2.8 Practice Questions

1. Why does the domain differ for different functions?
2. How do we determine the domain of a function defined by an equation?
3. Explain why the domain of [latex]\text{}f\left(x\right)=\sqrt[3]{x}\text{}[/latex] is different from the domain of [latex]\text{}f\left(x\right)=\sqrt[]{x}[/latex].
4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?
5. How do you graph a piecewise function?

Odd Number Verbal Solutions

For the following exercises, find the domain of each function using interval notation.

6. [latex]f\left(x\right)=-2x\left(x-1\right)\left(x-2\right)[/latex]
7. [latex]f\left(x\right)=5-2{x}^{2}[/latex]
8. [latex]f\left(x\right)=3\sqrt{x-2}[/latex]
9. [latex]f\left(x\right)=3-\sqrt{6-2x}[/latex]
10. [latex]f\left(x\right)=\sqrt{4-3x}[/latex]
11. [latex]f\left(x\right)=\sqrt{x^2+4}[/latex]
12. [latex]f\left(x\right)=\sqrt[3]{1-2x}[/latex]
13. [latex]f\left(x\right)=\sqrt[3]{x-1}[/latex]
14. [latex]f\left(x\right)=\frac{9}{x-6}[/latex]
15. [latex]f\left(x\right)=\frac{3x+1}{4x+2}[/latex]
16. [latex]f\left(x\right)=\frac{\sqrt{x+4}}{x-4}[/latex]
17. [latex]f\left(x\right)=\frac{x-3}{{x}^{2}+9x-22}[/latex]
18. [latex]f\left(x\right)=\frac{1}{{x}^{2}-x-6}[/latex]
19. [latex]f\left(x\right)=\frac{2{x}^{3}-250}{{x}^{2}-2x-15}[/latex]
20. [latex]\frac{5}{\sqrt{x-3}}[/latex]
21. [latex]\frac{2x+1}{\sqrt{5-x}}[/latex]
22. [latex]f\left(x\right)=\frac{\sqrt{x-4}}{\sqrt{x-6}}[/latex]
23. [latex]f\left(x\right)=\frac{\sqrt{x-6}}{\sqrt{x-4}}[/latex]
24. [latex]f\left(x\right)=\frac{x}{x}[/latex]
25. [latex]f\left(x\right)=\frac{{x}^{2}-9x}{{x}^{2}-81}[/latex]
26. Find the domain of the function [latex]\text{}f\left(x\right)=\sqrt{2{x}^{3}-50x}\text{}[/latex] by:
    1. using algebra.
    2. graphing the function in the radicand and determining intervals on the x-axis for which the radicand is nonnegative.

Odd Number Algebraic Solutions

For the following exercises, write the domain and range of each function using interval notation.

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

38. [latex]f\left(x\right)=\left\{\begin{array}{l}x+1\;if\;x\;<-2\\-2x-3\;if\;x\geq-2\end{array}\right.[/latex]
39. [latex]f\left(x\right)=\left\{\begin{array}{l}2x-1\;if\;x\;<1\\1+x\;if\;x\geq1\end{array}\right.[/latex]
40. [latex]f\left(x\right)=\left\{\begin{array}{l}x+1\;if\;x\;<0\\x-1\;if\;x>0\end{array}\right.[/latex]
41. [latex]f\left(x\right)=\left\{\begin{array}{l}3\;if\;x\;<0\\\sqrt x\;if\;x\geq0\end{array}\right.[/latex]
42. [latex]f\left(x\right)=\left\{\begin{array}{l}x^2\;if\;x<0\\1-x\;if\;x>0\end{array}\right.[/latex]
43. [latex]f\left(x\right)=\left\{\begin{array}{l}x^2\;if\;x\;<0\\x+2\;if\;x\geq0\end{array}\right.[/latex]
44. [latex]f\left(x\right)=\left\{\begin{array}{l}x+1\;if\;x\;<1\\x^3\;if\;x\geq1\end{array}\right.[/latex]
45. [latex]f\left(x\right)=\left\{\begin{array}{l}\left|x\right|\;if\;x<2\\1\;if\;x\geq2\end{array}\right.[/latex]

Odd Number Graphical Solutions

For the following exercises, given each function [latex]f[/latex], evaluate [latex]f\left(-3\right),\text{}f\left(-2\right),\text{}f\left(-1\right)[/latex], and [latex]f\left(0\right)[/latex].

46. [latex]f\left(x\right)=\left\{\begin{array}{l}x+1\;if\;x\;<-2\\-2x-3\;if\;x\geq-2\end{array}\right.[/latex]
47. [latex]f\left(x\right)=\left\{\begin{array}{l}1\;if\;x\;\leq-3\\0\;if\;x>-3\end{array}\right.[/latex]
48. [latex]f\left(x\right)=\left\{\begin{array}{l}-2x^2+3\;if\;x\;\leq-1\\5x-7\;if\;x>-1\end{array}\right.[/latex]

For the following exercises, given each function [latex]\text{}f\text{}[/latex], evaluate [latex]f\left(-1\right),\text{}f\left(0\right),\text{}f\left(2\right),\text{}[/latex] and [latex]\text{}f\left(4\right)[/latex].

49. [latex]f\left(x\right)=\left\{\begin{array}{l}7x+3\;if\;x\;<0\\7x+6\;if\;x\geq0\end{array}\right.[/latex]
50. [latex]f\left(x\right)=\left\{\begin{array}{l}x^2-2\;if\;x\;<2\\4+\left|x-5\right|\;if\;x\geq2\end{array}\right.[/latex]
51. [latex]f\left(x\right)=\left\{\begin{array}{l}5x\;if\;x\;<0\\3\;if\;0\leq x\geq3\\x^2\;if\;x>3\end{array}\right.[/latex]

For the following exercises, write the domain for the piecewise function in interval notation.

52. [latex]f\left(x\right)=\left\{\begin{array}{l}x+1\;if\;x\;<-2\\-2x-3\;if\;x\geq-2\end{array}\right.[/latex]
53. [latex]f\left(x\right)=\left\{\begin{array}{l}x^2-2\;if\;x\;<1\\-x^2+2\;if\;x>1\end{array}\right.[/latex]
54. [latex]f\left(x\right)=\left\{\begin{array}{l}2x-3\;if\;x\;<0\\-3x^2\;if\;x\geq2\end{array}\right.[/latex]

Odd Number Numeric Solutions

55. Graph [latex]\text{}y=\frac{1}{{x}^{2}}\text{}[/latex] on the viewing window [latex]\text{}\left[-0.5,-0.1\right]\text{}[/latex] and [latex]\text{}\left[0.1,\text{ }0.5\right]\text{}[/latex]. Determine the corresponding range for the viewing window. Show the graphs.
56. Graph [latex]\text{}y=\frac{1}{x}\text{}[/latex] on the viewing window [latex]\text{}\left[-0.5,-0.1\right]\text{}[/latex] and [latex]\text{}\left[0.1,\text{ }0.5\right]\text{}[/latex]. Determine the corresponding range for the viewing window. Show the graphs.

Odd Number Technology Solutions

57. Suppose the range of a function [latex]\text{}f\text{}[/latex] is [latex]\text{}\left[-5,\text{ }8\right]\text{}[/latex]. What is the range of [latex]\text{}|f\left(x\right)|[/latex]?
58. Create a function in which the range is all nonnegative real numbers.
59. Create a function in which the domain is [latex]\text{}x>2[/latex].

Odd Number Extension Solutions