  We determine all degree-4 rational functions f(X) in F_q(X) which permute
P^1(F_q), and answer two questions of Ferraguti and Micheli about the number of
such functions and the number of equivalence classes of such functions up to
composing with degree-one rational functions. We also determine all degree-8
rational functions f(X) in F_q(X) which permute P^1(F_q) in case q is
sufficiently large, and do the same for degree 32 in case either q is odd or
f(X) is a nonsquare. Further, for most other positive integers n<4096, for each
sufficiently large q we determine all degree-n rational functions f(X) in
F_q(X) which permute P^1(F_q) but which are not compositions of lower-degree
rational functions in F_q(X). Some of these results are proved by using a new
Galois-theoretic characterization of additive (linearized) polynomials among
all rational functions, which is of independent interest.
