  This paper deals with the analogue of Inhomogeneous Diophantine Approximation
in function fields. The inhomogeneous approximation constant of a Laurent
series $\theta\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ with respect
to $\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be
$c(\theta,\gamma)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle
N\theta - \gamma \rangle|$. We show that for every $\theta$ there exists
$\gamma$ such that $c(\theta,\gamma)\geq q^{-2}$, and find a sufficient
condition on $\theta$ which forces $c(\theta,\gamma) \leq q^{-2}$ for every
$\gamma$. Given $\theta$, we prove that the set
$BA_{\theta}=\left\{\gamma\in\mathbb
F_q\left(\left(\frac{1}{t}\right)\right)\;:\; c(\theta,\gamma)>0\right\}$ has
full Hausdorff dimension. Our methods allow us to solve the case of vectors in
$\mathbb F_q\left(\left(\frac{1}{t}\right)\right)^d$ as well. Our results offer
a strengthening to analogues of results for real inhomogeneous approximation.
