1. Proof that F is injective. Suppose that two different elements in the domain of F have the same image; that is, if x and y are elements of the domain of F and F(x)= F(y). We need to show that x=y. Let F(x) = F(y). This means that y1= x1/ (x1 + x2 + x3) = y1, y2= x2/ (x1 + x2 + x3) = y2 and y3= x3/ (x1 + x2 + x3) = y3.Now adding (1), (2), and (3), we have:y1+y2+y3= x1/ (x1 + x2 + x3) + x2/ (x1 + x2 + x3) + x3/ (x1 + x2 + x3)But this is equal to 1, therefore,x1 + x2 + x3= y1 + y2 + y3 = 1, or, equivalently, y1= 1- y2 - y3x1= (1- y2 - y3)(x1 + x2 + x3) = (1-y2 - y3), x2= y2(x1 + x2 + x3) = y2, and x3= y3(x1 + x2 + x3) = y3Thus, we have constructed an element of the domain of F, with different elements of the domain, that have the same image. Therefore, F is injective.
2. Find the investment directly without appealing to the Inverse Function Theorem. F(x1,x2,x3) = (y1,y2,y3)So, x1= (1- y2 - y3), x2= y2, and x3= y3Thus, the inverse of F is F-1(y1,y2,y3)= ((1-y2-y3),y2,y3)3. The domain of F. The domain of F is the set of all three-tuples, F(x1,x2,x3) where 0 ≤ xi ≤ ∞, and where at least one xi is positive. That is, Dom (F)={(x1,x2,x3)|x1≥0, x2≥0, x3≥0 and (x1,x2,x3)≠(0,0,0)}4. The range of F, which is the domain of F-1. The range of F is the set of all three-tuples, F(y1,y2,y3) where 0 ≤ yi ≤ 1, and where at least one yi is positive.
That is, Rng(F)={(y1,y2,y3)|y1≥0, y2≥0, y3≥0 and y1+y2+y3=1}5. We have J_f(x) = ∣∣ ∂(y1,y2,y3) /∂(x1,x2,x3) ∣∣= ∣∣ ∂y1/∂x1 ∂y1/∂x2 ∂y1/∂x3 ∂y2/∂x1 ∂y2/∂x2 ∂y2/∂x3 ∂y3/∂x1 ∂y3/∂x2 ∂y3/∂x3 ∣∣= ∣∣ (1 / (x1 + x2 + x3) ) - (x1/ (x1 + x2 + x3)2) - (x1/ (x1 + x2 + x3)2) 0 1 / (x1 + x2 + x3) - (x2/ (x1 + x2 + x3)2) 0 0 1 / (x1 + x2 + x3) - (x3/ (x1 + x2 + x3)2) ∣∣= (1 / (x1 + x2 + x3) )((1 - y2 - y3) (1 - y2 - y3 - y3) - y2 (1 - y2 - y3) - y3(1 - y2 - y3)) = (1 / (x1 + x2 + x3) )((1 - y2 - y3 - y2 + 2y2y3 + y2 - y3 - 2y2y3 + y3 - y2y3 - y3 + y2y3)) = 1 / (x1 + x2 + x3) which is non-zero in the domain of F. Therefore, J_f(x) ≠ 0 when x ∈ Dom (F).
Know more about Inverse Function Theorem:
https://brainly.com/question/31423480
#SPJ11
Romberg integration for approximating -', $(x) dx gives R21 = 6 and R22 = 6.28 then Ru = 0.35 2.15 5.16 4.53
The required value of Ru is 7.71
Given that Romberg integration for approximating `∫(x) dx` gives `R21 = 6` and `R22 = 6.28`. Then, we need to find `Ru`.
We know that Romberg integration formula is given by:
R_kj = (4^j * R_(k+1),j-1 - R_k,j-1) / (4^j - 1), where, `R_kj` denotes the approximation of integral using `2^(k-1) x 2^(j-1)` points and `R_k,j-1` denotes the approximation of integral using `2^(k-1) x 2^(j-2)` points.
Now, we are given that:
`R21 = 6` and `R22 = 6.28`, we need to calculate `Ru`.
For this, let's use the Romberg integration formula as follows:
`R31 = (4^1 * R22 - R21) / (4^1 - 1)`
Substituting the given values, we get:
`R31 = (4 * 6.28 - 6) / 3 = 7.56 / 3 = 2.52`
Similarly,`R32 = (4^1 * R32 - R31) / (4^1 - 1)`
Substituting the given values, we get:
`R32 = (4 * 6.28 - 2.52) / 3 = 23.12 / 3 = 7.71`
Therefore, `Ru = R32 = 7.71`.Hence, the required value of `Ru` is `7.71`.
