Suppose that we have
$E [f(Ma+x)] = c^Ta $, for all $a \in \mathbb{R}^d$
where $M \in \mathbb{R}^{n,d}$ is fixed, $c \in \mathbb{R}^d$ is fixed, and $x$ is a normal random variable in $\mathbb{R}^n$ with mean 0 and variance $\sigma^2 I$. Is it possible to prove that function $f$ must be linear to satisfy the above equation, i.e. $f(y) = l^T y$?