3.7E: Exercises
- Page ID
- 30306
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
Use the Vertical Line Test
In the following exercises, determine whether each graph is the graph of a function.
1. ⓐ
![The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 3, 0), (3, 0), (0, negative 3), and (0, 3).](https://math.libretexts.org/@api/deki/files/34702/CNX_IntAlg_Figure_03_06_201_img_new.jpg?revision=1)
ⓑ
- Answer
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ⓐ no ⓑ yes
2. ⓐ
![The figure has an s-shaped curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The s-shaped curved line goes through the points (negative 1, 1), (0, 0), and (1, 1).](https://math.libretexts.org/@api/deki/files/34704/CNX_IntAlg_Figure_03_06_203_img_new.jpg?revision=1)
ⓑ
3. ⓐ
![The figure has a parabola opening right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (negative 2, 2), and (2, 2).](https://math.libretexts.org/@api/deki/files/34706/CNX_IntAlg_Figure_03_06_205_img_new.jpg?revision=1)
ⓑ
- Answer
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ⓐ no ⓑ yes
4. ⓐ
![The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 2, 0), (negative 4, 5), and (negative 4, negative 5). The curved line on the right goes through the points (2, 0), (4, 5), and (4, negative 5).](https://math.libretexts.org/@api/deki/files/34708/CNX_IntAlg_Figure_03_06_207_img_new.jpg?revision=1)
ⓑ
Identify Graphs of Basic Functions
In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.
5. \(f(x)=3x+4\)
- Answer
-
ⓐ
ⓑ \( D:(-\inf ,\inf ),\space R:(-\inf ,\inf ) \)
6. \(f(x)=2x+5\)
7. \(f(x)=−x−2\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
8. \(f(x)=−4x−3\)
9. \(f(x)=−2x+2\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
10. \(f(x)=−3x+3\)
11. \(f(x)=\frac{1}{2}x+1\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
12. \(f(x)=\frac{2}{3}x−2\)
13. \(f(x)=5\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ), R:{5}\)
14. \(f(x)=2\)
15. \(f(x)=−3\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R: {−3}\)
16. \(f(x)=−1\)
17. \(f(x)=2x\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
18. \(f(x)=3x\)
19. \(f(x)=−2x\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ), R:(-\inf ,\inf )\)
20. \(f(x)=−3x\)
21. \(f(x)=3x^2\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:[0,\inf )\)
22. \(f(x)=2x^2\)
23. \(f(x)=−3x^2\)
- Answer
-
ⓐ
ⓑ \( D: (-\inf ,\inf ),\space R:(-\inf ,0]\)
24. \(f(x)=−2x^2\)
25. \(f(x)=12x^2\)
- Answer
-
ⓐ
ⓑ \(D: (-\inf ,\inf ),\space R:[-\inf ,0)\)
26. \(f(x)=\frac{1}{3}x^2\)
27. \(f(x)=x^2−1\)
- Answer
-
ⓐ
ⓑ \(D: (-\inf ,\inf ),\space R:[−1, \inf )\)
28. \(f(x)=x^2+1\)
29. \(f(x)=−2x^3\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
30. \(f(x)=2x^3\)
31. \(f(x)=x^3+2\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ), R:(-\inf ,\inf )\)
32. \(f(x)=x^3−2\)
33. \(f(x)=2\sqrt{x}\)
- Answer
-
ⓐ
ⓑ \(D:[0,\inf ), R:[0,\inf )\)
34. \(f(x)=−2\sqrt{x}\)
35. \(f(x)=\sqrt{x-1}\)
- Answer
-
ⓐ
ⓑ \(D:[1,\inf ), R:[0,\inf )\)
36. \(f(x)=\sqrt{x+1}\)
37. \(f(x)=3|x|\)
- Answer
-
ⓐ
ⓑ \(D:[ −1,−1, \inf ), R:[−\inf ,\inf )\)
38. \(f(x)=−2|x|\)
39. \(f(x)=|x|+1\)
- Answer
-
ⓐ
ⓑ \(D:(-\inf ,\inf ), R:[1,\inf )\)
40. \(f(x)=|x|−1\)
Read Information from a Graph of a Function
In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation.
41.
- Answer
-
\(D: [2,\inf ),\space R: [0,\inf )\)
42.
43.
- Answer
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\(D: (-\inf ,\inf ),\space R: [4,\inf )\)
44.
45.
- Answer
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\(D: [−2,2],\space R: [0, 2]\)
46.
In the following exercises, use the graph of the function to find the indicated values.
47.
ⓐ Find: \(f(0)\).
ⓑ Find: \(f(12\pi)\).
ⓒ Find: \(f(−32\pi)\).
ⓓ Find the values for \(x\) when \(f(x)=0\).
ⓔ Find the \(x\)-intercepts.
ⓕ Find the \(y\)-intercepts.
ⓖ Find the domain. Write it in interval notation.
ⓗ Find the range. Write it in interval notation.
- Answer
-
ⓐ \(f(0)=0\) ⓑ \((\pi/2)=−1\)
ⓒ \(f(−3\pi/2)=−1\) ⓓ \(f(x)=0\) for \(x=−2\pi,-\pi,0,\pi,2\pi\)
ⓔ \((−2\pi,0),(−\pi,0),\) \((0,0),(\pi,0),(2\pi,0)\) \((f)(0,0)\)
ⓖ \([−2\pi,2\pi]\) ⓗ \([−1,1]\)
48.
ⓐ Find: \(f(0)\).
ⓑ Find: \(f(\pi)\).
ⓒ Find: \(f(−\pi)\).
ⓓ Find the values for \(x\) when \(f(x)=0\).
ⓔ Find the \(x\)-intercepts.
ⓕ Find the \(y\)-intercepts.
ⓖ Find the domain. Write it in interval notation.
ⓗ Find the range. Write it in interval notation
49.
ⓐ Find: \(f(0)\).
ⓑ Find: \(f(−3)\).
ⓒ Find: \(f(3)\).
ⓓ Find the values for \(x\) when \(f(x)=0\).
ⓔ Find the \(x\)-intercepts.
ⓕ Find the \(y\)-intercepts.
ⓖ Find the domain. Write it in interval notation.
ⓗ Find the range. Write it in interval notation.
- Answer
-
ⓐ \(f(0)=−6\) ⓑ \(f(−3)=3\) ⓒ \(f(3)=3\) ⓓ \(f(x)=0\) for no x ⓔ none ⓕ \(y=6\) ⓖ \([−3,3]\)
ⓗ \([−3,6]\)
50.
ⓐ Find: \(f(0)\).
ⓑ Find the values for \(x\) when \(f(x)=0\).
ⓒ Find the \(x\)-intercepts.
ⓓ Find the \(y\)-intercepts.
ⓔ Find the domain. Write it in interval notation.
ⓕ Find the range. Write it in interval notation
Writing Exercises
51. Explain in your own words how to find the domain from a graph.
52. Explain in your own words how to find the range from a graph.
53. Explain in your own words how to use the vertical line test.
54. Draw a sketch of the square and cube functions. What are the similarities and differences in the graphs?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
![The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “use the vertical line test”, “identify graphs of basic functions”, and “read information from a graph”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved](https://math.libretexts.org/@api/deki/files/23400/CNX_IntAlg_Figure_03_06_219_img_new.jpg?revision=1)
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?