Answer
$[1,3)$
Work Step by Step
Let $f(x)=\sqrt {x-1}$ and $g(x)=\dfrac{1}{\sqrt{3-x}}$
In order to find the domain we will have to set the equation inside the square root greater than and equal to zero.
we have, $\sqrt {x-1} \geq 0$ and $\sqrt{3-x} \geq 0$
or, $x-1 \geq 0$ and $(\sqrt{3-x})^2 \geq (0)^2$
or, $x \geq 1$ and $3-x \geq 0$
or, $x \geq 1$ and $x \lt 3$
Thus, Domain for $\dfrac{f(x)}{g(x)}$ will be a set of interval to both domains, that is, $[1,3)$
