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Computing Values of Functions

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly out output.  An example of a commonly used function that related each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x). The input variables are often referred to as the arguments of the function.

Functions are very important in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions are defined by a formula or an algorithm that tells how to compute the output for a given input. Some functions are given by a graph of the function. A function can also be described implicitly or as a solution of a differential equation.

The input and output of a function can be expressed as an ordered pair. These pairs are ordered so that the first element is the input and the second is the output. If the input and outputs are real numbers, this ordered pair is viewed as the Cartesian coordinates of a point on a graph of the function.

A function is defined by its set of inputs. These inputs are called the domain. A set containing the outputs is called the codomain or range. The set of all paired inputs and outputs are called the functions graph.

A function f with domain X and codomain Y is commonly denoted by

F: X→Y

The elements of X are called arguments of f. For each argument X there is a corresponding unique Y in the codomain called the function value at x. 

Euler and Modified Euler's method

I need help using Euler an the improved Euler methods: (look at attachment for better formula display) 1. Consider the initial value problem y' = 2xy, y(1) = 1. Use the Euler's method and improved Euler's method with h = 0.1and h = 0.05 to obtain approximate values of the solution at x = 1.5. At each step compare the approxim

Functions, Limits & Continuity

1. Find the doubling time of an investment earning 8% interest if interest is compounded continuously. 2. Find the limit of f(x) = (7x+7)/(4x+4) as x approaching positive infinity and negative infinity. 3. Find the slope of the function's graph at the given point, then find an equation for the line tangent to the graph th

Stability and Lyapunov functions

Following is the problem that I solved the first part: Consider the system x' = f(x), where f: R^2 into R^2, is defined by: f(x) = [ (x1)^3 + (x1) (x2)^2 ] [ (x1)^2 (x2) + (x2)^3 ] a- Find all equilibrium points of the system. b- Use an appropriate Lyapunov function to determine the stability of the equilibr

Statistics Variable Identified Populations

A campaign conducts a poll of 1500 registered voters, and finds that 54% intend to vote for candidate A, 32% intend to vote for candidate B (their candidate), and 14% are undecided. Undecided voters, and those intending to support candidate A, are asked what single issue might cause them to support candidate B. The topic of ca

Z x Z functions and range

Consider the set of integers Z. The set Z x Z consists of all ordered pairs of integers. In symbols, Z x Z={(x,y):x,y∈Z} For example, (2,-5) , (0,0) , (-127,10) , and (-5,2) are all distinct elements of Z x Z. Notice that the order matters; (2,-5) and (-5,2) are different elements of Z x Z. Now let?s define

Time projections and tables

Three individual steps to be taken before the service could begin: (1) write instructions and procedures, (2) select techniques to operate the equipment, and (3) procure the equipment. It would be possible to save time on the project, by paying some premiums to complete certain activities faster than the normal schedule listed b

Cauchy sequences

5.2.1. Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are equivalent sequences of rationals, then (a_n)[n=1,infinity] is a Cauchy sequence if and only if (b_n)[n=1,infinity] is a Cauchy sequence. 5.2.2. Let epsilon > 0. Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are eventually epsilon-close, then (a_n)

Xbar and R charts, Is the process in control?

A manufacturing company produces rods. Each rod is required to be 20 millimeters in diameter. Each hour, random samples of size n = 4 rods are measured to check process control. Five hours of observations yielded the following: Diameter Time Rod 1 Rod 2 Rod 3 Rod 4 9 A.M. 19.8 20.4 19.9 20.3 10 A.M. 20.1 20.2 19.9 19.8


You are going to invest $20,000 in a portfolio consisting of assets X, Y, Z as follows: Asset Annual Return probability beta proportion X 10% 0.50 1.2 0.333 Y 8% 0.25 1.6 0.333 Z

Quantitative Method

I need help on this question- A quantitative method may be sensitive to changes in the parameters involved in comparing alternatives. A careful analyst conducts a sensitivity analysis to see if the situation's outcome is susceptible to small changes in such parameters (say 5%). Is Decision Analysis susceptible to such a s

Examine several newspapers and magazines

Examine several newspapers and magazines and describe at least three examples of functions that appear. What is the domain and range of each function? EXAMPLE OF ASSIGNMENT: The amount of money you pay for a gym membership is a function, because you pay a fee each month in addition to the initial fees for joining. In thi


Prove that an entire function that omits the value 0 and 1 is a constant.


