80 Chapter 1 Foundations for Functions 1. Order 1. − 5, -2, 0.95, - √3 , and 1 from least to greatest. Then classify each number by the subsets of the real numbers to which it belongs. Rewrite each set in the indicated notation. 2. interval notation 3. (-∞, 12 ; set-builder notation Identify the property demonstrated by each equation. 4. x + y = y + x 5. 9 · 2 + 9 · 7 = 9 · (2 + 7) 6. x = (1)x 7. A company manufactures square windows that come in three sizes: 6 square feet, 8 square feet, and 15 square feet. Estimate the side length of each window to the nearest tenth of a foot. Then identify which window is the largest one that could fit in a wall with a width of 3 feet. Simplify each expression. 8. -2 √3 + √75 9. √24 - √54 10. √22 · √55 11. 2 (x + 1) + 9x 12. 5x - 5y - 7x + y 13. 12x + 4 ( x + y ) - 6y Simplify each expression. Assume all variables are nonzero. 14. 8a 2 b 5 (-2a 3 b 2 ) 15. 28u -2 v 3 _ 4u 2 v 2 16. (5x 4 y -3 ) -2 17. ( 3x 2 y _ xy 2 ) -1 18. German shepherds are often used as police dogs because they have 2.25 × 10 8 smell receptors in their nose. Humans average only 5 × 10 6 smell receptors in their nose. How many times as great is the number of smell receptors in a German shepherd’s nose as that in a human’s nose? Give the domain and range for each relation. Then tell whether each relation is a function. 19. x 10 9 8 9 10 y 2 4 6 8 10 20. For each function, evaluate f (-2), f ( 1 _ 2 ) , and f (0). 21. f (x) = -4x 22. f (x) = -3 x 2 + x 23. f (x) = √x + 3 24. The table shows how the distance from the top of a building to the horizon depends on the building’s height. Graph the relationship from building height to horizon distance, and identify which parent function best describes the data. Then use your graph to estimate the distance to the horizon from the top of a building with a height of 80 m. Horizon Distances Height of Building (m) 5 10 20 40 100 Distance to Horizon (km) 8.0 11.3 15.9 22.5 35.6
Transcript
80 Chapter 1 Foundations for Functions
1. Order 1. − 5 , -2, 0.95, - √ � 3 , and 1 from least to greatest. Then classify each number by
the subsets of the real numbers to which it belongs.
Identify the property demonstrated by each equation.
4. x + y = y + x 5. 9 · 2 + 9 · 7 = 9 · (2 + 7) 6. x = (1) x
7. A company manufactures square windows that come in three sizes: 6 square feet, 8 square feet, and 15 square feet. Estimate the side length of each window to the nearest tenth of a foot. Then identify which window is the largest one that could fit in a wall with a width of 3 feet.
Simplify each expression. Assume all variables are nonzero.
14. 8a 2 b 5 (- 2a 3 b 2 ) 15. 28u -2 v 3 _ 4u 2 v 2
16. ( 5x 4 y -3 ) -2
17. ( 3x 2 y
_ xy 2
) -1
18. German shepherds are often used as police dogs because they have 2.25 × 10 8 smell receptors in their nose. Humans average only 5 × 10 6 smell receptors in their nose. How many times as great is the number of smell receptors in a German shepherd’s nose as that in a human’s nose?
Give the domain and range for each relation. Then tell whether each relation is a function.
19. x 10 9 8 9 10
y 2 4 6 8 10
20.
For each function, evaluate f (-2) , f (
1 _ 2
)
, and f (0) .
21. f (x) = -4x 22. f (x) = -3 x 2 + x 23. f (x) = √ ��� x + 3
24. The table shows how the distance from the top of a building to the horizon depends on the building’s height. Graph the relationship from building height to horizon distance, and identify which parent function best describes the data. Then use your graph to estimate the distance to the horizon from the top of a building with a height of 80 m.
1. A mall employee is dressing a mannequin. There are 6 pairs of shoes, 4 types of jeans, and 8 sweaters. Using 1 of each, how many ways can the mannequin be dressed?
2. How many ways can you award first, second, and third place to 8 contestants?
3. How many ways can a group of 3 students be chosen from a class of 30?
4. Four cards are randomly selected from a standard deck of 52 playing cards. What is the probability that the cards are all jacks, all queens, or all kings?
5. The table shows the results of tossing 2 coins. Find the HH HT TH TT
3 6 5 6experimental probability of tossing 2 tails.
