(x)

I've discovered over the past 8 years teaching Calculus, that my students don't really have any idea what f(x) means. Don't get me wrong - they know that "f(x) is the same as y" and "f(2) means plug 2 in for x" - those things have been drilled into them. But what they don't understand is how to work with function notation if there is no equation involved - i.e. from a table or from a graph. Since I'm the Pre-Calculus teacher, I decided to tackle this issue myself when I reviewed functions. And it has proved to be a challenge! Some kids get it right away, but some REALLY struggle with it. And I'm having a hard time putting my finger on what the problem is exactly. For example, I gave my students this table to fill out -

They had a really hard time filling it out. One problem was they had trouble applying the order of operations correctly - "Should I apply the change to x and then find the corresponding y, or find y and then apply the change to it?" But a lot of their problem was that they simply didn't understand the top two rows of the table and how they related to the other rows. I always get them to understand it by the time they get through Calculus with me (if I had a nickel for every time I say f(x) gives you y-values, f'(x) gives you slope, and f''(x) gives you concavity....). But I wonder how much faster (and better) they would understand reasoning from graphs and reasoning from tabular data if they got this skill down a little better while they were in Pre-cal (or earlier).... Anyone else experience this? Found a good way to address it? Should we address it from the beginning of function notation in Algebra I? When we're reviewing it in Algebra II? As I'm doing it now in Pre-Cal?