Composite function to be nonconcave

Question

Given that f is a concave function and g is a concave, but not necessarily increasing function, is it possible that g o f is nonconcave? If so, what can be a good example?

Answer 1

Let $f,g \colon \def\R{\mathbf R}\R \to \R$ be given by $f(x) = -x^2$, $g(x) = -x$, then $f,g$ are concave, but $(g \circ f)(x) = -(-x^2) = x^2$ is not.