  Bounded potentials are good models to represent the effective two-body
interaction in some colloidal systems, such as dilute solutions of polymer
chains in good solvents. The simplest bounded potential is that of penetrable
spheres, which takes a positive finite value if the two spheres are overlapped,
being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model
is far from trivial, since interactions are not restricted to nearest neighbors
and so its exact solution is not known. In this paper we first derive the exact
correlation functions of penetrable-rod fluids to second order in density at
any temperature, as well as in the high-temperature and zero-temperature limits
at any density. Next, two simple analytic theories are constructed: a
high-temperature approximation based on the exact asymptotic behavior in the
limit $T\to\infty$ and a low-temperature approximation inspired by the exact
result in the opposite limit $T\to 0$. Finally, we perform Monte Carlo
simulations for a wide range of temperatures and densities to assess the
validity of both theories. It is found that they complement each other quite
well, exhibiting a good agreement with the simulation data within their
respective domains of applicability and becoming practically equivalent on the
borderline of those domains. A perspective on the extension of both approaches
to the more realistic three-dimensional case is provided.