To know more about Romberg integration, visit the link : https://brainly.com/question/32552511
#SPJ11
8. The probability that a mature hen will lay an egg on a given day is 0.80. Hannah has 6 hens. Using the table, what is the probability that at
least 2 of the hens will lay eggs on a given day?
4
Number of Eggs
Probability
0
0.000064
1
0.002
2.
0.015
3
0.082
5
0.393
6
?
0.246
Clara visits the aquarium while on vacation. The aquarium is 2 1/2 from her hotel. She walks 1/4 mile to the bus stop, take the bus for 1 3/4 miles, and walks the rest of the way to the aquarium.
Answer:
1/2
Step-by-step explanation:
Ben weighs 41.9 kg. His older brother weighs 57.6 kg. How much more does his older brother weigh?
Step-by-step explanation:
[tex]57.6 - 41.9 = 5.7[/tex]
The rate of expenditure for maintenance of a particular machine is given by M'(x) =12x Squareroot x^2 +5, where x is time measured In years Total maintenance costs throw the second year are $105 Find the total maintenance function Select one A M(x) = 12(x^2 + 5)^3/2 - 93 B M(x) = 12(x^2 + 5)^3/2 - 3 C M(x) = 4(x^2 + 5)^3/2 - 93 D M(x) = 4(x^2 + 5)^3/2 -3
The total maintenance function for the given rate of expenditure is [tex]M(x) = 12(x^2 + 5)^{(3/2)} - 93[/tex].
The rate of expenditure for maintenance is given by [tex]M'(x) = 12x\sqrt{x^2 + 5}[/tex], where x represents time measured in years. To find the total maintenance function, we need to integrate M'(x) with respect to x.
Integrating M'(x) gives us the antiderivative [tex]M(x) = \int12x\sqrt{x^2 + 5} dx[/tex]. By applying the power rule of integration and substituting u = x^2 + 5, we can simplify the integral.
After simplification, we obtain [tex]M(x) = 4(x^2 + 5)^{(3/2)} - 93[/tex]. Therefore, the total maintenance function is [tex]M(x) = 12(x^2 + 5)^{(3/2)} - 93[/tex].
Hence, the correct option is A: [tex]M(x) = 12(x^2 + 5)^{(3/2)} - 93[/tex], which represents the total maintenance function based on the given rate of expenditure.
Learn more about function here:
https://brainly.com/question/30721594
#SPJ11
consider the following time series. t 1 2 3 4 5 yt 6 11 9 14 15
The given time series data is represented by the pairs (t, y(t)): (1, 6), (2, 11), (3, 9), (4, 14), and (5, 15).
The time series data provided consists of values of the dependent variable y at different time points t. In this case, we have the values of y at time points 1, 2, 3, 4, and 5. The corresponding values of y are 6, 11, 9, 14, and 15, respectively.
Time series data is commonly used in various fields, such as economics, finance, and engineering, to analyze patterns and trends over time. By examining the values in the given time series, one can identify any trends, seasonality, or other patterns that may be present in the data.
To know more about time series click here: brainly.com/question/29818565
#SPJ11
A coin shows heads with probability p. Let Xn be the number of flips required to obtain a run of n consecutive heads. Show that E(Xn) = Ek=p-k.
To show that E(Xn) = Ek=p-k, we can use the concept of conditional expectation and the law of total expectation.
How to show that E(Xn) = Ek=p-k.Let's define X as the number of flips required to obtain a run of n consecutive heads.
We can express X as the sum of two random variables: the number of flips required to obtain the first head (H) and the number of additional flips required to obtain a run of n-1 consecutive heads (Xn-1).
We can write the equation as:
X = H + Xn-1
Taking the expectation on both sides, we have:
E(X) = E(H + Xn-1)
Using the linearity of expectation, we can rewrite this as:
E(X) = E(H) + E(Xn-1)
The expected number of flips required to obtain the first head is simply the reciprocal of the probability of getting heads on a single flip, which is 1/p.
So, E(H) = 1/p.
Next, let's consider the expectation of Xn-1. Since Xn-1 represents the number of additional flips required to obtain a run of n-1 consecutive heads, it is equivalent to Xn but with a reduced value of n.
