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For the functions f(x)=x−7 and g(x)=x+2, find (fg)(x) and the domain of (fg)(x)

fg(x)=x−7x+2. The domain of f is the solution to x−7≥0. Solving gives [7,∞). The domain of g is all real numbers and the zero is x+2=0⟺x=−2. The domain is the intersection of the domains of f and g excluding the zeros of g. This gives [7,∞)

Download LATEX 

\begin{question}For the functions $f(x)=\sqrt{x - 7}$ and $g(x)=x + 2$, find $\left(\frac{f}{g}\right)(x)$ and the domain of $\left(\frac{f}{g}\right)(x)$
    \soln{10cm}{$\frac{f}{g}\left(x\right)=\frac{\sqrt{x - 7}}{x + 2}$. The domain of $f$ is the solution to $x - 7\geq 0$.  Solving gives $[7,\infty)$.  The domain of $g$ is all real numbers and the zero is $x + 2=0  \iff x= -2$.  The domain is the intersection of the domains of $f$ and $g$ excluding the zeros of $g$. This gives $[7,\infty)$}

\end{question}

Download Question and Solution EnvironmentLATEX
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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

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HTML for Canvas 
<p> <p>For the functions  <img class="equation_image" title=" \displaystyle f(x)=\sqrt{x - 7} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%5Csqrt%7Bx%20-%207%7D%20" alt="LaTeX:  \displaystyle f(x)=\sqrt{x - 7} " data-equation-content=" \displaystyle f(x)=\sqrt{x - 7} " />  and  <img class="equation_image" title=" \displaystyle g(x)=x + 2 " src="/equation_images/%20%5Cdisplaystyle%20g%28x%29%3Dx%20%2B%202%20" alt="LaTeX:  \displaystyle g(x)=x + 2 " data-equation-content=" \displaystyle g(x)=x + 2 " /> , find  <img class="equation_image" title=" \displaystyle \left(\frac{f}{g}\right)(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%5Cfrac%7Bf%7D%7Bg%7D%5Cright%29%28x%29%20" alt="LaTeX:  \displaystyle \left(\frac{f}{g}\right)(x) " data-equation-content=" \displaystyle \left(\frac{f}{g}\right)(x) " />  and the domain of  <img class="equation_image" title=" \displaystyle \left(\frac{f}{g}\right)(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%5Cfrac%7Bf%7D%7Bg%7D%5Cright%29%28x%29%20" alt="LaTeX:  \displaystyle \left(\frac{f}{g}\right)(x) " data-equation-content=" \displaystyle \left(\frac{f}{g}\right)(x) " /> </p> </p>
HTML for Canvas
<p> <p> <img class="equation_image" title=" \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x - 7}}{x + 2} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bf%7D%7Bg%7D%5Cleft%28x%5Cright%29%3D%5Cfrac%7B%5Csqrt%7Bx%20-%207%7D%7D%7Bx%20%2B%202%7D%20" alt="LaTeX:  \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x - 7}}{x + 2} " data-equation-content=" \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x - 7}}{x + 2} " /> . The domain of  <img class="equation_image" title=" \displaystyle f " src="/equation_images/%20%5Cdisplaystyle%20f%20" alt="LaTeX:  \displaystyle f " data-equation-content=" \displaystyle f " />  is the solution to  <img class="equation_image" title=" \displaystyle x - 7\geq 0 " src="/equation_images/%20%5Cdisplaystyle%20x%20-%207%5Cgeq%200%20" alt="LaTeX:  \displaystyle x - 7\geq 0 " data-equation-content=" \displaystyle x - 7\geq 0 " /> .  Solving gives  <img class="equation_image" title=" \displaystyle [7,\infty) " src="/equation_images/%20%5Cdisplaystyle%20%5B7%2C%5Cinfty%29%20" alt="LaTeX:  \displaystyle [7,\infty) " data-equation-content=" \displaystyle [7,\infty) " /> .  The domain of  <img class="equation_image" title=" \displaystyle g " src="/equation_images/%20%5Cdisplaystyle%20g%20" alt="LaTeX:  \displaystyle g " data-equation-content=" \displaystyle g " />  is all real numbers and the zero is  <img class="equation_image" title=" \displaystyle x + 2=0  \iff x= -2 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%202%3D0%20%20%5Ciff%20x%3D%20-2%20" alt="LaTeX:  \displaystyle x + 2=0  \iff x= -2 " data-equation-content=" \displaystyle x + 2=0  \iff x= -2 " /> .  The domain is the intersection of the domains of  <img class="equation_image" title=" \displaystyle f " src="/equation_images/%20%5Cdisplaystyle%20f%20" alt="LaTeX:  \displaystyle f " data-equation-content=" \displaystyle f " />  and  <img class="equation_image" title=" \displaystyle g " src="/equation_images/%20%5Cdisplaystyle%20g%20" alt="LaTeX:  \displaystyle g " data-equation-content=" \displaystyle g " />  excluding the zeros of  <img class="equation_image" title=" \displaystyle g " src="/equation_images/%20%5Cdisplaystyle%20g%20" alt="LaTeX:  \displaystyle g " data-equation-content=" \displaystyle g " /> . This gives  <img class="equation_image" title=" \displaystyle [7,\infty) " src="/equation_images/%20%5Cdisplaystyle%20%5B7%2C%5Cinfty%29%20" alt="LaTeX:  \displaystyle [7,\infty) " data-equation-content=" \displaystyle [7,\infty) " /> </p> </p>