I have an expression involving matrices, of the form:

$$f(k)=x^T A_k^{-1}A x$$

where $x$ is a $1\times N$ vector, $A_k = A + k I$ and $A$ is an $N\times N$ matrix ($A_k$ is invertible for all $k$) and $k>0$. It is known that $f(k)$ is real and monotonic increasing, but nothing more. I need to further analyze the behaviour of $f(k)$ and the simplest way would be to plugin my matrices and plot it. However, this becomes computationally intensive for large $N$ as it involves calculating the inverses repeatedly.

One thought I had was to create a one-to-one map from $f(k)$ to an equivalent algebraic expression $g(k)$, which is easy to evaluate and investigate analytically. Indeed, if the expression had involved traces of inverses, one could've used the Stieltjes transform for some insight, but that doesn't seem likely here.

My question is: Are there general approaches/references I can look at to learn how to tackle such problems?