3.6E: 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. ⓐ

ⓐ no ⓑ yes

2. ⓐ

3. ⓐ

ⓐ 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. $$f(x)=3x+4$$

ⓑ $$D:(-\inf ,\inf ),\space R:(-\inf ,\inf )$$

6. $$f(x)=2x+5$$

7. $$f(x)=−x−2$$

ⓑ $$D:(-\inf ,\inf ),\space R:(-\inf ,\inf )$$

8. $$f(x)=−4x−3$$

9. $$f(x)=−2x+2$$

ⓑ $$D:(-\inf ,\inf ),\space R:(-\inf ,\inf )$$

10. $$f(x)=−3x+3$$

11. $$f(x)=\frac{1}{2}x+1$$

ⓑ $$D:(-\inf ,\inf ),\space R:(-\inf ,\inf )$$

12. $$f(x)=\frac{2}{3}x−2$$

13. $$f(x)=5$$

ⓑ $$D:(-\inf ,\inf ), R:{5}$$

14. $$f(x)=2$$

15. $$f(x)=−3$$

ⓑ $$D:(-\inf ,\inf ),\space R: {−3}$$

16. $$f(x)=−1$$

17. $$f(x)=2x$$

ⓑ $$D:(-\inf ,\inf ),\space R:(-\inf ,\inf )$$

18. $$f(x)=3x$$

19. $$f(x)=−2x$$

ⓑ $$D:(-\inf ,\inf ), R:(-\inf ,\inf )$$

20. $$f(x)=−3x$$

21. $$f(x)=3x^2$$

ⓑ $$D:(-\inf ,\inf ),\space R:[0,\inf )$$

22. $$f(x)=2x^2$$

23. $$f(x)=−3x^2$$

ⓑ $$D: (-\inf ,\inf ),\space R:(-\inf ,0]$$

24. $$f(x)=−2x^2$$

25. $$f(x)=12x^2$$

ⓑ $$D: (-\inf ,\inf ),\space R:[-\inf ,0)$$

26. $$f(x)=\frac{1}{3}x^2$$

27. $$f(x)=x^2−1$$

ⓑ $$D: (-\inf ,\inf ),\space R:[−1, \inf )$$

28. $$f(x)=x^2+1$$

29. $$f(x)=−2x^3$$

ⓑ $$D:(-\inf ,\inf ),\space R:(-\inf ,\inf )$$

30. $$f(x)=2x^3$$

31. $$f(x)=x^3+2$$

ⓑ $$D:(-\inf ,\inf ), R:(-\inf ,\inf )$$

32. $$f(x)=x^3−2$$

33. $$f(x)=2\sqrt{x}$$

ⓑ $$D:[0,\inf ), R:[0,\inf )$$

34. $$f(x)=−2\sqrt{x}$$

35. $$f(x)=\sqrt{x-1}$$

ⓑ $$D:[1,\inf ), R:[0,\inf )$$

36. $$f(x)=\sqrt{x+1}$$

37. $$f(x)=3|x|$$

ⓑ $$D:[ −1,−1, \inf ), R:[−\inf ,\inf )$$

38. $$f(x)=−2|x|$$

39. $$f(x)=|x|+1$$

ⓑ $$D:(-\inf ,\inf ), R:[1,\inf )$$

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.

$$D: [2,\inf ),\space R: [0,\inf )$$

42.

43.

$$D: (-\inf ,\inf ),\space R: [4,\inf )$$

44.

45.

$$D: [−2,2],\space R: [0, 2]$$

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.

ⓐ $$f(0)=0$$ ⓑ $$(\pi/2)=−1$$
ⓒ $$f(−3\pi/2)=−1$$ ⓓ $$f(x)=0$$ 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.

ⓐ $$f(0)=−6$$ ⓑ $$f(−3)=3$$ ⓒ $$f(3)=3$$ ⓓ $$f(x)=0$$ 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?

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