[Community Question] Linear-algebra: How to find $f(x)$ given $g(x)$ and $f \circ g$

One of our user asked:

i know how to find $g(x)$ if $f(x)$ and $f \omicron g (x)$ is known

let say

$$f \circ g (x) = \frac{x}{x-2}$$

and

$$g(x) = x- 2$$

find $f(x)$

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