the volume of the solid is x²/2R.
Let x be the length of the base of the rectangle.
The volume of the solid is given by:
V = ∫R 2x dx
= 2∫R x dx
= 2[x²/2]∫R dx
= x²/2 ∫R dx
= x²/2 (R - 0)
= x²/2 R
The volume of the solid is given by the integral of the cross sectional area of the solid. The cross sectional area is a rectangle whose base is x and the height is twice the length of the base. Therefore, the area of the cross section is 2x. The volume of the solid is calculated by integrating the area over the range of the variable, which in this case is R. The integral of 2x over the range R is 2x times R (2x*R). This can be simplified to x squared over two times R (x^2/2*R). Therefore, the volume of the solid is x squared over two times R.
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create a matrix of dimension 300 x 7, called `bin.matrix`, whose columns contain the 7 vectors we've created, in order of the success probabilities of their underlying binomial distributions (0.2 through 0.8). hint: use `cbind()`.
The creation of the bin.matrix using cbind() allows us to organize and manipulate our data in a way that is conducive to statistical analysis.
A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, with a constant probability of success.
First, let's define what a binomial distribution is. For example, if we toss a fair coin 10 times, the number of heads that come up is described by a binomial distribution.
Now, let's talk about the cbind() function. cbind() is short for "column bind". It is used to combine vectors or matrices into a single matrix, where each input vector or matrix becomes a column in the final matrix.
In this exercise, we have created 7 vectors, each with a different success probability for a binomial distribution, ranging from 0.2 to 0.8. We will use cbind() to combine these vectors into a matrix called bin.matrix.
The dimension of the bin.matrix is
=> 300 x 7.
This means that the matrix has 300 rows and 7 columns. Each row represents a trial, and each column represents a different success probability. For example, the first column might represent a trial with a success probability of 0.2, while the second column might represent a trial with a success probability of 0.3, and so on.
By creating this matrix, we can easily perform statistical analyses on the data contained within it. We can calculate summary statistics, such as the mean and variance, for each column, or we can compare the distributions of the different columns to see if there are any significant differences between them.
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home realty claims that it can sell a detached, residential house faster than any other realty company. with the aim of examining home's claim, you sample customers who sold a detached, residential house through home and record the selling times (in days) of the houses. your data are summarized in the following histogram.
The proportion of selling times in the sample that are greater than or equal to 10 days is 0.85.
As per the data given:
HOME Realty claims that it can sell a detached, residential house faster than any other realty company.
With the aim of examining HOME's claim, you sample 20 customers who sold a detached.
The data has been summarized in the histogram(attached at the end of solution)
Here we have to determine the proportion of selling times in the sample that are greater than or equal to 10 days.
Selling Time Frequency
0 - 10 3
10 - 20 4
20 - 30 7
30 - 40 3
40 - 50 3
Frequencies for which selling times are 10 or greater
= 4 + 7 + 3 + 3
= 17
Total frequencies = 3 + 4 + 7 + 3 + 3
= 20
Proportion for which selling times are 10 or greater
= [tex]$\frac{17}{20}[/tex]
= 0.85
Therefore the proportion is 0.85.
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HOME Realty claims that it can sell a detached, residential house faster than any other realty company. With the aim of examining HOME's claim, you sample 20 customers who sold a detached, residential house through HOME and record the selling times (in days) of the houses. Your data are summarized in the following histogram(attached at the end of solution):
Based on the histogram, find the proportion of selling times in the sample that are greater than or equal to 10 days. Write your answer as a decimal, and do not round your answer.
#7
An insect population after x months can be modeled by
the function g(x) = 11(1.2). Which statement is the
best interpretation of one of the values in this function?
The insect population increased by 12 insects each month
The insect population decreased by 12 insects each month
The insect population increased at a rate of 20% each month
The insect population decreased at a rate of 20% each
month
+
the shop has 1100 ounces of cranberry juice and 680 ounces of passion fruit juice.if the juices are used to make crazy cranberry smoothies, which juice will run out first?how much of the other juice will be left over?
Answer: The passion fruit will run out first and 420 ounces left.
Step-by-step explanation: If you put the same amount of cranberry and passion fruit in each juice than they will be using an equal amount each time. We know that there are more ounces of cranberry than passion fruit which already tells us that there will be more cranberry when the passion fruit runs out. In order to find how much is left we can simply subtract the amount of cranberry and passion fruit.
1100-680=420
The law of sines states that the ratio of the ____ of an angle to the length of the side ________ that angle is ________ in any triangle.
