2.3.1: Exercises 2.3
- Page ID
- 63330
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
What does it mean if \(x=2\) is in the domain of \(f(x)\)?
- Answer
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It means that \(2\) is a valid input for the function \(f\).
Exercise \(\PageIndex{2}\)
What does it mean if \(x=4\) is not in the domain of \(f(x)\)?
- Answer
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It means that \(4\) is not a valid input for the function \(f\).
Exercise \(\PageIndex{3}\)
T/F: The domain of \(f(g(x))\) depends only on the domain of \(g(x)\). Explain.
- Answer
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False; it depends on both the domain of \(f(x)\) and the domain of \(g(x)\).
Exercise \(\PageIndex{4}\)
T/F: The domain of \(\frac{f(x)}{g(x)}\) depends only on where \(g(x)=0\). Explain.
- Answer
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False; if \(g(x)=0\), \(\frac{f(x)}{g(x)}\) is not defined, but \(f(x)\) may not be defined everywhere
Problems
In exercises \(\PageIndex{5}\) - \(\PageIndex{7}\), express the domain of the given function using interval notation.
Exercise \(\PageIndex{5}\)
\(x\leq 4\) and \(x>-6\)
- Answer
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\(x \in (-6,4]\)
Exercise \(\PageIndex{6}\)
\(-3\leq x \leq 10\)
- Answer
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\(x \in [-3,10]\)
Exercise \(\PageIndex{7}\)
\(x > 4\) or \(-2>x\)
- Answer
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\(x \in (-\infty,-2)\cup (4,\infty)\)
In exercises \(\PageIndex{8}\) - \(\PageIndex{11}\), write each statement using inequalities.
Exercise \(\PageIndex{8}\)
\(x \in [3,4)\cup (4,\infty)\)
- Answer
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\(3 \leq x <4\) or \(x>4\)
Exercise \(\PageIndex{9}\)
\(x \in [-2,4)\)
- Answer
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\(-2 \leq x < 4\)
Exercise \(\PageIndex{10}\)
\(x \in (5,6] \cup [7,8)\)
- Answer
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\(5 < x \leq 6\) or \(7 \leq x <8\)
Exercise \(\PageIndex{11}\)
\(x \in (5,6] \cup [7,8)\)
- Answer
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\(5 < x \leq 6\) or \(7 \leq x <8\)
In exercises \(\PageIndex{12}\) - \(\PageIndex{22}\), express the domain of the given function using interval notation.
Exercise \(\PageIndex{12}\)
\(\displaystyle \frac{\sqrt{x+11}}{x-11}\)
- Answer
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\(x \in [-11,11) \cup (11,\infty)\)
Exercise \(\PageIndex{13}\)
\(\displaystyle \frac{\ln{(x-6)}}{2x-26}\)
- Answer
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\(x \in (6,13)\cup (13, \infty)\)
Exercise \(\PageIndex{14}\)
\(\displaystyle \frac{2t}{\sqrt{t-5}}\)
- Answer
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\(t \in (5,\infty)\)
Exercise \(\PageIndex{15}\)
\(\displaystyle \ln{(\sqrt{x+3})}\)
- Answer
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\(x \in (-3,\infty)\)
Exercise \(\PageIndex{16}\)
\(\displaystyle \theta^3+4\theta^2-2\theta+\pi\)
- Answer
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\(\theta \in (-\infty,\infty)\)
Exercise \(\PageIndex{17}\)
\(\displaystyle \frac{\log_3{(x-4)}}{\log_3{(2x)}}\)
- Answer
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D: \((4,\infty)\)
Exercise \(\PageIndex{18}\)
\(\displaystyle \frac{x}{\log_2{(2x-1)}}\)
- Answer
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D: \((\frac{1}{2},1)\cup(1,\infty))\)
Exercise \(\PageIndex{19}\)
\(\displaystyle f(x) = \ln{(4-x^2)}\)
- Answer
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D: \((-2,2)\)
Exercise \(\PageIndex{20}\)
\(\displaystyle f(x) = \ln{(x^2-4)}\)
- Answer
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D: \((-\infty,-2) \cup(2, \infty)\)
Exercise \(\PageIndex{21}\)
\(\displaystyle f(x) = \sqrt{(x+3)^2-4}\)
- Answer
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D: \((-\infty,-5] \cup [-1, \infty)\)
Exercise \(\PageIndex{22}\)
\(\displaystyle f(x) = \sqrt[3]{(x-2)^3 +1}\)
- Answer
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D: \((-\infty,\infty)\)