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Find the domain of \(\displaystyle f(x) = \sqrt{5 x + 10} + 9\). 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 \(\displaystyle 5 x + 10 \geq 0\). Isolating \(\displaystyle x\) gives \(\displaystyle 5 x \geq -10\) Dividing to finish isolating \(\displaystyle x\) gives \(\displaystyle x \geq -2\). The domain is \(\displaystyle \left[-2, \infty\right)\)
\begin{question}Find the domain of $f(x) = \sqrt{5 x + 10} + 9$. 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 $5 x + 10 \geq 0$. Isolating $x$ gives $5 x \geq -10$ Dividing to finish isolating $x$ gives $x \geq -2$. The domain is $\left[-2, \infty\right)$} \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{5 x + 10} + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Csqrt%7B5%20x%20%2B%2010%7D%20%2B%209%20" alt="LaTeX: \displaystyle f(x) = \sqrt{5 x + 10} + 9 " data-equation-content=" \displaystyle f(x) = \sqrt{5 x + 10} + 9 " /> . 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 5 x + 10 \geq 0 " src="/equation_images/%20%5Cdisplaystyle%205%20x%20%2B%2010%20%5Cgeq%200%20" alt="LaTeX: \displaystyle 5 x + 10 \geq 0 " data-equation-content=" \displaystyle 5 x + 10 \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 5 x \geq -10 " src="/equation_images/%20%5Cdisplaystyle%205%20x%20%5Cgeq%20-10%20" alt="LaTeX: \displaystyle 5 x \geq -10 " data-equation-content=" \displaystyle 5 x \geq -10 " /> 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 \geq -2 " src="/equation_images/%20%5Cdisplaystyle%20x%20%5Cgeq%20-2%20" alt="LaTeX: \displaystyle x \geq -2 " data-equation-content=" \displaystyle x \geq -2 " /> . The domain is <img class="equation_image" title=" \displaystyle \left[-2, \infty\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%5B-2%2C%20%5Cinfty%5Cright%29%20" alt="LaTeX: \displaystyle \left[-2, \infty\right) " data-equation-content=" \displaystyle \left[-2, \infty\right) " /> </p> </p>