We consider discrete best approximation problems in the framework of tropical algebra, which focuses on semirings and semifields with idempotent addition. Given a set of samples from input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function by tropical Puiseux polynomial and rational functions. We describe a solution approach that transforms the problem into the best approximation of linear vector equations. Application of this approach yields a direct analytical solution for the polynomial approximation problem and an iterative algorithmic solution for approximation by rational functions. As an illustration, we present results of the best Chebyshev approximation by piecewise linear functions.