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

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. $f\left(x\right)=-2x\left(x - 1\right)\left(x - 2\right)$

7. $f\left(x\right)=5 - 2{x}^{2}$

8. $f\left(x\right)=3\sqrt{x - 2}$

9. $f\left(x\right)=3-\sqrt{6 - 2x}$

10. $f\left(x\right)=\sqrt{4 - 3x}$

11. $f\left(x\right)=\sqrt{{x}^{2}+4}$

12. $f\left(x\right)=\sqrt[3]{1 - 2x}$

13. $f\left(x\right)=\sqrt[3]{x - 1}$

14. $f\left(x\right)=\frac{9}{x - 6}$

15. $f\left(x\right)=\frac{3x+1}{4x+2}$

16. $f\left(x\right)=\frac{\sqrt{x+4}}{x - 4}$

17. $f\left(x\right)=\frac{x - 3}{{x}^{2}+9x - 22}$

18. $f\left(x\right)=\frac{1}{{x}^{2}-x - 6}$

19. $f\left(x\right)=\frac{2{x}^{3}-250}{{x}^{2}-2x - 15}$

20. $\frac{5}{\sqrt{x - 3}}$

21. $\frac{2x+1}{\sqrt{5-x}}$

22. $f\left(x\right)=\frac{\sqrt{x - 4}}{\sqrt{x - 6}}$

23. $f\left(x\right)=\frac{\sqrt{x - 6}}{\sqrt{x - 4}}$

24. $f\left(x\right)=\frac{x}{x}$

25. $f\left(x\right)=\frac{{x}^{2}-9x}{{x}^{2}-81}$

26. Find the domain of the function $f\left(x\right)=\sqrt{2{x}^{3}-50x}$ 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. $f(x)=\begin{cases}{x}+{1}&\text{ if }&{ x }<{ -2 } \\{-2x - 3}&\text{ if }&{ x }\ge { -2 }\\ \end{cases}$ 39. $f\left(x\right)=\begin{cases}{2x - 1}&\text{ if }&{ x }<{ 1 }\\ {1+x }&\text{ if }&{ x }\ge{ 1 } \end{cases}$ 40. $f\left(x\right)=\begin{cases}{x+1}&\text{ if }&{ x }<{ 0 }\\ {x - 1 }&\text{ if }&{ x }>{ 0 }\end{cases}$

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

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

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

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

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

54. $f\left(x\right)=\begin{cases}{ 2x - 3 }&\text{ if }&{ x }<{ 0 }\\{ -3}{x}^{2}&\text{ if }&{ x }\ge{ 2 }\end{cases}$ 55. Graph $y=\frac{1}{{x}^{2}}$ on the viewing window $\left[-0.5,-0.1\right]$ and $\left[0.1,0.5\right]$. Determine the corresponding range for the viewing window. Show the graphs. 56. Graph $y=\frac{1}{x}$ on the viewing window $\left[-0.5,-0.1\right]$ and $\left[0.1,\text{ }0.5\right]$. Determine the corresponding range for the viewing window. Show the graphs. 57. Suppose the range of a function $f$ is $\left[-5,\text{ }8\right]$. What is the range of $|f\left(x\right)|?$ 58. Create a function in which the range is all nonnegative real numbers. 59 .Create a function in which the domain is $x>2$.

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

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