For the function f(x) = x^2 +3x / 2x^2 + 7x +3 find the following, and use it to graph the function.

Find: a)(2pts) Domain

b)(2pts) Intercepts

c)(2pts) Symmetry

d) (2pts) Asymptotes

e)(4pts) Intervals of Increase or decrease

f) (2pts) Local maximum and local minimum values

g)(4pts) Concavity and Points of inflection and

h)(2pts) Sketch the curve

Consider the following Markov chain with P{X0 = 2} = 0.6 and P{X0 = 4} = 0.4:

1 2 3 4 5 6

1 0 0 0 0 1 0

2 .2 .05 0 .6 0 .15

3 0 0 .8 0 0 .2

4 0 .6 0 .2 0 .2

5 1 0 0 0 0 0

6 0 0 .7 0 0 .3

a. What is P{X1 = 4, X2 = 6 | X0 = 2}?

b. What is P{X2 = 6 | X0 = 2}? What is P{X18 = 6 | X16 = 2}?

c. What is P{X0 = 2, X1 = 4, X2 = 6}?

d. What is P{X1 = 4, X2 = 6}?

Why is it not good to directly store the hash of password in a file? How can the use of salt enhance password security?

7. (10 pts.) Amelia visits several islands. The individuals on each these islands are either truth-tellers or liars. Amelia is interested in finding out whether everyone on a given island is a truth-teller, or whether everyone is liar, or whether there are both truth-tellers and liars. (Note: it only takes one liar for an island’s inhabitants to include both truth-tellers and liars.) Amelia is also interested in finding someone from whom to bum a cigarette. (Yes, she shouldn’t smoke, but she does anyway and intends to stop soon.) In each of the following questions, use complete English sentences to present the reasoning that backs up your answer. By the way, in this problem, and indeed every problem, you should remember that any universally generalized conditional whose antecedent-predicate is true of nothing is true. Amelia has already visited two of these islands. Now she visits two more.

a. On the third island, Amelia asks whether anyone on the island smokes. Everyone answers: “If I smoke, then everyone smokes.” What can Amelia conclude about the third island? b. On the fourth island, in response to the same question, everyone replies, “Some of us smoke, but I do not.” What can Amelia conclude about the fourth island?

b. On the fourth island, in response to the same question, everyone replies, “Some of us smoke, but I do not.” What can Amelia conclude about the fourth island?

Question 2: Pocket Politics (any resemblance to recent events is purely coincidental) Long time ago, in a country far far away, a Totally Racist Unqualified Malicious President ruled the land with an iron fist. His rivals had to do everything in their power to free the people from his terrible regime and make the country great again. Many people offered to take the TRUMP down, but six wise and brave men and women stood out from the rest:
1. Joe “Busy Hands” Bye-then (J) 2. Bernie “Crazy Eyes” Slanders (B) 3. Elizabeth “Pocahontas” Warden (E) 4. Tulsi “Go Land Crabs!” Globbard (T) 5. Mike “Mini Me” Broomfield (M) 6. Pete “Father-of-Chickens” Boot-a-Judge (P)
In order for the good people to make the right choice, the candidates have to gather in the town hall for a night of sword fights and verbal altercations. You are in charge of organizing the grand event.

Q1. Since no clear ranking could be established at the previous fights, the candidates are now paired up tournament style and each pair fights it out until someone gives up. How many ways do you have to divide the six candidates into three pairs? (Hint: How many ways can you select one pair? How many ways does it leave you to select the second pair? Then the third? Don’t forget to eliminate redundancy due to order).

1. . Let A = {1,2,3,4,5}, B = {1,3,5,7,9}, and C = {2,6,10,14}.

a. Compute the following sets: A∪B, A∩B, B∪C, B∩C, A\B, B\A.

b. Compute the following sets: A∩(B∪C), (A∩B)∪(A∩C), A∪(B∩C), (A∪B)∩(A∪C).

c. Prove that A∪B = (A\B)∪(A∩B)∪(B\A).

2. Let C0 = {3n : n ∈ Z} = {...,−9,−6,−3,0,3,6,9,...} C1 = {3n+1 : n ∈ Z} = {...,−8,−5,−2,1,4,7,10,...} C2 = {3n+2 : n ∈ Z} = {...,−7,−4,−1,2,5,8,11,...}.

a. Prove that the sets C0, C1, and C2 are pairwise disjoint.

b. Compute C0 ∪C1 ∪C2.

