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Let q be a prime power, $$\mathbb {F}_q$$ be the finite field of order q and $$\mathbb {F}_q(x)$$ be the field of rational functions over $$\mathbb {F}_q$$. In this paper we classify and count all rational functions $$\varphi \in \mathbb {F}_q(x)$$ of degree 3 that induce a permutation of $$\mathbb {P}^1(\mathbb {F}_q)$$. As a consequence of our classification, we can show that there is no complete permutation rational function of degree 3 unless $$3\mid q$$ and $$\varphi $$ is a polynomial.

Full classification of permutation rational functions and complete rational functions of degree three over finite fields