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]f\left(x\right)=\sqrt[3]{x}[/latex] is different from the domain of [latex]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?

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]f\left(x\right)=\sqrt{2{x}^{3}-50x}[/latex] by:

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

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Domain: ________ Range: ________

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

38. [latex]f(x)=\begin{cases}{x}+{1}&\text{ if }&{ x }<{ -2 } \\{-2x - 3}&\text{ if }&{ x }\ge { -2 }\\ \end{cases}[/latex] 39. [latex]f\left(x\right)=\begin{cases}{2x - 1}&\text{ if }&{ x }<{ 1 }\\ {1+x }&\text{ if }&{ x }\ge{ 1 } \end{cases}[/latex] 40. [latex]f\left(x\right)=\begin{cases}{x+1}&\text{ if }&{ x }<{ 0 }\\ {x - 1 }&\text{ if }&{ x }>{ 0 }\end{cases}[/latex]

41. [latex]f\left(x\right)=\begin{cases}{3} &\text{ if }&{ x } <{ 0 }\\ \sqrt{x}&\text{ if }&{ x }\ge { 0 }\end{cases}[/latex] 42. [latex]f\left(x\right)=\begin{cases}{x}^{2}&\text{ if }&{ x } <{ 0 }\\ {1-x}&\text{ if }&{ x } >{ 0 }\end{cases}[/latex]

43. [latex]f\left(x\right)=\begin{cases}{x}^{2}&\text{ if }&{ x }<{ 0 }\\ {x+2 }&\text{ if }&{ x }\ge { 0 }\end{cases}[/latex] 44. [latex]f\left(x\right)=\begin{cases}x+1& \text{if}& x<1\\ {x}^{3}& \text{if}& x\ge 1\end{cases}[/latex] 45. [latex]f\left(x\right)=\begin{cases}|x|&\text{ if }&{ x }<{ 2 }\\ { 1 }&\text{ if }&{ x }\ge{ 2 }\end{cases}[/latex] For the following exercises, given each function [latex]f[/latex], evaluate [latex]f\left(-3\right),f\left(-2\right),f\left(-1\right)[/latex], and [latex]f\left(0\right)[/latex]. 46. [latex]f\left(x\right)=\begin{cases}{ x+1 }&\text{ if }&{ x }<{ -2 }\\ { -2x - 3 }&\text{ if }&{ x }\ge{ -2 }\end{cases}[/latex] 47. [latex]f\left(x\right)=\begin{cases}{ 1 }&\text{ if }&{ x }\le{ -3 }\\{ 0 }&\text{ if }&{ x }>{ -3 }\end{cases}[/latex]

48. [latex]f\left(x\right)=\begin{cases}{-2}{x}^{2}+{ 3 }&\text{ if }&{ x }\le { -1 }\\ { 5x } - { 7 } &\text{ if }&{ x } > { -1 }\end{cases}[/latex]
For the following exercises, given each function [latex]f[/latex], evaluate [latex]f\left(-1\right),f\left(0\right),f\left(2\right)[/latex], and [latex]f\left(4\right)[/latex].

49. [latex]f\left(x\right)=\begin{cases}{ 7x+3 }&\text{ if }&{ x }<{ 0 }\\{ 7x+6 }&\text{ if }&{ x }\ge{ 0 }\end{cases}[/latex] 50. [latex]f\left(x\right)=\begin{cases}{x}^{2}{ -2 }&\text{ if }&{ x }<{ 2 }\\{ 4+|x - 5|}&\text{ if }&{ x }\ge{ 2 }\end{cases}[/latex] 51. [latex]f\left(x\right)=\begin{cases}5x& \text{if}& x<0\\ 3& \text{if}& 0\le x\le 3\\ {x}^{2}& \text{if}& x>3\end{cases}[/latex]
For the following exercises, write the domain for the piecewise function in interval notation.

52. [latex]f\left(x\right)=\begin{cases}{x+1}&\text{ if }&{ x }<{ -2 }\\{ -2x - 3}&\text{ if }&{ x }\ge{ -2 }\end{cases}[/latex] 53. [latex]f\left(x\right)=\begin{cases}{x}^{2}{ -2 }&\text{ if}&{ x }<{ 1 }\\{-x}^{2}+{2}&\text{ if }&{ x }>{ 1 }\end{cases}[/latex]

54. [latex]f\left(x\right)=\begin{cases}{ 2x - 3 }&\text{ if }&{ x }<{ 0 }\\{ -3}{x}^{2}&\text{ if }&{ x }\ge{ 2 }\end{cases}[/latex] 55. Graph [latex]y=\frac{1}{{x}^{2}}[/latex] on the viewing window [latex]\left[-0.5,-0.1\right][/latex] and [latex]\left[0.1,0.5\right][/latex]. Determine the corresponding range for the viewing window. Show the graphs. 56. Graph [latex]y=\frac{1}{x}[/latex] on the viewing window [latex]\left[-0.5,-0.1\right][/latex] and [latex]\left[0.1,\text{ }0.5\right][/latex]. Determine the corresponding range for the viewing window. Show the graphs. 57. Suppose the range of a function [latex]f[/latex] is [latex]\left[-5,\text{ }8\right][/latex]. What is the range of [latex]|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]x>2[/latex].

60. The cost in dollars of making [latex]x[/latex] items is given by the function [latex]C\left(x\right)=10x+500[/latex].

61. The height [latex]h[/latex] of a projectile is a function of the time [latex]t[/latex] it is in the air. The height in feet for [latex]t[/latex] seconds is given by the function [latex]h\left(t\right)=-16{t}^{2}+96t[/latex]. What is the domain of the function? What does the domain mean in the context of the problem?