For the functions $LaTeX: \displaystyle f(x)=\sqrt{x + 7}$ and $LaTeX: \displaystyle g(x)=x - 12$ , find $LaTeX: \displaystyle \left(\frac{f}{g}\right)(x)$ and the domain of $LaTeX: \displaystyle \left(\frac{f}{g}\right)(x)$

$LaTeX: \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x + 7}}{x - 12}$ . The domain of $LaTeX: \displaystyle f$ is the solution to $LaTeX: \displaystyle x + 7\geq 0$ . Solving gives $LaTeX: \displaystyle [-7,\infty)$ . The domain of $LaTeX: \displaystyle g$ is all real numbers and the zero is $LaTeX: \displaystyle x - 12=0 \iff x= 12$ . The domain is the intersection of the domains of $LaTeX: \displaystyle f$ and $LaTeX: \displaystyle g$ excluding the zeros of $LaTeX: \displaystyle g$ . This gives $LaTeX: \displaystyle [-7,12)\cup (12,\infty)$