1.0E: Exercises
- Page ID
- 10622
This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.
Exercise \(\PageIndex{1}\)
| \(x\) | \(y\) | \(x\) | \(y\) |
| -3 | 9 | 1 | 1 |
| -2 | 4 | 2 | 4 |
| -1 | 1 | 3 | 9 |
| 0 | 0 |
- Answer
-
a. Domain = {\(−3,−2,−1,0,1,2,3\)}, range = {\(0,1,4,9\)}
b. Yes, a function
Exercise \(\PageIndex{2}\)
| \(x\) | \(y\) | \(x\) | \(y\) |
| 1 | -3 | 1 | 1 |
| 2 | -2 | 2 | 2 |
| 3 | -1 | 3 | 3 |
| 0 | 0 |
- Answer
-
a. Domain = {\(0,1,2,3\)}, range = {\(−3,−2,−1,0,1,2,3\)}
b. No, not a function
Exercise \(\PageIndex{3}\)
| \(x\) | \(y\) | \(x\) | \(y\) |
|---|---|---|---|
| 3 | 3 | 15 | 1 |
| 5 | 2 | 21 | 2 |
| 8 | 1 | 33 | 3 |
| 10 | 0 |
- Answer
-
a. Domain = {\(3,5,8,10,15,21,33\)}, range = {\(0,1,2,3\)}
b. Yes, a function
For the following exercises, find the values for each function, if they exist, then simplify.
a. \(f(0)\) b. \(f(1)\) c. \(f(3)\) d. \(f(−x)\) e. \(f(a)\) f. \(f(a+h)\)
Exercise \(\PageIndex{4}\)
\(f(x)=5x−2\)
- Answer
-
a. \(−2\) b. \(3\) c. \(13\) d. \(−5x−2\) e. \(5a−2\) f. \(5a+5h−2\)
Exercise \(\PageIndex{5}\)
\(f(x)=\frac{2}{x}\)
- Answer
-
a. Undefined b. \(2\) c. \(23\) d. \(−\frac{2}{x}\) e \(\frac{2}{a}\) f. \(\frac{2}{a+h}\)
Exercise \(\PageIndex{6}\)
\(f(x)=\sqrt{6x+5}\)
- Answer
-
a. \(\sqrt{5}\) b. \(\sqrt{11}\) c. \(\sqrt{23}\) d. \(\sqrt{−6x+5}\) e. \(\sqrt{6a+5}\) f. \(\sqrt{6a+6h+5}\)
Exercise \(\PageIndex{7}\)
\(f(x)=|x−7|+8\)
- Answer
-
a. \(15\) b. \(14\) c. \(12\) d. \(|x+7|+8\) e. \(|a−7|+8\) f. \(|a+h−7|+8\)
Exercise \(\PageIndex{8}\)
\(f(x)=\frac{x−2}{3x+7}\)
- Answer
-
a. \frac{-2}{7} b. \(-.1\) c. \(\frac{1}{17}\) d. \(-\frac{x+2}{-3x+7}\) e \(\frac{a−2}{3a+7}\) f. \(\frac{a+h−2}{3a+3h+7}\)
Exercise \(\PageIndex{9}\)
\(f(x)=9\)
- Answer
-
a. 9 b. 9 c. 9 d. 9 e. 9 f. 9
For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
Exercise \(\PageIndex{10}\)
\(g(x)=\sqrt{8x−1}\)
- Answer
-
\(x≥\frac{1}{8};y≥0;x=\frac{1}{8}\); no y-intercept
Exercise \(\PageIndex{11}\)
\(f(x)=−1+\sqrt{x+2}\)
- Answer
-
\(x≥−2;y≥−1;x=−1;y=−1+\sqrt{2}\)
Exercise \(\PageIndex{12}\)
\(g(x)=\frac{3}{x−4}\)
- Answer
-
\(x≠4;y≠0\); no x-intercept; \(y=−\frac{3}{4}\)
Exercise \(\PageIndex{13}\)
\(g(x)=\sqrt{\frac{7}{x−5}}\)
- Answer
-
\(x>5;y>0\); no intercepts
Exercise \(\PageIndex{14}\)
\(f(x)=\frac{x}{x^2−16}\)
- Answer
-
\(x≠\pm 4\); \(x=0,y=0\)
Contributors and Attributions
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.
Pamini Thangarajah (Mount Royal University, Calgary, Alberta, Canada)