3. Let R>0 be the set of positive real numbers, that is, R>0 = {x ∈ R : x > 0}.

Prove that {e x : x ∈ R} = R>0 and {logx : x ∈ R>0} = R.

SupposeG=〈a〉is a cyclic group of order 12.

Find all of the proper subgroups of G, and list their elements. Find all the generators of each subgroup. Explain your reasoning.

Give an example of a nonabelian group G of order n and a subgroup H of order k. Then list all of the cosets of G/H. where n = 24 and k = 3.

A bottle of

milkmilk

initially has a temperature of

7575degrees°F.

It is left to cool in a refrigerator that has a temperature of

4545degrees°F.

After 10 minutes the temperature of the

milkmilk

is

5757degrees°F.

a. Use​ Newton's Law of​ Cooling,

Upper T equals Upper C plus left parenthesis Upper T 0 minus Upper C right parenthesis e Superscript ktT=C+T0−Cekt​,

to find a model for the temperature of the

milkmilk​,

T​,

after t minutes.

T	​=	Upper C plus left parenthesis Upper T 0 minus Upper C right parenthesis e Superscript ktC+T0−Cekt
T	​=	45 plus left parenthesis 30 right parenthesis e Superscript nothing t45+(30)e t	left arrow←	Solve for k and enter the answer
		​(Round to four decimal​ places.)

b. What is the temperature of the

milkmilk

after 15​ minutes?

Tequals=nothingdegrees°F

​(Type an integer. Round to nearest​ degree.)

c. When will the temperature of the

milkmilk

be

5252degrees°​F?

tequals=nothing

minutes

A fishing boat leaves a dock at 2:00pm and travels due east at a speed of 20km/h. Another boat has been heading due north at 15 km/h and reaches the same dock at 3:00pm. At what time were the two boats closest together?

1) Assume that in 2014 the number of vehicle sales in the Ukraine was 232 thousand and in 2019 it was 108 thousand. a) determine the average rate of change (slope) in the number of vehicle sales from 2014 to 2019. Include the units. b) if x is the number of years since 2014 and z(x) is the number of vehicles sold, write the equation of the line through these two points. c) Assuming z(x) is a linear function, use the equation to predict the number of vehicles sold in 2022.

2) A cyclist traveled 12 kilometers per hour faster than an in-line skater. In the time it took the cyclist to travel 52.5 kilometers, the skater had gone 22.5 kilometers. Determine the speed of the skater.

3) The wexler family and the Santamaria family each used their sprinklers last month. The water output rate for the Wexler's family sprinkler was 20 gallons per hour. The water output rate for the Santamaria's sprinkler was 23 gallons per hour. The families used their sprinklers for a combined total of 40 hours, resulting in a total water output of 875 gallons. How many hours was each family's sprinkler used?

This is the full question

1. Consider a‘‘duel’’ between two players. Let’s call these players H and D.Now,we have historical information on each because this is not their first duel. H will kill at long range with probability 0.3 and at short range with probability 0.8. D will kill at long range with probability 0.4 and at short range with probability 0.6. Let’s consider a system that awards 10 points for a kill for each player. Build a payoff matrix by computing the expected values as the payoff for each player. Solve the game.

Determine the equation of a sine or cosine function for the vertical position, in metres, of a rider on a Ferris Wheel, after a certain amount of time, in seconds. The maximum height above the ground is 26 metres and the minimum height is 2 metres. The wheel completes one turn in 60 seconds. Assume that the lowest point is at time = 0 seconds.

r(t)=sinti+costj.

- Sketch the plane curve represented by r and include arrows indicating its orientation.

- Sketch the position vector r(t), the velocity vector r′(t), and the acceleration vector r′′(t) for the two times t = π/2, 5π/4 , putting the initial points of the velocity and acceleration vectors at the terminal points of the position vectors.

- Prove that the vectors r(t) and r′(t) are orthogonal for every t.

12. (a) Is the subset { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a subgroup of the direct product group ( V x Z18 )? (V is the Klein four group.) Carefully explain or justify your answer.

(b) Is the subgroup { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a normal subgroup of the direct product group ( V x Z18 )? Carefully explain or justify your answer.