Let k f₁, ¼, f_r Î \mathbb F_q^k(x)f{_1}, \ldots, f{_r} \in {\mathbb F}_{q^k}(x) be a system of rational functions forming a strongly linearly independent set over a finite field \mathbb F_q{\mathbb F}_q . Let g₁, ¼, g_r Î \mathbb F_q\gamma_1, \ldots, \gamma_r \in {\mathbb F}_q be arbitrarily prescribed elements. We prove that for all sufficiently large extensions \mathbb F_q^km{\mathbb F}_{q^{km}} , there is an element x Î \mathbb F_q^km\xi \in {\mathbb F}_{q^{km}} of prescribed order such that Tr_{\mathbb Fq^km /\mathbb Fq}(f_i(x))=g_i{\rm Tr}_{{\mathbb F}_{q^{km} }/{\mathbb F}_q}(f_i(\xi))=\gamma_i is the relative trace map from \mathbb F_q^km{\mathbb F}_{q^{km}} onto \mathbb F_q{\mathbb F}_q We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.