An algebra problem by Gazar Khalid

We know:

$\text{Domain of } \left(\dfrac{f}{g}\right)(x) = \text{Domain of } f(x) \cap \text{Domain of } g(x)$

The domain of $f(x)$ :

$\begin{aligned} \text{Domain of } f(x) \implies \sqrt{x + 6} &\geq 0 \\ x &\geq -6 \\ x &\in [-6, \infty) \end{aligned}$

The domain of $g(x)$ :

$\begin{aligned} \text{Domain of } g(x) \implies \dfrac{x+9}{x-1} &\blue{(x-1) \ne 0} \\ x \in \R - \{1\} &\blue{\text{Not defined for x-1}} \end{aligned}$

Now intersection of values is:

$x \in [-6, \infty) \cap x \in \R - \{1\} \quad \implies \quad x \in [-6, \infty) - \{1\}$ $x \in \boxed{[-6, 1) \cup (1, \infty)}$

The domain of $f(x)$ is $x \in [-6, \infty)$ . The domain of $g(x)$ is $x \in (-\infty, 1) \cup (1, \infty)$ . Therefore, the domain of $\dfrac {f(x)}{g(x)}$ is $x \in \boxed{[-6, 1) \cup (1, \infty)}$ .