In this paper, we study rational functions of $q$-Racah type and a multivariate extension, using representation theory of $\mathcal U_q(\mathfrak{ sl}_2)$. Eigenfunctions of twisted primitive elements in $\mathcal U_q(\mathfrak{ su}_2)$ can be expressed in terms of $q^{ -1}$-Krawtchouk polynomials. Using this, we show that overlap coefficients of solutions of a generalized eigenvalue problem (GEVP) and an eigenvalue problem (EVP) can be expressed in terms of a rational function of $_4\varphi_3$-type. With help of the quantum algebra, we derive (bi)orthogonality relations as well as a GEVP for these functions. Furthermore, using this new algebraic interpretation, we can exploit the co-algebra structure of $\mathcal U_q(\mathfrak{ sl}_2)$ to find a multivariate extension of these rational functions and derive biorthogonality relations and GEVPs for the multivariate functions. Then we repeat this procedure for the non-compact quantum algebra $\mathcal U_q(\mathfrak{ su}_{ 1,1})$, where the $q^{ -1}$-Al-Salam--Chihara polynomials play the role of the $q^{ -1}$-Krawtchouk polynomials. As an application of the multivariate rational functions, we show that they appear as duality functions for certain interacting particle systems.
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