Using the law of total expectation, we can express E(Xn-1) as a conditional expectation:
E(Xn-1) = E(E(Xn-1 | Xn-1 > 1))
In other words, the expected value of Xn-1 can be obtained by conditioning on the event that the first flip is not a head.
Since the first flip is not a head, we need to start over and obtain a new run of n consecutive heads.
Therefore, the expectation of Xn-1 conditioned on Xn-1 > 1 is equal to E(Xn).
Putting it all together, we have:
E(X) = E(H) + E(Xn-1)
= 1/p + E(Xn)
Simplifying further, we get:
E(X) = 1/p + E(Xn)
Now, notice that E(X) is equal to E(Xn) when n = 1.
Therefore, we can write:
E(X) = 1/p + E(X) (since E(X1) = E(X))
Solving for E(X), we find:
E(X) = 1/p
Finally, let's substitute k = n-1 into the expression:
E(Xn) = 1/p
Since k = n-1, we have:
E(Xn) = 1/p = Ek=p-k
Therefore, we have shown that E(Xn) = Ek=p-k.
Learn more about probability at https://brainly.com/question/13604758
#SPJ4
Eiko is wearing a magic ring that increases the power of her healing spell by 30\%30%30, percent. Without the ring, her healing spell restores HHH health points.
Answer:
Step-by-step explanation:
Answer:
130/100 H and (3/10 + 1) H
Step-by-step explanation:
Hope this helps!
what’s the measure of angle B?
Answer:
60°
Step-by-step explanation:
∠B=(240-120)/2=120/2=60°
Jayden is trying to pick out an outfit for the first day of school. He can choose from 2
pairs of pants, 3 t-shirts, and 6 pairs of shoes. How many different outfits does
Jayden have to choose from?
Answer:
36
Step-by-step explanation:
Use the counting principle.
2 * 3 * 6 = 36
Answer: 36
Answer: 30 outfits
Step-by-step explanation:
2 x 5 x 3 =30
find the value of x... assume segments that appear tangent are tangent.
Answer:
x=12
Step-by-step explanation:
a^2 + b^2 = c^2
5^2 + b^2 = 13^2
25 + b^2 = 169
b^2= 144
b=12 Therefore,
x=12
The lines below are parallel. If the slope of the green line is -3, what is the
slope of the red line?
m =
Answer:
-3
Step-by-step explanation:
parallel lines are congruent which means that they equal the same thing
You flip a coin. What is P(not tails)? 50%
Please help me :(((((((
Answer:
The answer is B
Step-by-step explanation:
Each smaller rectangle is being multiplied by a so 2a + 3a + 4a would give you the total area of the entire large rectangle.
What is the sum of the interior angles of a regular dodecagon (12 sided polygon)? Round to
the nearest thousandth.
Answer:
1800°
Step-by-step explanation:
(12-2)x180 = 1800
According to this partial W-2 form, how much money was paid in FICA taxes? Use the partial sample of a W-2 form to answer a question. $823.73 $4345.89 $6817.08 $11,162.97
The amount of money paid in FICA taxes cannot be determined based on the given options.
To determine the amount of money paid in FICA taxes from the partial W-2 form, we would need to look for specific entries related to FICA taxes. Typically, the W-2 form provides information such as Social Security wages and Medicare wages, which are used to calculate the corresponding FICA taxes.
The FICA tax consists of two components: the Social Security tax and the Medicare tax. The Social Security tax is calculated based on a fixed percentage (e.g., 6.2%) of the individual's Social Security wages, up to a certain income threshold. The Medicare tax is calculated based on a different fixed percentage (e.g., 1.45%) of the individual's Medicare wages, with no income threshold.
Without access to the specific entries on the partial W-2 form related to Social Security wages and Medicare wages, it is not possible to determine the exact amount of money paid in FICA taxes.
To learn more about amount click here :
brainly.com/question/31009592
#SPJ11
PLEASEEEEE ILL MARK BRAINIEST
Answer:
5. 18 that is what i think the answers are
6. 13.75
5 2 11 8 35 32
Find 0. Round to the nearest degree.
A. 20°
B. 69°
C. 21°
D. 70°
Answer:
21
Step-by-step explanation:
Answer:
B
Step-by-step explanation:
Hope this helped, and please mark as brainliest! <3
Evaluate the function at the given value.
h(x) = 1/3.6^x
what is h(2)?
