Iteration of the function \(f_ \lambda (z)=\lambda + z+\tan (z), z \in \mathbb {C}\) is investigated in this article for \(\lambda \in \mathbb {C}\) . It is proved that for every \(\lambda \) , the Fatou set of \(f_\lambda \) has a completely invariant Baker domain \(B_\lambda \) ; we call it the primary Fatou component. The rest of the article deals with \(f_\lambda \) when it is topologically hyperbolic. For all real \(\lambda \) or \(\lambda \) such that \( \lambda = k\pi +i \lambda _2\) for some integer k and \(0< \lambda _2<1\) , the Fatou set of \(f_\lambda \) is the union of \(B_\lambda \) and another completely invariant Baker domain. It is proved that if \(|2+\lambda ^2|<1\) , then the Fatou set is the union of \(B_\lambda \) and infinitely many invariant attracting domains (along with their pre-images). Each such attracting domain U has exactly one invariant access to infinity and is unbounded in a special way; \(\{\Im (z): z\in U\}\) is unbounded whereas for every \(z_0\in U\) , \(\{\Re (z): z \in U ~\text{ and }~\Im (z)>\Im (z_0)\}\) is bounded. If \(\Im (\lambda )> \sqrt{2}+ \sinh ^{-1}(1)\) then it is found that the primary Fatou component is the only Fatou component and the Julia set is disconnected. For every natural number k, there exists a complex number \(\lambda \) namely, \(\lambda =k\pi +i\frac{\pi }{2}\) such that the Fatou set of \(f_\lambda \) has k many wandering domains with distinct grand orbits. These wandering domains are found to be escaping. The Fatou set of \(f_{k\pi + i\frac{\pi }{2}}\) is the union of \(B_\lambda \) and the grand orbits of these k wandering domains.