Each letter of the alphabet is written on a card. The cards are placed into a bag. Determine whether the events are independent or dependent, and find the indicated probability.
6. The letter D is drawn, replaced in the bag, and then the letter J is drawn.
7. Three vowels are drawn without replacement.
A card is drawn from a bag containing the 9 cards shown. Find each probability.
8. selecting a C or an even number
9. selecting an odd number or a multiple of 3
10. The probability distribution for the number of absent students on any given day for a certain class is given. Find the expected number of absent students.
The number of known satellites of the planets in the solar system (as of 2005) is given.
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
Moons 0 0 1 2 63 33 27 13 1Source: NASA Planetary Data System, 2005
11. Make a box-and-whisker plot of the data. Find the interquartile range.
12. Is 63 an outlier? Explain.
13. Identify the outlier in the following data set: 93, 107, 110, 103, 98, 95, 12, 111, 128, 99, 114, and 90. Describe how the outlier affects the mean and the standard deviation.
14. Use the Binomial Theorem to expand (3x + y) 4 .
The probability of winning a carnival game is 15%. Elaine plays 10 times.
15. Find the probability that Elaine will win 2 times.
16. Find the probability that Elaine will win at least 2 times.
Write a possible explicit rule for the nth term of each sequence.
3. -4, -2, 0, 2, 4, . . . 4. 54, 18, 6, 2, 2 _ 3
, . . .
Expand each series and evaluate.
5. k = 1
4
5 k 3 6. k = 1
7
(-1) k+1 (k)
Find the 9th term of each arithmetic sequence.
7. -19, -13, -7, -1, . . . 8. a 2 = 11.6 and a 5 = 5
9. Find 2 missing terms in the arithmetic sequence 125, , , 65.
Find the indicated sum for each arithmetic series.
10. S 20 for 4 + 7 + 10 + 13 + . . . 11. k = 1
12
(-9k + 8)
12. The front row of a theater has 16 seats and each subsequent row has 2 more seats than the row that precedes it. How many seats are in the 12th row? How many seats in total are in the first 12 rows?
Find the 10th term of each geometric sequence.
13. 3 _ 256
, 3 _ 64
, 3 _ 16
, 3 _ 4
, . . . 14. a 4 = 2 and a 5 = 8
15. Find the geometric mean of 4 and 25.
Find the indicated sum for each geometric series.
16. S 6 for 2 + 1 + 1 _ 2
+ 1 _ 4
+ . . . 17. k = 1
6
250 (- 1 _ 5
) k-1
18. You invest $1000 each year in an account that pays 5% annual interest. How much is the first $1000 you invested worth after 10 full years of interest payments? How much in total do you have in your account after 10 full years?
Find the sum of each infinite geometric series, if it exists.
7. Tim and Kim took 4.6 hours to complete a 25.3 mile kayaking trip. If they want to paddle for 3 hours on their next trip, how far should they plan to go?
Graph.
8. y = 5 _ 3
x - 4 9. 6x + 8y = 24 10. 6x + 2y < 10
Write the equation of each line in slope-intercept form.
11. passing through (9, 12) and (7, 2)
12. parallel to 9x - 5y = 8 and through (-10, 2)
13. perpendicular to y = - 2 __ 7 x + 3 and through (6, 4)
14. The Spanish Club is selling T-shirts and hats and would like to raise at least $2400. It sells T-shirts for $15 and hats for $8. Write and graph an inequality representing the number of T-shirts and hats the club must sell to meet its goal.
Let g (x) be the indicated transformation(s) of f (x) = x. Write the rule for g (x) .
15. vertical stretch by a factor of 4 16. horizontal translation 6 units right
17. horizontal compression by a factor of 1 __ 6 followed by a vertical shift 4 units down
18. A consumer group is studying how hospitals are staffed. Here are the results from eight randomly selected hospitals in a state.
Full-Time Hospital Employees
Hospital Beds 23 29 35 42 46 54 64 76
Full-Time Employees
69 95 118 126 123 178 156 176
a. Make a scatter plot of the data with hospital beds as the independent variable.
b. Find the correlation coefficient and the equation of the line of best fit. Draw the line of best fit on your scatter plot.
c. Predict the number of beds in a hospital with 80 full-time employees.
19. Solve ⎜12 + 4x⎟ - 6 = 26.
Solve and graph.
