Let $f:\mathbb{R}^{m \times n} \rightarrow \mathbb{R}^{m \times n}$ be defined as follows:
$$f(X)=X\left(\int_0^\infty e^{Mt}(NX+X^TN^T+PP^T)e^{M^T t}dt\right)^{-1}$$
where $M \in \mathbb{R}^{n \times n}$ is such that its eigenvalues have negative real part, $N \in \mathbb{R}^{n \times m}$ and $PP^T$ is positive definite. Is $f$ injective? Does the answer depend on $M,N,P?$ Bonus Question: is $f$ surjective?