Q:

Diatonic harmonic function of chord inversions

I'm trying to make my way through the classical harmony, and I'm making good progress, but since my last composition lesson there's been something which is not entirely clear to me.
So my teacher is a good creative advisor, but when it comes to actually teaching theory... he can do better. He told me that when making these harmony exercises with a given bassline, you have to determine what the function - tonic, subdominant, dominant - of the chord on that note should be. The way I understood it, is that you only want a tonic in root position at the beginning and the end (an exercise is usually 8 bars long) and inbetween should be a succession of subdominant and dominant functioning chords. You want to group chords within the same function as much as possible, so if I'd analyze a bassline that way, I would get something like this:

Now what isn't entirely clear to me is the way that the function of chords change when you invert them.
So for example, I has the function of a tonic, but I-6/4 has the function of a dominant, as in the cadential 6/4 chord. VI and VI-6 can have the function of a tonic, but VI-6/4 is more like a subdominant chord. So obviously chords that have the root of I, IV or V as bass take over that function, but what exactly gives a chord a T/SD/D function/feel?
I think I got most of this covered, but I'm just making sure if I'm understanding this correctly :)

Edit:
This is the way I have currently harmonized this bassline:

Just saying that I've only had chord degrees I to VI and the dominant-seventh chord yet, so that's all I can use now. :)

A:

Your understanding is close. One way to look at things is that the function of a chord is determined by its use. The figured bass approach treats a chord as primarily being intervals above a bass while the (more modern) Rameauian (I just wanted use all the vowels in a row) treats chords a permutations of a chord with the root in the bass. Both approaches are correct but they look at things from different points of view. In some cases one POV may seem a better explanation. In the cadential 64, the upper notes act like they are intervals moving over a bass (one gets a sequence of parallel thirds in the upper two voices) while in an arpeggio, a 64 sounds like a rearrangement of the chord notes. 
I think of inversions as providing a method of writing smooth bass lines without changing chord roots. This generally works with first inversion chords. A progression like C-G-C is fine with the bass jumping around (and taking care of voice leading in the rest of the phrase) as is C-G63-C where the bass is just a passing tone between two root position chords. The second is smoother if that is what one wants. One can even get nice root movement by sixths using the Pachelbel Canon progression (or something similar) with C-G6-a-e6-F-C6-d; this is part of the "rule of the octave" studies predating Pachelbel. However, in various compositions of that time (and I cannot remember any specific one) there were other possibilities, C-G6-a-C64-F-C6-d6-G which cycles nicely back to the beginning. The e6 of the first example is replaced by a "passing" C64 in the second. Depending on the melody and the rest of the phrase, one of these may be preferred. 

