The first inversion of a chord is considered to have the same function as the chord in root position. This is esp. true for the primary triads: tonic, subdominant and dominant.
That said, the first inversion can help affirm the function of a secondary triad (substitution chord).
For instance, in C major, both Am and Em are substitution chords of C, so both Am and Em can have tonic function. But Am is also a substitution chord of the subdominant F, so Am can also have subdominant function. The first inversion of Am, c-a-e has the C in the root, and because the c is in the bass, it will be more effective as tonic than it will be as subdominant.
The second inversion of a chord is not considered to have the same function. That's because the second inversion has a (dissonant) fourth interval that needs a specific resolution.
The most abundant second inversion chord is on the first degree, and it will typically work as a suspension leading to the dominant chord. So in C major, the 2nd inversion would be g-c-e and it will typically go to G (g-b-d). Even though the 2nd inversion g-c-e is a chord of C, it does not have tonic function. You will notice immediately if you try to substitute the final chord of a song for its second inversion - it doesn't sound finished, because the second inversion does not have tonic function.
With regard to voicing: voicing is the exact placement of the notes, including any omissions and doublings of notes. The inversion is a more general concept that only says something about which chord tone appears as bass tone. Any inversion of a chord can be voiced in many different ways. But once the inversion is set, the voicing does not change the function of the chord - it merely colors it.