Answer:
h(x)=6^x/3 is what I got hope I could help
According to a recent survey, women chat on their mobile phones more than do men (CNN.com, August 25, 2010). To determine if the same patterns also exist in colleges, Irina takes a random sample of 40 male students and 40 female students in her college. She finds that female students chatted for a sample average of 820 minutes per month, with a sample standard deviation of 160 minutes. Male students, on the other hand, chatted for a sample average of 760 minutes per month, with a sample standard deviation of 240 minutes. It is not reasonable to assume equal population variances. Test the hypothesis whether women chat on their phones more than men at the 5% significance level.
A report on a study in which each of 6 workers was provided with both a conventional shovel and a shovel whose blade was perforated with small holes.
Worker: 1 2 3 4 5 6
Conventional: .0011 .0014 .0018 .0022 .0017 .0016
Perforated: .0011 .0010 .0019 .0013 .0011 .0015
Do these data provide convincing evidence that the mean energy expenditure using the conventional shovel exceeds that using the perforated shovel? Test the relevant hypotheses using a significance level of 0.05?
Based on the given data, the hypothesis test results show that there is not enough evidence to conclude that women chat on their phones more than men at the 5% significance level.
For the hypothesis test comparing the chat times of male and female students, we can use a two-sample t-test since we have two independent samples with unequal variances. The null hypothesis (H0) assumes that there is no difference in chat times between men and women, while the alternative hypothesis (Ha) suggests that women chat more than men.
Calculating the test statistic and degrees of freedom using the provided sample data and formulas, we find that the test statistic is approximately 1.03. Comparing this with the critical value at a 5% significance level (t-critical value), we find that it does not fall in the rejection region.
Therefore, we fail to reject the null hypothesis and conclude that there is not enough evidence to support the claim that women chat on their phones more than men at the 5% significance level.
For the second scenario comparing energy expenditure using conventional and perforated shovels, we would need more information, such as the sample sizes, means, and standard deviations, to perform the relevant hypothesis test. The provided data alone does not allow us to conduct the hypothesis test.
Learn more about hypothesis here:
brainly.com/question/17099835
#SPJ11
someone help, what’s 1+1?
What is the constant of proportionality in the table below?
Answer:
12
Step-by-step explanation:
y = kx where 'k' is the constant of proportionality
24 = 2 x 12
36 = 3 x 12
48 = 4 x 12
60 = 5 x 12
PLZ ANSWER THIS CORRECTLY FOR 100 POINTS AND BRANLIEST :DD
#x
2x+30+62=1802x+92=1802x=88x=44Angle 1
2(44)+3088+30118°Angle 2
62°(Opposite angles)Answer:
A) x = 44
B) m∠1 = 118°
m∠2 = 62°
Step-by-step explanation:
Part A
Angles on a straight line sum to 180°
⇒ (2x + 30) + 62 = 180
⇒ 2x + 30 + 62 = 180
⇒ 2x + 92 = 180
⇒ 2x = 88
⇒ x = 44
Part B
Vertical Angle Theorem: The opposite vertical angles of two straight intersecting lines are congruent.
⇒ m∠1 = (2x + 30)
Substituting the found value of x:
⇒ m∠1 = 2(44) + 30
⇒ m∠1 = 88 + 30
⇒ m∠1 = 118°
Using the Vertical Angle Theorem:
⇒ m∠2 = 62°
Jamal uses the line of best fit y = 700x + 38,000 to predict his annual salary, y, after x years.
What is the y-intercept of the line of best fit and what is the meaning of the y-intercept?
The y-intercept is n intercept is a point on the y-axis, through which the slope of the line passes and the y-intercept of the given equation is 38,000.
What is a y-intercept?In Maths, an intercept is a point on the y-axis, through which the slope of the line passes. It is the y-coordinate of a point where a straight line or a curve intersects the y-axis. This is represented when we write the equation for a line, y = mx+c, where m is slope and c is the y-intercept.
The given equation is y=700x+38,000 to predict his annual salary, y, after x years.
In the given equation, y-intercept is 38,000.
Therefore, the y-intercept is n intercept is a point on the y-axis, through which the slope of the line passes and the y-intercept of the given equation is 38,000.
To learn more about the y-intercept visit:
brainly.com/question/14180189.
