- Why does the domain differ for different functions?
- How do we determine the domain of a function defined by an equation?
- 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].
- When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?
- 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:
b. graphing the function in the radicand and determining intervals on the x-axis for which the radicand is nonnegative.
For the following exercises, write the domain and range of each function using interval notation.
27.
Domain: ________ Range: ________
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
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].
B. What is the cost of making 25 items?
C. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, [latex]C\left(x\right)?[/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?