If \(\varphi \in \mathbb{C}(z)\) is a rational function of degree \(d \geq 2\), then \(\varphi\) defines a holomorphic map \(\mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})\), where \(\mathbb{P}^1(\mathbb{C})\) is the complex projective space of dimension 1. Let \(\varphi^n\) be the \(n\)-th iterate of \(\varphi\). The authors start by proving the following theorem. Let \(\varphi \in \mathbb{C}(z)\) be a rational function of degree \(d \geq 2\). If \(n \geq 4\), then \(\varphi^{-n}(\infty)\) contains at least 3 distinct points unless \(\varphi^2 = \varphi \circ \varphi\) is a polynomial. In combination with Picard's theorem, they use this result to prove a 1993 theorem of Silverman: Let \(\varphi \in \mathbb{C}(z)\) be a rational function of degree \(d \geq 2\). If \(f: \mathbb{C} \to \mathbb{P}^1(\mathbb{C})\) is a holomorphic map, and if the image of \(\varphi^n(f)\) omits \(\infty\) for some integer \(n \geq 4\) (where \(\varphi^2\) is not a polynomial), then \(f\) must be a constant. With the help of Nevalinna's theorem for meromorphic functions, the authors prove an interesting quantitative version of this result, and they also show that Silverman's theorem still remains true for \(n = 2\) and \(n = 3\), with 7 exceptions.