The law of sines states that the ratio of the Sine of an angle to the length of the side opposite that angle is constant in any triangle.
The Law of Sines is defined as a relationship that applies to all triangles, not just right triangles.
The law states that the ratio of length of side of a triangle to sine of angle opposite that side is constant for all sides and angles in the triangle.
So , for any triangle, ratio of length of one side to the sine of angle opposite that side is = ratio of length of any other side to the sine of the angle opposite that side.
The Sine Law is also useful for solving problems which involves the angles and sides of triangles, especially when not all sides and angles are known.
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Gravetter/Wallnau/Forzano, Essentials - Chapter 3 - End-of-chapter question 24 A sample yielded the following scores: 2, 3, 4, 4,5,5,5,6,6,7 n = 10 Assume that the scores are measurements of a discrete variable and find the median Median = ___Assume that the scores are measurements of a continuous variable and find the median by locating the precise midpoint of the distribution (Use two decimal places.) Median = ___
Answer:
Step-by-step explanation:
24. Gravetter/Wallnau/Forzano, Essentials - Chapter 3 - End-of-chapter question 24
A sample yielded the following scores:
2, 3, 4, 4, 5, 5, 5, 6, 6, 7
n= 10
Assume that the scores are measurements of a discrete variable and find the median.
Median =5Correct
Points:
1 / 1
Median = 5
To find the median for a set of numbers with an even quantity the two middle numbers are added together and divided by 2.
n=102+(n = 102+1)2
= 5th score + 6th score2
= 5 + 52
= 102=5
Assume that the scores are measurements of a continuous variable and find the median by locating the precise midpoint of the distribution. (Use two decimal places.)
Median = 4.83Correct
The median for the discrete variable is 5 and for continuous variable is 5.
What is median?Median in statistics is the middle value of a list of data when arranged in order. It is a measure of central tendency that separates the data into two halves: the upper half and the lower half.
Given that: scores = 2, 3, 4, 4, 5, 5, 5, 6, 6, 7
a) The scores are in ascending order; then the median which is the middle number will be:
median = (5th term + 6 term)/2
= (5+5)/2
= 10/2
= 5
b) Use the cumulative percentage table ;
X frequency cumulative frequency cumulative percentage
2 1 1 10%
3 1 2 20%
4 2 4 40%
5 3 7 70%
6 2 9 90%
7 1 10 100%
Fraction = the no. required to reach 50%/ no. in that interval
= 1/3
Since we have a total of 4 boxes when we reach the value of 4.5
The median of the continuous variable = 4.5 + (1/3)
= 4.833
≈ 5
Hence, the median for the discrete variable is 5 and for continuous variable is 5.
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If f(x) = ln(x), what is the transformation that occurs if g(x) = ln(x + 2)
The transformation from ln ( x ) to ln ( x + 2 ) is horizontal transformation or shift.
What kinds of functions can be transformed?
Transformations can be divided into three categories: translations, reflections, and dilations. There are two types of transformations that can be done to a function: vertical (which affects the y-values) and horizontal (affects the x-values).
The four transformation variables are included in the equation of a function that is shown below (a, b, h, and k).
f(x) = ln(x)
if g(x) = ln(x + 2)
Then,
The transformation from ln(x) to ln(x+2) is horizontal transformation or shift. As 2 units are added in g(x) the function is horizontally shifted by 2 units in left-hand direction.
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in a sample of 28 cups of coffee at the local coffee shop, the temperatures were normally distributed with a mean of 182.5 degrees with a sample standard deviation of 14.1 degrees. what would be the 95% confidence interval for the temperature of your cup of coffee? homework help: 5ve. confidence intervals with sample standard deviation links to an external site.(1:37) group of answer choices (177.03, 187.97) (168.40, 196.60) (177.28, 187.72) (148.40, 176.60)
The Confidence interval starts at 177.28 and the Confidence interval ends at 187.72. Thus option c is correct.
The given data is as follows:
Number of coffee, N = 28
Mean, µ = 182.5
Standard deviation, α = 14.1
Confidence Level, CL = 95%
z-value for 95% confidence interval = 1.96
The confidence interval is calculated by using the formula,
Confidence interval = μ ± Ζ (α/√n )
Confidence interval = 182.5 ± 1.96*(14.1/√28)
Confidence interval = 182.5 ± 5.22
Confidence interval = (177.28, 187.72)
Therefore we can conclude that the Confidence interval starts at 177.28 and the Confidence interval ends at 187.72.
