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Questions: Algebra BusinessCalculus
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For the functions \(\displaystyle f(x)=\sqrt{x + 6}\) and \(\displaystyle g(x)=x - 11\), find \(\displaystyle \left(\frac{f}{g}\right)(x)\) and the domain of \(\displaystyle \left(\frac{f}{g}\right)(x)\)
\(\displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x + 6}}{x - 11}\). The domain of \(\displaystyle f\) is the solution to \(\displaystyle x + 6\geq 0\). Solving gives \(\displaystyle [-6,\infty)\). The domain of \(\displaystyle g\) is all real numbers and the zero is \(\displaystyle x - 11=0 \iff x= 11\). The domain is the intersection of the domains of \(\displaystyle f\) and \(\displaystyle g\) excluding the zeros of \(\displaystyle g\). This gives \(\displaystyle [-6,11)\cup (11,\infty)\)
\begin{question}For the functions $f(x)=\sqrt{x + 6}$ and $g(x)=x - 11$, find $\left(\frac{f}{g}\right)(x)$ and the domain of $\left(\frac{f}{g}\right)(x)$
\soln{10cm}{$\frac{f}{g}\left(x\right)=\frac{\sqrt{x + 6}}{x - 11}$. The domain of $f$ is the solution to $x + 6\geq 0$. Solving gives $[-6,\infty)$. The domain of $g$ is all real numbers and the zero is $x - 11=0 \iff x= 11$. The domain is the intersection of the domains of $f$ and $g$ excluding the zeros of $g$. This gives $[-6,11)\cup (11,\infty)$}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>For the functions <img class="equation_image" title=" \displaystyle f(x)=\sqrt{x + 6} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%5Csqrt%7Bx%20%2B%206%7D%20" alt="LaTeX: \displaystyle f(x)=\sqrt{x + 6} " data-equation-content=" \displaystyle f(x)=\sqrt{x + 6} " /> and <img class="equation_image" title=" \displaystyle g(x)=x - 11 " src="/equation_images/%20%5Cdisplaystyle%20g%28x%29%3Dx%20-%2011%20" alt="LaTeX: \displaystyle g(x)=x - 11 " data-equation-content=" \displaystyle g(x)=x - 11 " /> , 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><p> <p> <img class="equation_image" title=" \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x + 6}}{x - 11} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bf%7D%7Bg%7D%5Cleft%28x%5Cright%29%3D%5Cfrac%7B%5Csqrt%7Bx%20%2B%206%7D%7D%7Bx%20-%2011%7D%20" alt="LaTeX: \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x + 6}}{x - 11} " data-equation-content=" \displaystyle \frac{f}{g}\left(x\right)=\frac{\sqrt{x + 6}}{x - 11} " /> . 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 + 6\geq 0 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%206%5Cgeq%200%20" alt="LaTeX: \displaystyle x + 6\geq 0 " data-equation-content=" \displaystyle x + 6\geq 0 " /> . Solving gives <img class="equation_image" title=" \displaystyle [-6,\infty) " src="/equation_images/%20%5Cdisplaystyle%20%5B-6%2C%5Cinfty%29%20" alt="LaTeX: \displaystyle [-6,\infty) " data-equation-content=" \displaystyle [-6,\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 - 11=0 \iff x= 11 " src="/equation_images/%20%5Cdisplaystyle%20x%20-%2011%3D0%20%20%5Ciff%20x%3D%2011%20" alt="LaTeX: \displaystyle x - 11=0 \iff x= 11 " data-equation-content=" \displaystyle x - 11=0 \iff x= 11 " /> . 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 [-6,11)\cup (11,\infty) " src="/equation_images/%20%5Cdisplaystyle%20%5B-6%2C11%29%5Ccup%20%2811%2C%5Cinfty%29%20" alt="LaTeX: \displaystyle [-6,11)\cup (11,\infty) " data-equation-content=" \displaystyle [-6,11)\cup (11,\infty) " /> </p> </p>