Show that a function admits at most one maximizer or provide a counter example.

Let $g$ be a strictly concave, increasing, and differentiable function defined on the real line and consider the map $f$ defined on the non-negative reals by $ f(x) = (x/(1+x))g(a-x) + (1/(1+x))g(-x)$, where $a > 0$ is fixed. Observe that $f(x)$ is a convex combination of $g$ evaluated at two district point. I am trying to show that this map admits at most one maximizer. I have made several attempts (taking derivatives, trying to establish concavity or quasi-concavity), but to no avail. I also can't seem to find a counter example. Thanks!