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Describe the translation needed to obtain the graph of g(x) = 7^x-4 from the graph of (x) = 7^x
The translation needed to obtain the graph of g(x) = 7^x-4 from the graph of (x) = 7^x is 4 units right
How to determine the translation from the functionFrom the question, we have the following parameters that can be used in our computation:
g(x) = 7^x-4
f(x) = 7^x
Using the above as a guide, we have the following:
g(x) = f(x - 4)
This represents a translation to the right by 4 units
Hence, the translation is 4 units right
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i need with this question
The height of the wall is 14 feet
What are similar triangles?Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.
Therefore ;
TH/ JS = TV/VS
4/JS = 6/21
21× 4 = 6 JS
JS = 21× 4/6
JS = 84/6
JS = 14 feet
therefore the height of the wall is 14 feet
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Find x, y, and z such that x³+y³+z³=k, for each k from one to 100.
Answer:
The answer is:-
Step-by-step explanation:
x^3+y^3+z^3=k
xyz^9=k
xyz/9=k
xyz=9k
Hence, k=9 and, x, y and z are 1.
y=2x-1 (2,3) (5,11)(-1,-3) determine which of the following points work for the given information algebraically
Emily parents put $1,500 in her bank account for college tuition. At an interest rate of 8.25% compounded annually. What will be the total balance after 2 years?
(A) $1,314.08
(B) $2,385.72
(C) $3,273.50
(D) $1,757.71
Answer:(A) 1.314.08
Step-by-step explanation:
Inflation
11.9 A cube has edge lengths that are each 100 centimeters. What is the volume of
the cube?
A
B
1 cubic centimeter
1 cubic meter
100 cubic centimeters
100 cubic meters
The volume of the cube is 1 cubic meter.
Option B is the correct answer.
What is a cube?It is a polygon having six faces.
The volume of a cube is side³.
Example:
The volume of a cube of 3 cm in length is 27 cm³.
We have,
Side length = 100 cm
Now,
Volume.
= Side³
= 100³
= 1000000 cm³ _____(1)
Now,
1 m = 100 cm
1 m³ = 1000000 cm³ _____(2)
From (1) and (2),
Volume = 1 cubic meter.
Thus,
The volume of the cube is 1 cubic meter.
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in exercises 9 and 10, (a) for what values of h is v3 in span {v 1, v2}, and (b) for what values of h is {v 1, v2, v3 } linearly dependent? justify each answerV1 = (1 -3 2)V2 = (-3 9 6)V3 = (5 -7 h)
9. a)The required value of h is 4
9.b) The required value of h is 4
10.a)The required value of h is -6
10.b) They are linearly dependent for all values of h.
How to find the vector?Here the given vectors are:
v1 = [tex]\left[\begin{array}{ccc}1\\-3\\2&\end{array}\right][/tex]
v2 = [tex]\left[\begin{array}{ccc}-3\\10\\-6&\end{array}\right][/tex]
v3 = [tex]\left[\begin{array}{ccc}2\\-7\\h&\end{array}\right][/tex]
The vectors v1 and v2 are not scalar multiples of one another so they are not linearly dependent.
[tex]\left[\begin{array}{ccc}1&-3&2\\-3&10&-7\\2&-6&h\end{array}\right][/tex]= 0
1(10h -42) -3(-14 +3h) +2 (18-20) =0
10h -9h-4 =0
h = 4
Hence the value of h is 4 so that the vector v3 is in span{v1,v2}
b) The given vectors will be linearly dependent if h =4
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Question Help Error Analysis On a math test a student, Jake, has to identify all the coefficients and constants of the expression 9m +4+y. Jake says that 9 is a coefficient and 4 is a constant. Identify all the coefficients and constants of the expression. What error might Jake have made? The coefficient(s) of the expression is/are. (Use a comma to separate answers as needed.)
The error which Jake made is likely to be a result of confusion or oversight in identifying the variable y as having a coefficient of 1.
What error might Jake have made?The coefficients of an expression are the numerical factors that multiply the variables, while the constants are the numerical values that do not contain any variables.
In the expression 9m + 4 + y, the coefficients are 9 and 1 (which is the implied coefficient for y since there is no visible numerical factor), and the constant is 4.
So Jake correctly identified 9 as a coefficient and 4 as a constant, but missed the implied coefficient of 1 for the variable y. The error that Jake made is likely to be a result of confusion or oversight in identifying the variable y as having a coefficient of 1, rather than assuming it is a constant or neglecting it entirely.
