Mixing Non-Traidic Harmony With Traids.

Hello

I'm a little confused with mixing non-triadic chords in with traditional functional triadic progressions. How important is the root tone of the more non-triadic chords? For example, in the key of C major if I use a non-triadic chord that has a root tone of F then does this still function as a typical FMaj chord would in a progression simply because the root tone is still F? A few books I have on contemporary harmony go into how to figure of what the root tone is in a ambiguous non-triadic chord, but I don't really understand why the root tone matters when apparently these chords are "non-functional" anyway.

For another example so you understand what I mean let's say in CMaj I start with the chords: CMaj-Amin-E (tones stacked as E-B-A-D, root of chord is E), what is the significance that the root of this chord is E? could it just be though of as a more dissonant alternative to the Mediant triad and function the same way in a progression?

Thanks for your time

Just because its not functional in the traditional sense doesnt mean the relationship it has with other chords is unimportant. In some cases, knowing the roots of an ambiguous chords allows us to figure out the larger progressions at hand. Be it chromatic mediants, a chord progression that follows a very particular pattern, figuring out if the roots are moving in half steps or whole steps, or if the roots are mimicking traditional functions while the quality of the chords are not... and so on and so forth. 

Remember, when analyzing post-tonal music (contemporary music that breaks from traditions) you must use a variety of tools to analyze. G quartal chord going to a Cdim or something like that may not be functional in the traditional sense, but the fact that the roots still move in fifths may be a piece to a bigger puzzle. 

There are some fun online chord dictionary tools where you can select the notes in a chord and the program will tell you the name of the chord.  …but sometimes it gives you more than one answer because more than one answer is actually possible…  How your ear interprets the function of a chord all has to do with the larger context.  All the music theory ever written is an attempt to put into words the rules of what our ear already knows intuitively.  

What makes the melody feel like the melody?  Keeping in mind that the melody can be handed off from one part to another?  What makes a root feel like a root?  Your ear generally knows if you listen.  And from that you can extrapolate to say that a chord is in the second inversion, or whatever.  

Just listen to impressionism - Debussy is a perfect example.