How to calculate a matrix as a function of a matrix

Question

Say, we have a matrix

$$f=\begin{bmatrix} x& 2x\\ 3x& x+2 \end{bmatrix}.$$

Say we have a matrix another matrix represented by

$$g=\begin{bmatrix} 5x& 6x \\ 4x& x-1 \end{bmatrix}$$

Is there a way to calculate $$f(g(x))?$$ If so, how would it be performed in this example? I'm assuming it wouldn't be a strict multiplication, rather some use of a chain rule. I am not exactly sure how to proceed.

Answer 1

Supposing that $x\in \Bbb R$ then you have $f:\Bbb R\to \Bbb R^{2\times 2}$ and $g:\Bbb R\to \Bbb R^{2\times 2}$.

If you want $f(g(x))$ you should have the image of $g$ inside the domain of $f$, what is not possible because a subset of $\Bbb R^{2\times 2}$ is not inside of $\Bbb R$.