Please solve the following 20 problems and provide detailed steps.

Matlab help

Two related questions. First - I need a good way to display error bars on a scatter plot. I have a set of x and y data, that I have attached as a text file along with y+err and y-err data (denoted +/-). The err values vary from point to point, and +err and -err are not necessarily the same. Second - I am using the data to

Functions and equations and measurement

This Discussion Question will concentrate on functions and graphs. Understanding the definitions of words is the essence of mathematics. When we understand the meaning of words, finding a solution is much easier because we know what task the problem is asking us to complete. a. Part 1 b. In your own words, define the wo

Analytic functions

Which of the following real functions u(x,y) of two real variables are the real parts of an analytic function f(z) with z=x+iy a) u(x,y) = x^3-y^3 b) u(x,y)=x^2-y^2+y IN each case compute f(z) if it exists

Functions, find the domain and range of a function and asymptotes

1. Break the function down by answering the following: a. How many terms are in the equation? b. Give the coefficients and the constants (label them). c. What is the degree of the function? d. What are the intercepts (x and y)? Give the (x,y) coordinate for both. e. What is the domain? Function: f (x) = 4x^5 + 3x^4 - 2

Floor and ceiling functions

Let F(x): R -> Z be the floor function and C(x): R -> Z be the ceiling function. Select which of the following function equations are true (select zero or more answers). a. C(x) - F(x) = 1 for all x. b. C(F(x)) = F(x) for all x c. C(F(x)) = F(C(x)) - 1 for all x d. F(F(x)) = F(x) for all x

Functions and Subsets

Theorem 4.16 Let f: A --> B, C and D be subsets of A, and E and F be subsets of B. Then b) f(the union of C and D) = the union of f(C) and f(D) d) f^-1(the union of E and F) = the union of f^-1(E) and f^-1(F)

Pointwise Operations of Functions

2 1. If f (x)= 2X - X -3 and g(x)= X+1, find f + g. 2. if f(x) = 3X-4 and g (x) = X+2 , find f + g 3. if f (x)= 3X-4 and g(x)= X+2 find f * g 4. if f(x)= 3X and g (x)= X-5 find (f*g) (X) 5. if f(x) = 5X+2 an g (x) = 3X-4 find (f*g) (x) factor 2 4X + 81 - 36

Tempered Distribution

Show that exp(x) is not a tempered distribution. Please justify your steps. Thank you

Greatest Common Factors

Find the GCF: (12,18), (16,20) (44,153) I tried to do on my own, I'm not sure if I understand what I am doing. If you can would it be possible to describe in details exactly what I should be doing. Thanks.


Please see the attached file for the fully formatted problems. 2. Suppose that f, g, h are three functions defined on (a, b) and c (a, b). Assume that f and h are differentiable at c, f(c) = h(c), and f(x) g(x) h(x) for all x (a, b). Prove that g is also differentiable at c and (c) = (c) = (c).

Population Growth Problems

In a galaxy far, far away (my professor writes his own practice problems), on the planet Xylor, a herd of 100 Tybars was introduced for breeding. After 5 years, the herd had increased to 500. If the rate of herd growth is assumed to be directly proportional to the number of Tybars present on Xylor at any time t: a. How many T


Describe the method used to estimate the sum, difference, product, or quotient. Then describe a better method and revise the estimate. 1) 362 - 118 ≈ 400 - 100 = 300 2) 19 x 2 ≈ 20 x 0 = 0 3) 148 + 138 ≈ 100 + 100 = 200 4) 305 ÷ 16 ≈ 300 ÷ 20 = 15