For the functions $LaTeX: \displaystyle f(x)=\sqrt{x - 7}$ and $LaTeX: \displaystyle g(x)=x + 2$ , find $LaTeX: \displaystyle \left(\frac{f}{g}\right)(x)$ and the domain of $LaTeX: \displaystyle \left(\frac{f}{g}\right)(x)$

$LaTeX: \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x - 7}}{x + 2}$ . 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 $LaTeX: \displaystyle x + 2=0 \iff x= -2$ . 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 $LaTeX: \displaystyle [7,\infty)$