\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Find the domain of \(\displaystyle f(x) = \sqrt{- 17 x - 9} - 10\). Write the solution in interval notation.
The domain of a radical function is the solution to the inequality radicand \(\displaystyle \geq 0\). The inequality is \(\displaystyle - 17 x - 9 \geq 0\). Isolating \(\displaystyle x\) gives \(\displaystyle - 17 x \geq 9\) Dividing to finish isolating \(\displaystyle x\) gives \(\displaystyle x \leq - \frac{9}{17}\). The domain is \(\displaystyle \left(-\infty, - \frac{9}{17}\right]\)
\begin{question}Find the domain of $f(x) = \sqrt{- 17 x - 9} - 10$. Write the solution in interval notation.
\soln{9cm}{The domain of a radical function is the solution to the inequality radicand $\geq 0$. The inequality is $- 17 x - 9 \geq 0$. Isolating $x$ gives $- 17 x \geq 9$ Dividing to finish isolating $x$ gives $x \leq - \frac{9}{17}$. The domain is $\left(-\infty, - \frac{9}{17}\right]$}
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the domain of <img class="equation_image" title=" \displaystyle f(x) = \sqrt{- 17 x - 9} - 10 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Csqrt%7B-%2017%20x%20-%209%7D%20-%2010%20" alt="LaTeX: \displaystyle f(x) = \sqrt{- 17 x - 9} - 10 " data-equation-content=" \displaystyle f(x) = \sqrt{- 17 x - 9} - 10 " /> . Write the solution in interval notation. </p> </p><p> <p>The domain of a radical function is the solution to the inequality radicand <img class="equation_image" title=" \displaystyle \geq 0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cgeq%200%20" alt="LaTeX: \displaystyle \geq 0 " data-equation-content=" \displaystyle \geq 0 " /> . The inequality is <img class="equation_image" title=" \displaystyle - 17 x - 9 \geq 0 " src="/equation_images/%20%5Cdisplaystyle%20-%2017%20x%20-%209%20%5Cgeq%200%20" alt="LaTeX: \displaystyle - 17 x - 9 \geq 0 " data-equation-content=" \displaystyle - 17 x - 9 \geq 0 " /> . Isolating <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> gives <img class="equation_image" title=" \displaystyle - 17 x \geq 9 " src="/equation_images/%20%5Cdisplaystyle%20-%2017%20x%20%5Cgeq%209%20" alt="LaTeX: \displaystyle - 17 x \geq 9 " data-equation-content=" \displaystyle - 17 x \geq 9 " /> Dividing to finish isolating <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> gives <img class="equation_image" title=" \displaystyle x \leq - \frac{9}{17} " src="/equation_images/%20%5Cdisplaystyle%20x%20%5Cleq%20-%20%5Cfrac%7B9%7D%7B17%7D%20" alt="LaTeX: \displaystyle x \leq - \frac{9}{17} " data-equation-content=" \displaystyle x \leq - \frac{9}{17} " /> . The domain is <img class="equation_image" title=" \displaystyle \left(-\infty, - \frac{9}{17}\right] " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28-%5Cinfty%2C%20-%20%5Cfrac%7B9%7D%7B17%7D%5Cright%5D%20" alt="LaTeX: \displaystyle \left(-\infty, - \frac{9}{17}\right] " data-equation-content=" \displaystyle \left(-\infty, - \frac{9}{17}\right] " /> </p> </p>