20. 16 ≤ 24 - 8x _ 5
21. ⎜3x - 9⎟ > 12 22. 3 ⎜12 - 4x⎟ + 4 ≤ 28
23. A pollster predicts the actual percent p of a population that favors a political candidate by using a sample percent s plus or minus 3%. Write an absolute-value inequality for p.
24. Translate f (x) = ⎜x⎟ so that its vertex is at (4, -2) . Then graph.
25. Find g (x) if f (x) = ⎜2x⎟ - 3 is stretched horizontally by a factor of 3 and reflected across the x-axis.
a207se_c02cht_0170.indd 170 10/27/05 3:26:51 PM
236 Chapter 3 Linear Systems
Solve each system by using a graph and a table.
1. �
� x - y = -4
3x - 6y = -12
2. �
� y = x - 1
x + 4y = 6
3.
x - y = 3
2x + 3y = 6
Classify each system and determine the number of solutions.
4. �
� 6y = 9x
8x + 4y = 20
5. �
� 12x + 3y = -9
-y - 4x = 3
6. �
� 3x - 9y = 21
6 = x - 3y
Use substitution or elimination to solve each system of equations.
7. �
� y = x - 2
x + 5y = 20
8. �
� 5x - y = 33
7x + y = 51
9. �
� x + y = 5
2x + 5y = 16
Graph each system of inequalities.
10. �
� 2y - 4x ≥ 4
y - x ≥ 1
11. �
� x + y ≥ 3
y - 4 ≤ 0
12. Chemistry A chemist wants to mix a new solution with at least 18% pure salt. The chemist has two solutions with 9% pure salt and 24% pure salt and wants to make at most 250 mL of the new solution. Write and graph a system of inequalities that can be used to find the amounts of each salt solution needed.
13. Minimize the objective function P = 5x + 9y under the following constraints. �
�
x ≥ 0
y ≥ 0
y ≤ 2x + 1
y ≤ -3x + 6
Graph each point in three-dimensional space.
14. (2, -1, 3) 15. (0, -1, 3) 16. (-2, 1, -1)
Business Use the following information and the table for Problems 17 and 18.A plumber charges $50 for repairing a leaking faucet, $150 for installing a sink, and $200 for an emergency situation. The plumber’s total income was exactly $1000 for each day shown in the table.
17. Write a linear equation in three variables to represent this situation.
18. Complete the table for the possible numbers of tasks each day.
Solve each system of equations using elimination, or state that the system is inconsistent or dependent.
Use the data from the table to answer the questions.
1. Display the data in the form of matrix A.
2. What are the dimensions of the matrix?
3. What is the value of the matrix entry with address a31?
4. What is the address of the entry that has a value of 2?
Evaluate, if possible.
E =
2 3
-1 0
4 1
�
F =
4 -2 0
-1 1 -2 �
G =
2 -1
3 1
�
H =
-2 1
3 0
5 -1
�
J =
1 -5 6
�
K =
7
0
-2
�
5. E + F 6. EF 7. FE
8. H 2 9. G 3 10. FK
Use a matrix to transform �PQR.
11. Translate �PQR 2 units up and 1 unit right.
12. Enlarge �PQR by a factor of 3 _ 2
.
13. Use
0 2
2 0
�
to transform �PQR. Describe the image.
Find the determinant of each matrix.
14.
4 0
0 -3
�
15.
0.25 1
2 8
�
16.
3
-2
-1
-1 �
17.
1
3
2
-2
-1
1
3
-3
5
�
18. Use Cramer’s rule to solve
x + 2y = 1
3x - y = 10
19. Use Cramer’s rule to solve
�
�
x + 3z = 3 + 2y
3x + 22 = y + 3z
2x + y + 5z = 8
Find the inverse, if it exists.
20.
2 0.7
4 1.4
�
21.
3 -1
1 3
�
22.
3 1
2 -1
�
23.
3 2 -1
2 3 -5 1 4 2
�
24. The cost of 2.5 pounds of figs and 1.5 pounds of dates is $14.42. The cost of 3.5 pounds of figs and 1 pound of dates is $16.91. Use a matrix operation to find the price of each per pound.
Write the matrix equation for each system, and solve, if possible.
25.
6x + y = 2
3x - 2y + 1 = 0
26.
5x - 2y = 3
2.5x - y = 1.5
27.
x + 2y = 3.5
3x = 2.7 + y
28. �
�
2x - z = 3 + y
x + 2 = y + 5
4z + x + y = 1
Write the augmented matrix, and use row reduction to solve, if possible.
