Let $f: Y \subset \mathbb{R}^n \to \mathbb{R}^m$ be a non-constant function and $g: \mathbb{R}^m_+ \times Y \to \mathbb{R}$ be defined as follows: $$g(\mathbf{x},\mathbf{y}) = \mathbf{x}^T f(\mathbf{y})$$
I am looking for conditions on $f$ and $Y$ under which $g$ is guaranteed to be convex in $(\mathbf{x},\mathbf{y})$ over the set $\mathbb{R}^m_+ \times Y$.
I have made attempts at proving that this function can never be convex but failed. I have also failed at finding an example of an $f$ for which $g$ is convex. Any clues?