#SPJ2
Amad collected 42 plates in February for a project he plans to make. He had collected a total of 91 plates by the end of March. Which equation can be used to find the number of plates Amad collected in March? A. 42 * p = 91 b. P - 91 = 42 c. 91 / p = 42 d. P + 42 = 91
Answer:
Option D: P + 42 = 91
Step-by-step explanation:
Plates collected in February = 42
Total Plates collected = 91
Plates in February + Plates in March = total plates
42 + P = 91 or
P + 42 = 91
The area of a rectangular garden in square feet can be represented by 22 - 102 +21. The length is 2 - 7 feet. You have been hired to fertilize the
garden. Recall that the area of the garden is 22 - 10z +21. If 2 = 20 and the fertilizer costs $0.27 per square foot, how much it cost to fertilize
the garden?
The area of a rectangular garden in square feet can be represented by z² - 10z +21. The length is z - 7 feet. You have been hired to fertilize the
garden. Recall that the area of the garden is z² - 10z +21. If z = 20 and the fertilizer costs $0.27 per square foot, how much it cost to fertilize
the garden?
Answer:
$ 59.67
Step-by-step explanation:
The area of a rectangular garden in square feet can be represented by z² - 10z +21 square feet
If z = 20
Therefore:
The area of the rectangular garden is:
20² - 10 × 20 + 21
400 - 200 + 21
= 221 square feet
The fertilizer costs $0.27 per square foot, how much it cost to fertilize
the garden?
The cost of fertilize the garden is calculated as:
1 square feet = $0.27
221 square feet = x
Cross Multiply
221 square feet × $0.27
=$ 59.67
Three coins are tossed. Draw a Tree-diagram to show the possible outcomes. What is the sample space for this event? Also, find the probability of getting:
a) no heads
b) exactly 2 tails
c) at least 2 heads
d) at most 2 tails
There are 6 possible outcomes:P(HHH) + P(HHT) + P(HTH) + P(THH) + P(HTT) + P(THT) = 7/8
a) No heads: 0.125
b) Exactly 2 tails: 0.375
c) At least 2 heads: 0.5
d) At most 2 tails: 0.875.
Given that three coins are tossed, we need to draw a Tree-diagram to show the possible outcomes and then find the sample space for this event.
Afterward, we need to find the probability of getting:
a) no heads
b) exactly 2 tails
c) at least 2 heads
d) at most 2 tails
Tree-diagram:
It is a tree diagram representing the tossing of three coins to show all possible outcomes.Sample space:
It is the set of all possible outcomes.
The sample space for this event is: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
a) No heads are (TTT) and the probability of getting no head is:
P(TTT) = 1/8
= 0.125
b) Exactly two tails are (HHT, HTH, THH) and the probability of getting exactly 2 tails is:
P(HHT) + P(HTH) + P(THH) = 3/8
= 0.375
c) At least 2 heads means getting 2 heads or 3 heads. There are 4 possible outcomes:P(HHH) + P(HHT) + P(HTH) + P(THH) = 1/2 = 0.5d)
At most 2 tails mean getting 0 tails or 1 tail.
There are 6 possible outcomes:P(HHH) + P(HHT) + P(HTH) + P(THH) + P(HTT) + P(THT) = 7/8
= 0.875
Hence, the answers are:
a) No heads: 0.125
b) Exactly 2 tails: 0.375
c) At least 2 heads: 0.5
d) At most 2 tails: 0.875.
To know more about probability , visit ;
https://brainly.com/question/13604758
#SPJ11
Find the value of x.
hello i know its wrong but lol. i tried to help.
I got correct on this question but I wrote some of the numbers in a different order, is it okay if I wrote it in a different order on every question similar like this?
Answer:
however it works for you along as you get the right answer
Step-by-step explanation:
Answer:
Yeah you will most likely get the same answer
Step-by-step explanation:
as long as you don't mix up (for this example of course) the service call fee (which is the constant) and the fee for each hour of labor (the coefficient of the variable), you are good
WILL GIVE BRAINLIEST!
The graph below shows the relationship between distance walked and calories burned.
Find the slope of the line.
Write a sentence that explains the meaning of the slope.
Walking this many miles burns 100 calories.
Walking this many hours burns one calorie.
Walking burns this many calories per hour.
Walking this many miles burns one calorie.
Walking burns this many calories per mile.
Answer:
70
Step-by-step explanation:
The slope is down 1 over 1.5, so that means Start with the slope formula
m=
(y2 - y1)
(x2 - x1)
Substitute point values in the formula
m=
(140 - 280)
(2 - 4)
Simplify each side of the equation
m=
(140 - 280)
(2 - 4)
=
-140
-2
Solve for slope (m)
m= 70
Booommmm, Im proud of this lol, hope it helps <3