16.3E: Exercises
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Practice Makes Perfect
Use the Vertical Line Test
In the following exercises, determine whether each graph is the graph of a function.
1. ⓐ
ⓑ
- Answer
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ⓐ no ⓑ yes
2. ⓐ
ⓑ
3. ⓐ
ⓑ
- Answer
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ⓐ no ⓑ yes
4. ⓐ
ⓑ
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
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ⓐ
ⓑ \( D:(-\inf ,\inf ),\space R:(-\inf ,\inf ) \)
6. \(f(x)=2x+5\)
7. \(f(x)=−x−2\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
8. \(f(x)=−4x−3\)
9. \(f(x)=−2x+2\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
10. \(f(x)=−3x+3\)
11. \(f(x)=\frac{1}{2}x+1\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
12. \(f(x)=\frac{2}{3}x−2\)
13. \(f(x)=5\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ), R:{5}\)
14. \(f(x)=2\)
15. \(f(x)=−3\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R: {−3}\)
16. \(f(x)=−1\)
17. \(f(x)=2x\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
18. \(f(x)=3x\)
19. \(f(x)=−2x\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ), R:(-\inf ,\inf )\)
20. \(f(x)=−3x\)
21. \(f(x)=3x^2\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:[0,\inf )\)
22. \(f(x)=2x^2\)
23. \(f(x)=−3x^2\)
- Answer
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ⓐ
ⓑ \( D: (-\inf ,\inf ),\space R:(-\inf ,0]\)
24. \(f(x)=−2x^2\)
25. \(f(x)=12x^2\)
- Answer
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ⓐ
ⓑ \(D: (-\inf ,\inf ),\space R:[-\inf ,0)\)
26. \(f(x)=\frac{1}{3}x^2\)
27. \(f(x)=x^2−1\)
- Answer
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ⓐ
ⓑ \(D: (-\inf ,\inf ),\space R:[−1, \inf )\)
28. \(f(x)=x^2+1\)
29. \(f(x)=−2x^3\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ),\space R:(-\inf ,\inf )\)
30. \(f(x)=2x^3\)
31. \(f(x)=x^3+2\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ), R:(-\inf ,\inf )\)
32. \(f(x)=x^3−2\)
33. \(f(x)=2\sqrt{x}\)
- Answer
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ⓐ
ⓑ \(D:[0,\inf ), R:[0,\inf )\)
34. \(f(x)=−2\sqrt{x}\)
35. \(f(x)=\sqrt{x-1}\)
- Answer
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ⓐ
ⓑ \(D:[1,\inf ), R:[0,\inf )\)
36. \(f(x)=\sqrt{x+1}\)
37. \(f(x)=3|x|\)
- Answer
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ⓐ
ⓑ \(D:[ −1,−1, \inf ), R:[−\inf ,\inf )\)
38. \(f(x)=−2|x|\)
39. \(f(x)=|x|+1\)
- Answer
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ⓐ
ⓑ \(D:(-\inf ,\inf ), R:[1,\inf )\)
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
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\(D: [2,\inf ),\space R: [0,\inf )\)
42.
43.
- Answer
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\(D: (-\inf ,\inf ),\space R: [4,\inf )\)
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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
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ⓐ \(f(0)=0\) ⓑ \((\pi/2)=−1\)
ⓒ \(f(−3\pi/2)=−1\) ⓓ \(f(x)=0\) 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
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ⓐ \(f(0)=−6\) ⓑ \(f(−3)=3\) ⓒ \(f(3)=3\) ⓓ \(f(x)=0\) 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.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?