  In this paper, applied strictly monotonic increasing scaled maps, a kind of
well-conditioned linear barycentric rational interpolations are proposed to
approximate functions of singularities at the origin, such as $x^\alpha$ for
$\alpha \in (0,1)$ and $\log(x)$. It just takes $O(N)$ flops and can achieve
fast convergence rates with the choice the scaled parameter, where $N$ is the
maximum degree of the denominator and numerator. The construction of the
rational interpolant couples rational polynomials in the barycentric form of
second kind with the transformed Jacobi-Gauss-Lobatto points. Numerical
experiments are considered which illustrate the accuracy and efficiency of the
algorithms. The convergence of the rational interpolation is also considered.
