1.1.E: Exercises
- Page ID
- 157102
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.1 Exercises
The amount of garbage, \(G\), produced by a city with population \(p\) is given by \(G = f(p)\). \(G\) is measured in tons per week, and \(p\) is measured in thousands of people.
- The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\).
- Explain the meaning of the statement \(f(5) = 2\).
The number of cubic yards of dirt, \(D\), needed to cover a garden with area \(a\) square feet is given by \(D = g(a)\).
- A garden with area 5000 \(ft^2\) requires 50 cubic yards of dirt. Express this information in terms of the function \(g\).
- Explain the meaning of the statement \(g(100) = 1\).
Select all of the following graphs which represent \(y\) as a function of \(x\).
Select all of the following graphs which represent \(y\) as a function of \(x\).
Select all of the following tables which represent \(y\) as a function of \(x\).
a.
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x |
5 |
10 |
15 |
|
y |
3 |
8 |
14 |
b.
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x |
5 |
10 |
15 |
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y |
3 |
8 |
8 |
c.
|
x |
5 |
10 |
10 |
|
y |
3 |
8 |
14 |
Select all of the following tables which represent \(y\) as a function of \(x\).
a.
|
x |
2 |
6 |
13 |
|
y |
3 |
10 |
10 |
b.
|
x |
2 |
6 |
6 |
|
y |
3 |
10 |
14 |
c.
|
x |
2 |
6 |
13 |
|
y |
3 |
10 |
14 |
Given the function \(g(x)\) graphed here,
- Evaluate \(g(2)\)
- Solve \(g(x) = 2\)
Given the function \(f(x)\) graphed here.
- Evaluate \(f(4)\)
- Solve \(f(x) = 4\)
Based on the table below,
- Evaluate \(f(3)\)
- Solve \(f(x) = 1\)
|
x |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
\(f(x)\) |
74 |
28 |
1 |
53 |
56 |
3 |
36 |
45 |
14 |
47 |
Based on the table below,
- Evaluate \(f(8)\)
- Solve \(f(x) = 7\)
|
x |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
\(f(x)\) |
62 |
8 |
7 |
38 |
86 |
73 |
70 |
39 |
75 |
34 |
For each of the following functions, evaluate: \(f(-2)\), \(f(-1)\), \(f(0)\), \(f(1)\), and \(f(2)\)
| 11. \(f(x) = 4-2x\) | 12. \(f(x) = 8 - 3x\) |
| 13. \(f(x) = 8x^2 - 7x + 3\) | 14. \(f(x) = 6x^2 -7x+4\) |
| 15. \(f(x) = 3 + \sqrt{x+3}\) | 16. \(f(x) = 4 - \sqrt[3]{x-2}\) |
| 17. \(f(x) = \frac{x-3}{x+1}\) | 18. \(f(x) = \frac{x-2}{x+2}\) |
Let \(f(t) = 3t+5\)
- Evaluate \(f(0)\)
- Solve \(f(t) = 0\)
Let \(g(p) = 6 - 2p\)
- Evaluate \(g(0)\)
- Solve \(g(p) = 0\)
Using the graph shown,
- Evaluate \(f(c)\)
- Solve \(f(x) = p\)
- What are the coordinates of points \(L\) and \(K\)?
Match each graph with its equation.
| a. \(y=x\) | b. \(y = x^3\) | \(y = \sqrt[3]{x}\) | d. \(y = \frac{1}{x}\) |
| e. \(y = x^2\) | f. \(y = \sqrt{x}\) | g. \(y = |x|\) | h. \(y = \frac{1}{x^2}\) |
For Exercises \(\PageIndex{23}-\PageIndex{24}\), write the domain and range of each graph as an inequality.
Find the domain of each function.
| 25. \(f(x) = 3\sqrt{x-2}\) | 26. \(f(x) = 5\sqrt{x+3}\) |
| 27. \(f(x) = \frac{9}{x-6}\) | 28. \(f(x) = \frac{6}{x-8}\) |
| 29. \(f(x) = \frac{3x+1}{4x+2}\) | 30. \(f(x) = \frac{5x+3}{4x-1}\) |