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Given the function below, find p (-2).
p(x) = x² - 7x
Answer:
18
Step-by-step explanation:
to find p(-2), substitute it into the equation!
you can make this easier by factoring x out, so p(x) = x(x-7)
then, plug -2 in
p(-2)=-2(-2-7)
p(-2)=-2(-9)
p(-2)=18
Answer:
Step-by-step explanation:
p(x)=[tex]x^{2}-7x\\[/tex]
=>p(-2)=[tex](-2)^{2} -7.(-2)[/tex]
= 4+14=18
Coral runs down 23rd street. Each block is 1/4 mile long. She runs 2/3 of a block before she gets tired. What part of a mile does she run?
Answer:
1/6
Step-by-step explanation:
1/4x2/3 = 2/12 = 1/6
Using Company A's calling plan, the cost of an overseas phone call is a $0.95 connection fee plus 33 cents
per minute. If the total cost of the call is $11.84, how long is the phone call?
minutes
The total duration of the phone call was 33 minutes.
Given that, by using Company A's calling plan, the cost of an overseas phone call is a $0.95 connection fee plus 33 cents per minute. If the total cost of the call is $11.84 and we have to calculate the time how long was the phone call.
Let the number of minutes a phone call held be x.
Now, we will be representing the above given information in terms of linear equation and then solve for the value of x which will give us the number of minutes the phone call held.
[tex]Total[/tex] [tex]cost=[/tex] $[tex]0.95[/tex] [tex]+[/tex] $[tex]0.33x[/tex]
$[tex]11.84=[/tex] $[tex]0.95[/tex] [tex]+[/tex] $[tex]0.33x[/tex]
[tex]11.84=0.95+0.33x[/tex]
[tex]11.84-0.95=0.33x[/tex]
[tex]0.3=0.33x[/tex]
[tex]\frac{0.3}{0.33}=x[/tex]
[tex]33=x[/tex]
Therefore, the call will go 33 minutes long.
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Please help me with this math problem!
1) The model is R = 1820 - 19t
2) After six years R = 1706 barrels
3) There would be non more oil in 96 years.
How do you model a linear equation?We know that an equation can be used as a model. This implies that we can be able to use an equation to show how a variable is changing over time and that is what the problem that we have here is all about.
If at the current time we have 1820 billion barrels Then they decrease by 19 barrels per year and t is the number of years then the model is;
R = 1820 - 19t
After six years we would have;
R = 1820 - (19 * 6)
R = 1706 barrels
When there are no further barrels;
0 = 1820 - 19t
1820 = 19t
t = 1820/19
t = 96 years.
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Find the slope of the line that passes through each par of points(4,-4)(2-4)
how do these each of these coefficients (a, b, c) theoretically relate to quantities position x, velocity v and acceleration a?
The coefficients (a, b, c) in a polynomial equation is relate to the position x, velocity v, and acceleration a of an object.
Position (x): The position of the object can be represented by the x-coordinate of its position in space. This can be related to the coefficients in the equation by setting the x-coordinate equal to x and solving for the corresponding y-coordinate.
Velocity (v): The velocity of an object is the rate at which its position is changing. This can be related to the coefficients in the equation by taking the first derivative of the equation with respect to time. The result is a linear equation that represents the velocity as a function of time.
Acceleration (a): The acceleration of an object is the rate at which its velocity is changing. This can be related to the coefficients in the equation by taking the second derivative of the equation with respect to time. The result is a constant value that represents the acceleration of the object.
In summary, the coefficients (a, b, c) can be related to the position, velocity, and acceleration in equation for x, taking derivatives of the equation with respect to time, and interpreting the results as the position, velocity, and acceleration of the object.
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Rose drew a regression line for this paired data set. Her line passed through (1, 2) and (3, 5). What is the equation of Rose's regression line? Responses yˆ=6.8x+1.6 y hat equals 6.8 x plus 1.6 yˆ=1.5x+0.5 y hat equals 1.5 x plus 0.5 yˆ=0.66x−0.33 y hat equals 0.66 x plus 0.33 yˆ=2x+1 y hat equals 2 x plus 1
Answer: The equation of a line in the form of y = mx + b can be found using the slope-intercept form, where m is the slope of the line and b is the y-intercept. To find the slope, you can use the two points on the line: (1, 2) and (3, 5). The slope is found by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points.
m = (5 - 2) / (3 - 1) = 3 / 2 = 1.5
The y-intercept can be found by plugging in one of the points and the slope into the equation and solving for b. Let's use the point (1, 2).
