I would greatly appreciate it if someone could help me. Currently, I am learning about Authentic, Half, and Plagal Cadences. And I have to find them in composition, then tell the name. At first it seemed rather easy, but now I'm stuck. I found the root position, but the inversion it shows is not possible! I am so confused!
Assuming the tonic is F, this does appear to be a V6–I progression, which likely doesn't match your definition of V–I necessary for an authentic cadence.
It's possible that the person that gave this example is viewing that final sixteenth-note C as creating a bona fide root-position V, but that seems strange to me. Most people would view the C as an inner voice and the E as the "real" bass pitch, keeping this as a V6.
There are all kinds of different definitions for cadences; some textbooks would consider this an imperfect authentic cadence, since the V is in inversion. (I personally disagree with this assessment, but it's ultimately up to the textbook that you chose.)
Another possibility is the notion of a "contrapuntal cadence." In contrapuntal cadences, the V and/or I are in inversion. As such, they show a clear progression from V to I without the harmonic articulation of an authentic cadence. V6–I, since the V is in inversion, would be considered a contrapuntal cadence.
1))The first circled chord contains the notes CEG.
2))The next chord has F and A.
What is in the key signature for this example? Is it a B Flat? If so, you're likely in F Major.
So the second chord would be the I chord in F Major.
If you travel up the F Major scale-- F G A Bb C, C is the fifth scale degree and the place where the CEG chord is built.
What you are looking at is a V - I Cadence.
Therefore it is an authentic cadence!