Discrete mathematics function relation question. I just can not understand how partial functio works. for example, I think a) is not

total function, because y+5=3x is not always true for every domain. However, I do not know it is just partial function, but I do not know how to explain it.

For each of the following relations, determine whether the relation is a function, only a partial function, or not even a partial function.

If it’s a (total) function, write “total function”, and provide a proof that it meets both requirements.

If it’s partial but not total, write “partial, but not total function”, provide a proof that it meets the second requirement (uniqueness), and provide a counterexample to prove that it does not meet the first requirement (existence).

If it’s not even a partial function, write “not a partial function”, and give a counterexample to prove that it doesn’t meet the second requirement (uniqueness).

(a) The relation L on R defined by L = {(x, y) | y + 5 = 3x}.

(b) The relation P1 from Z to Z defined by P1 = {(x, y) | x · y = 6}.

(c) The relation P2 from R ∗ to R ∗ defined by P2 = {(x, y) | x · y = 6}. R ∗ is the set of all non-zero real numbers.

(d) The relation E on the set of all strings defined by L = {(s, t)| the length of s = the length of t}.

For some colleges, there are more applicants than there are openings. This requires colleges to implement a method for selecting students that utilizes criteria believed to predict how well a given student will perform in their college. For example, colleges will use a multiple regression equation that uses high school GPA, test score from the ACT or SAT (college entrance exams), and high school academic ranking (i.e., the quality of the high school) to predict how well a prospective student will do in their college. Describe at least 3 factors/variables you think colleges should use in selecting prospective students into their school. Your initial post must contain at least 10 sentences.

Management is planning to promote a new service in two media buys: flyers and online advertisements. A media budget of $1,500 is available for this promotional campaign. Based on past experience in promoting its other services, the following estimate of the relationship between sales and the amount spent on promotion in these two media is

                              S = -8F^2 - 22A^2 - 4FA + 12F + 32A

where

                             S = total sales in thousands of dollars

                             F = dollars spent on flyers

                             A = dollars spent on online advertising

Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $1,500 on total advertising. Determine the optimal amount to spend on flyers and online advertising.

How much should be allocated to flyers and online advertising?
(in 000s, two decimal places - include leading zero, no dollar sign).

a. Flyers:

b. Online Ads:

What is the optimal sales value generated?
(in 000s, two decimal places, no dollar sign).

c. Sales:

How much of the budget is unspent, based on your optimization?
(in 000s, two decimal places - include leading zero, no dollar sign).

d. Unspent budget:

Use the Laplace transform to solve

y'' + 4y' + 5y = 1, y(0)= 1, y'(0) = 2

how to prove the Existence of factorization in Euclidean domains

Find the exact solution : y''+9y'=cosπt, y(0)=0,y'(0）=1

QUESTION 1

1. In order to determine the average price of hotel rooms in Atlanta, a sample of 38 hotels were selected. It was determined that the average price of the rooms in the sample was $109.3. The population standard deviation is known to be $18. We would like to test whether or not the average room price is significantly different from $110.

   Compute the test statistic.

QUESTION 2

1. In order to determine the average price of hotel rooms in Atlanta, a sample of 39 hotels were selected. It was determined that the test statistic (z) was $-1.99. We would like to test whether or not the average room price is significantly different from $110. Population standard deviation is known to us.

   Compute the p-value.

QUESTION 3

1. In order to determine the average price of hotel rooms in Atlanta. Using a 0.1 level of significance, we would like to test whether or not the average room price is significantly different from $110. The population standard deviation is known to be $16. A sample of 64 hotels was selected. The test statistic (z) is calculated and it is -1.38.

   We conclude that the average price of hotel rooms in Atlanta is NOT significantly different from $110. (Enter 1 if the conclusion is correct. Enter 0 if the conclusion is wrong.)

QUESTION 4

1. In order to determine the average price of hotel rooms in Atlanta. Using a 0.1 level of significance, we would like to test whether or not the average room price is significantly different from $110. The population standard deviation is known to be $16. A sample of 64 hotels was selected. The p-value associated with the test statistic (z) is calculated and it is 0.03.

   We conclude that the average price of hotel rooms in Atlanta is NOT significantly different from $110. (Enter 1 if the conclusion is correct. Enter 0 if the conclusion is wrong.)

Suppose (A,∗) be an associative, unital, binary operation with inverses. Show that if|A|≤3,then in fact, (A,∗) isalsocommutative, even though we didn’t assume it at the beginning.

A photoconductor film is manufactured at a nominal thickness of 25 mils. The product engineer wishes to increase the mean speed of the film, and believes that this can be achieved by reducing the thickness of the film to 20 mils. Eight samples of each film thickness are manufactured in a pilot production process, and the film speed (in microjoules per square inch) is measured. For the 25-mil film, the sample data result is x-bar1=1.13 and s1=0.11, while for the 20-mil film, the data yield x-bar2=1.08 and s2=0.09. Note that an increase in film speed would lower the value of the observation in microjoules per square inch.

(a) Do the data support the claim that reducing the film thickness increases the mean speed of the film? Use a=0.10 and assume that the two population variances are equal and the underlying population of film speed is normally distributed. What is the P-value for this test? Round your answer to three decimal places (e.g. 98.765).

The data ___support/do not support___ the claim that reducing the film thickness increases the mean speed of the film. The P-value is _______.

(b) Find a 95% confidence interval on the difference in the two means that can be used to test the claim in part (a). Round your answers to four decimal places (e.g. 98.7654).

_______ <= mu1 - mu2 <= _______

Which one of the improper integrals below converges or diverges?
[int _a ^b] means integral from a to b, we use inf to indicate infinity.

a) [int _0 ^1] 1/x dx
b) [int _0 ^1] 1/x^(1/2) dx
c) [int _0 ^1] 1/x^2 dx

d) [int _1 ^inf] 1/x dx
e) [int _1 ^inf] 1/x^(1/2) dx
f) [int _1 ^inf] 1/x^2 dx

g) [int _1 ^inf] lnx / x^2 dx
h) [int _1 ^inf] lnx / x dx

i) [int _(-inf) ^inf] 1/(1+x^2) dx
j) [int _1 ^inf] exp(-x^2) dx
k) [int _1 ^inf] (sinx / x )^2 dx
l) [int _2 ^inf] 1 / sqrt(x^2 - 1) dx
m) [int _0 ^Pi/2] tanx dx

Complete the proof for the claim that any open ball B(x₀,r) in Euclidean space Rⁿ is homeomorphic to Rⁿ.

proof is given below the theorem. Show that suggested map g is in fact homeomorphism.

Theorem: Let X₀, X₁, and X₂ be topological spaces and let f: X₀ -> X₁ and g : X₁ -> X₂ be continuous functions. Then g∘f : X₀ -> X₂ is continuous.

proof : Suppose that V is open in X₂. Since g is continuous, g^-1(V) is open in X₁. Since f is continuous, f^-1(g^-1(V)) = (g∘f)^-1(V) is open in X₀. It follows that g∘f is continuous.

Prove: There are infinitely many primes congruent to 3 modulo 8. Hint: Consider N = (p₁p₂···p_r)² + 2.

A small business owner contributes $2,000 at the end of each quarter to a retirement account that earns 10% compounded quarterly. (a) How long will it be until the account is worth at least $150,000? (Round your answer UP to the nearest quarter.) 43 quarters (b) Suppose when the account reaches $150,000, the business owner increases the contributions to $4,000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.) $