2 = 1.5 * 1 + b
b = 2 - 1.5 = 0.5
So the equation of the regression line is:
y = 1.5x + 0.5
Therefore, the correct answer is: yˆ = 1.5x + 0.5
Step-by-step explanation:
1.
Solve for x. Round to one
decimal place (nearest
tenth).
X
48°
1
15
The length of the hypotenuse or the value of 'x' is 60.
What are trigonometric ratios in terms of a right-angle triangle?We know a right-angled triangle has three sides they are -: Hypotenuse,
Opposite and Adjacent.
We can remember SOH CAH TOA which is,
sin = opposite/hypotenuse, cos = adjacent/hypotenuse and
tan = opposite/adjacent.
From the given diagram respect to the reference angle of 37°,
Hypotenuse is 'x' and adjacent is 48.
And we know, cos = adjacent/hypotenuse.
Therefore, cos37° = 48/x.
x = 48/0.8.
x = 60.
Q. A right-angle triangle is shown, Find the value of x.
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pls help on these questions, im stuck on them. do 2-6 pls.
The area of each circle, to the nearest tenth is:
1. 19.6 cm²
2. 379.9 in.²
3. 28.3 mm²
4. 78.5 in.²
5. 132.7 cm²
6. 162.8 yd²
What is the Area of a Circle?The area of a circle is calculated as:
A = πr², where, r is the radius of the circle, and π is a constant.
1. r = 5/2 = 2.5 cm
Area = πr² = 3.14 * 2.5²
= 19.6 cm²
2. r = 11 in.
Area = πr² = 3.14 * 11²
= 379.9 in.²
3. r = 3 mm
Area = πr² = 3.14 * 3²
= 28.3 mm²
4. r = 10/2 = 5 in.
Area = πr² = 3.14 * 5²
= 78.5 in.²
5. r = 13/2 = 6.5 cm
Area = πr² = 3.14 * 6.5²
= 132.7 cm²
6. r = 7.2 yd
Area = πr² = 3.14 * 7.2²
= 162.8 yd²
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F and H are sets of real numbers defined as follows.
F ∩ H = [4, 9)
FUH= ∞
What is the interval?Generally, To find the intersection of F and H, denoted as F ∩ H, we need to identify the values that are in both sets F and H.
From the definitions of F and H, we have:
F = {v | v=4 or v>4}
H = {v | v<9}
The values that are in both F and H are those that satisfy both conditions, i.e., v=4 or v>4 and v<9. This is equivalent to just the condition v=4, since any value that satisfies v<−4 will also satisfy v>4 and v<9.
Therefore, F ∩ H = {v | 4}.
In interval notation, we can write this as:
F ∩ H = [4, 9)
To find the union of F and H, denoted as F ∪ H, we need to identify all the values that are in either set F or set H or both.
From the definitions of F and H, we have:
F = {v | v=4 or v>4}
H = {v | v<9}
this denotes
F= {v | v=4> x > ∞}
H = {v |-∞ <v<9}
FUH= ∞
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Please tell me the answer and explain how
The value of x is 60.
We can use the fact that the opposite angles of a parallelogram are equal to find the value of x. Angle BCD is therefore equal to angle ABD, and angle CBD is equal to angle CAD.
Based on the given angles, we can construct two equations:
Angle ABD + angle CBD = 100 (because angle BCD = angle ABD) Angle ABD + angle ACD = 80 (because angle CAD = angle CBD)
When we substitute the given values, we get:
x + 40 = 100 x + 20 = 80
When we solve for x in both equations, we get:
x = 60 x = 60.
We can be confident that our answer is correct because both equations give us the same value for x. As a result, the value of x is 60.
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A new battery’s voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that 90% of all batteries have acceptable voltages. Let Y denote the number of batteries that must be tested.
a. What is p(2), that isP ( Y = 2), ?
b. What is p(3)? [Hint: There are two different outcomes that result inY = 3 .]
c. To have Y=5, what must be true of the fifth battery selected? List the four outcomes for which Y= 5 and then determine p(5).
d. Use the pattern in your answers for parts (a)–(c) to obtain a general formula for p(y).
a) Probability, P(y = 2) = P(AA) is equals to the 0.81.
b) Probability, P(y = 3), is equals to the 0.162.
c) For P(y=5), 5ᵗʰ battery must be an A. four outcome for which Y= 5 , probability is equals to the 0.00324.
d) General formula for p(y) is P(y)= (y-1)(0.1)⁽ʸ⁻²⁾(0.9)².