29. Use the data from Items 1–4 above. Find the number of points assigned for finishing in first, second, and third places.
Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.
1. g (x) = (x + 1) 2 - 2 2. h (x) = - 1 _ 2
x 2 + 2
3. Use the following description to write a quadratic function in vertex form: f (x) = x 2 is vertically compressed by a factor of 1 __
2 and translated 6 units right to create g.
For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.
4. f (x) = -x 2 + 4x + 1 5. g (x) = x 2 - 2x + 3
6. The area A of a rectangle with a perimeter of 32 cm is modeled by the function A (x) = - x 2 + 16x, where x is the width of the rectangle in centimeters. What is the maximum area of the rectangle?
Find the roots of each equation by using factoring.
7. x 2 - 2x + 1 = 0 8. x 2 + 10x = -21
Solve each equation.
9. x 2 + 4x = 12 10. x 2 - 12x = 25
11. x 2 + 25 = 0 12. x 2 + 12x = -40
Write each function in vertex form, and identify its vertex.
13. f (x) = x 2 - 4x + 9 14. g (x) = x 2 - 18x + 92
Find the zeros of each function by using the Quadratic Formula.
15. f (x) = (x - 1) 2 + 7 16. g (x) = 2 x 2 - x + 5
17. The height h in feet of a person on a waterslide is modeled by the function h (t) = -0.025 t 2 - 0.5t + 50, where t is the time in seconds. At the bottom of the slide, the person lands in a swimming pool. To the nearest tenth of a second, how long does the ride last?
18. Graph the inequality y < x 2 - 3x - 4.
Solve each inequality.
19. - x 2 + 3x + 5 ≥ 7 20. x 2 - 4x + 1 > 1
For Exercises 21 and 22, use the table showing the average cost of LCD televisions at one store.
21. Find a quadratic model for the cost of a television given its size.
22. Use the model to estimate the cost of a 42 in. LCD television.
Perform the indicated operation, and write the result in the form a + bi.
Tell whether the function shows growth or decay. Then graph.
1. f (x) = 0. 4 x 2. f (x) = 1. 3 ( 2 _ 5
) x
3. f (x) = 7 _ 8
(1.1) x 4. f (x) = 50 (1 + 0.04)
x
5. Gina buys a car for $13,500. Assume that its value will decrease by about 15% per year. Write an exponential function to model the value of the car. Graph the function. When will the value fall below $3000?
Graph each function. Then write its inverse and graph.
6. f (x) = x - 1.06 7. f (x) = 5 _ 6
x - 1.06
8. f (x) = 1.06 - 5 _ 6
x 9. f (x) = 1 _ 4
(1.06 - 5 _ 6
x)
Write in the alternative form (exponential or logarithmic).
10. 16 1 _ 4
= 2 11. 16 -0.5 = 1 _
4
12. log 1 _ 4
64 = -3 13. log 81 1 _
3 = - 1 _
4
Use the given x-values to graph each function. Then write and graph its inverse. Describe the domain and range of the inverse function.
14. f (x) = ( 1 _ 4
) x
; x = -1, 0, 2, 4 15. f (x) = 2.5 x ; x = -1, 0, 1, 2, 3 16. f (x) = 5 -x ; x = -1, 0, 1, 2, 3
Simplify.
17. log 4 128 - log 4 8 18. log 2 12.8 + log 2 5
19. log 3 243 2 20. 5 log 5 x
Solve.
21. 3 x-1 = 729 x _ 2
22. 5 1.5-x ≤ 25
23. lo g 4 (x + 48) = 3 24. log (6 x 2 ) - log 2x = 1
25. The rate at which a liquid vitamin breaks down in the average human body can be modeled by y = D (0.95)
x , where y ml of the original dose D remains after x minutes.
How long will it take for an original dose of 15 ml to be reduced to less than 5 ml?
26. Plutonium Pu-239 has a half-life of about 24,000 years. The formula 1 __ 2 = e -kt relates
the half-life t to the decay constant k for a given substance. How much of a 100-gram quantity of plutonium will remain after 5 years?
27. f (x) = ln x is shifted 2 units left and 1 unit up and is vertically stretched by a factor of 3. Write the transformed function.
28. Use logarithmic regression to find the Population 50 62 78
Year 1 2 3function that models the population data in the table. In what year will the population exceed 100?
558 Chapter 7 Exponential and Logarithmic Functions
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