9. Given $f(x)=\frac{2 x+3}{x-1}$. If the graph of $y=g(x)$ is symmetric to the graph of $y=f^{-1}(x+1)$ about the line $y=x$, then the value of $g(3)$ is

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9. $\frac{7}{2}$.

From the given information, the inverse function of $y=f(x)$ is $f^{-1}(x)=\frac{x+3}{x-2}$. Therefore,
$$
f^{-1}(x+1)=\frac{x+4}{x-1} \text {. }
$$

Also, $g(x)$ is the inverse function of $f^{-1}(x+1)$, to find the value of $g(3)$, we need to solve the equation $3=\frac{x+4}{x-1}$, thus, $g(3)=\frac{7}{2}$.