A new battery’s voltage may be acceptable (A) or unacceptable (U). Assume that 90% of all batteries have acceptable voltages. So, probability of acceptance = 0.90 and
probability of unacceptance = 1 - 0.9 = 0.1
Let the number of batteries that must be tested be "Y". A particular flashlight requires two batteries, that is batteries will be independently selected.
a) For P(y = 2), both the batteries must be acceptable. Hence probability, P(y = 2)
= P(AA) = 0.90× 0.90 = 0.81
b)For P(y = 3 ) = P(AUA) + P(UAA)
=0.9× 0.1×0.9 + 0.1× 0.9× 0.9 = 0.162
C) For any(Y = n), nᵗʰ batterry should be acceptable, as search for two acceptable batteries will complete once second acceptable battery gets selected. Hence for P(y=5), 5ᵗʰ battery must be an A. four outcome for which Y
= 5 are (AUUUA) ,(UAUUA),(UUAUA),(UUUAA)
P(y=5) = P(AUUUA) + P(UAUUA) + P(UUAUA) + P(UUUAA)
= 4× 0.9² × 0.1³
= 0.00324
d) From above probabilities we conclude, the general formula for p(y) is p(y) =(y- 1)(0.1)⁽ʸ⁻²⁾(0.9)².
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Evaluate sec2 (pi/5) - tan2 (pi/5).
The value of sec2(pi/5) - tan2(pi/5) is equals to 1.
What is trigonometric identity ?
A trigonometric identity is a mathematical equation that relates the values of trigonometric functions of an angle. These identities are true for all values of the angle for which both sides of the equation are defined.
We can use the trigonometric identities to simplify this expression. Recall that:
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)
sin2(x) + cos2(x) = 1
Using these identities, we can rewrite the expression as:
sec2(pi/5) - tan2(pi/5)
= (1/cos2(pi/5)) - (sin2(pi/5)/cos2(pi/5))
= (1 - sin2(pi/5))/cos2(pi/5)
= cos2(pi/5)/cos2(pi/5) (using the identity sin2(x) + cos2(x) = 1)
= 1
Therefore, sec2(pi/5) - tan2(pi/5) = 1.
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Find a function whose domain is the set of all integers and whose target is the set of all positive integers that satisfies each set of properties.
(a) Neither one-to-one, nor onto.
(b) One-to-one, but not onto.
(c) Onto, but not one-to-one.
(d) One-to-one and onto.
a) This function maps even integers to the next odd integer, and maps odd integers to themselves. It is not one-to-one
b) This function maps positive integers to even positive integers, and maps negative integers to odd positive integers. It is one-to-one because different inputs always have different outputs.
c) This function simply maps every integer to itself. It is one-to-one because different inputs always have different outputs, and it is onto because every positive and negative integer is in its range.
d) This function simply maps every integer to itself. It is one-to-one because different inputs always have different outputs
(a) One example of a function that is neither one-to-one nor onto is:
[tex]f(x) = \begin{cases} x+1 & \text{if } x \text{ is even} \ x & \text{if } x \text{ is odd} \end{cases}[/tex]
This function maps even integers to the next odd integer, and maps odd integers to themselves. It is not one-to-one because, for example, f(2) = f(4) = 3. It is not onto because no positive even integer is in the range of f.
(b) One example of a function that is one-to-one but not onto is:
[tex]f(x) = \begin{cases} 2x & \text{if } x > 0 \ -2x+1 & \text{if } x \leq 0 \end{cases}[/tex]
This function maps positive integers to even positive integers, and maps negative integers to odd positive integers. It is one-to-one because different inputs always have different outputs. However, it is not onto because no odd positive integer is in the range of f.
(c) One example of a function that is onto but not one-to-one is:
[tex]f(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ -(x+1)/2 & \text{if } x \text{ is odd} \end{cases}[/tex]
This function maps even integers to positive integers, and maps odd integers to negative integers. It is onto because every positive and negative integer is in the range of f. However, it is not one-to-one because, for example, f(2) = f(-2) = 1.
(d) One example of a function that is both one-to-one and onto is:
f(x) = x
This function simply maps every integer to itself. It is one-to-one because different inputs always have different outputs, and it is onto because every positive and negative integer is in its range.
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