Why does it not make sense for a function to have more than one output for the same input? Provide examples.
First , define inputs and outputs ( independent and dependent variables) . Then, define ordered pairs and relations and give examples. Finally, define a function and talk about what particular type of relation is a function.
Can you find a real world relation that is not a function?
For example: The age of a person is the input and his annual salary is the output. Two people who are the same age ( input) can have two different salaries ( output).
Can you find a real - world function that has more than one input leading to the same output?
For example : the height of a person is the input and his weight is the output.
Two people who have different heights ( input) can have the same weight ( output).

Assume G is an Abelian group of order 144 and G contains at least 5 elements of order 72. (a) Determine the possible structures of G (i.e., write it as a direct product of cyclic groups). If there are more than one structures, list them all. (b) For each isomorphic class of G, how many elements of order 12 does it have?

Find a closed formula for each of the following sequences. Show all work and explain your answers.

(a) {1, 6, 17, 34, 57, 86, 121, . . .}, where a₀ = 1.

(b) a_n = 5a_n−1 + 4, a₀ = 2

(c) a_n = 10a_n−1 − 21a_n−2, a₀ = 6, a₁ = 26.

Scenario

Implementing the sieve of Eratosthenes algorithm to find all prime numbers up to a given limit.

Aim

Develop code for implementing the sieve of Eratosthenes.

Steps for Completion

1. Implement the isPrime() method of the SieveOfEratosthenes class that should return true if the number is prime, and false otherwise.

2. Consider building the sieve in the class constructor.

CODE GIVEN

public class SieveOfEratosthenes {

public SieveOfEratosthenes(int maxValue) {

// Build the sieve here

}

public boolean isPrime(int value) {

// Write your code here

}

}

Assume a closed economy without Government. However, there exists a financial sector that creates an array of financial assets on which both households and firms invest.

Let ? denote the average earnings from these financial assets. The consumption expenditure of the households is influenced by their wage income and the financial income and is given by ? = ?(?, ?); ?? > 0, ?? > 0, where ??, ?? are partial derivatives of consumption with respect to income ? and financial earnings ? respectively. Similarly, the real investment expenditure of firms is given by ? = ?(?, ?); ?? > 0, ?? < 0, where ??, ?? are the partial derivatives of the real investment with respect to income and financial earnings. Using either the Keynesian cross model or the Multiplier analysis, answer the following questions.

(i) Derive the relationship between output ? and financial earnings ?, and examine the analytical conditions under which the relationship is positive ( ?? ?? > 0) and negative ( ?? ?? < 0).

(ii) Describe why the scenario where the expansion in output driven by the rise in financial earnings, i.e. when ?? ?? > 0, could make the economy unstable and vulnerable to crisis?

Which of the following functions are one-to-one?

Group of answer choices

f;[−3,3]→[0,3],f(x)=9−x2f;[−3,3]→[0,3],f(x)=9−x2

f;R→R,f(x)=x3f;R→R,f(x)=x3

f;[0,∞]→[0,∞],f(x)=x2f;[0,∞]→[0,∞],f(x)=x2

Please provide proofs for parts i.)-iii.)

(i) Refer to the sequence in 1(ii). Show that with respect to the supremum norm on ?[0,1] this is a bounded sequence that has no convergent subsequence. (hint: What is the value of ‖?? − ??‖∞ if ? ≠ ??)

(ii) Refer to the sequence in 1(v). Show that this is a bounded sequence with respect to the 1-norm on ?[0,1] that has no convergent subsequence.

(iii) Let ℎ?(?) = sin??. Show that with respect to the 2-norm ?[0,2?], (ℎ?) is a bounded sequence that has no convergent subsequence. (This exercise shows that the Bolzano-Weierstrass Theorem does not generalise to ?[?,?] with any of the 3 “natural” norms on ?[?,?])

Note: sequences from 1ii.) and 1v.) are pointwise functions and are defined respectively below:

1ii.) For ? ≥ 2, define the function ?? on [0,1] by: ??(?) =( ??, if 0 ≤ ? ≤ 1/?)

(2-??, if 1/?< ? ≤ 2/n)

(0, if 2/?< ? ≤ 1)

1v.) Hn=n?? (and fn is defined as above)

Compute the Taylor series at x = 0 for ln(1+x) and for x cos x by repeatedly differentiating the function. Find the radii of convergence of the associated series.

3. Let X = {1, 2, 3, 4}. Let F be the set of all functions from X to X. For any relation R on X, define a relation S on F by: for all f, g ∈ F, f S g if and only if there exists x ∈ X so that f(x)Rg(x).

For each of the following statements, prove or disprove the statement.

(a) For all relations R on X, if R is reflexive then S is reflexive.

(b) For all relations R on X, if S is reflexive then R is reflexive.

(c) For all relations R on X, if R is symmetric then S is symmetric.

(d) For all relations R on X, if S is symmetric then R is symmetric.

Chapter 9 discussed the importance of stakeholder engagement in policy making. The author presented several benefits and an analysis of five cases in which stakeholder engagement added value to the policy making process. If you were leading a project to develop a comprehensive policy for managing pedestrian traffic flow in a popular downtown metropolitan district, what measures would you take to engage stakeholders in that project? Your answer should outline your suggestions and clearly explain why each one would add value.

Need 300 words with no plagrism

. Find the Laplace transform of the functions: f(t) = 3e^5t t^3 − 6e^−t t^4 g(t) = 5e^3t cos(4t) − 6e^2t sin(7t)

Sports Finance Questions

1. Compare and contrast various segments within the sport industry and how they handle financial issues.

2. Examine how sports facilities can become an economic engine for revenue generation.

3. Forecast the future of the sport industry based on changes in the sports broadcasting field.

Find a particular solution of

y'' + 2y' + y = e^-x [ ( 5 −2x )cos(x) − ( 3 + 3x )sin(x) ]

y_{p( x ) = ?}

_{Please show your work step by step. Thank you!}