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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. 

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Improving the convergence of a slowly convergent series

We want to evaluate log(2) by inserting x = 1 in the series: log(1+x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + x^7/7 - x^8/8 + x^9/9 - x^10/10 +... and take the first ten terms. But the result is rather poor. Show that by taking a linear combination of this truncated series with that of the function 1/(1+x) that eli

Remainder Theorem to Solve a Polynomial

1) Determine the value of k so that when P(x) = x(cubed) + kx(squared) - 2x(squared) + 1. Is divided by x + 2, the remainder is 5 2) Using long division, determine the remainder when P(x) is divided by x-3

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

Derivative of a ratio of complex functions

Show that the derivative of g(z) exists, and hence g(z) is complex analytic, at points where g(z) does not diverge, thereby proving that the points where g(z) diverges are the only singularities. 
g(z)= eiz/ z2 +4z+5

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

Rate of Change and Function Domain & Range

1. Taxes Along with incomes, people's charitable contributions have steadily increased over the past few years. The table below shows the average deduction for charitable contributions reported on individual income tax returns for the period 2007 to 2012. Find the average rate of change between 2009 and 2011. Year Ch

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

Fixed Point Iteration Formulas

The real root of x^3 - x - 1 is about 1.3. a) Construct a fixed point iteration formula for finding the root and prove that the formula will work. b) Construct another fixed point iteration formula and prove that it will not work.

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

An investor in the US bought UK securities

An investor in the United States bought a one-year British security valued at 196,000 British pounds. The U.S. dollar equivalent was $98,000. The British security earned 17 percent during the year, but the British pound depreciated five cents against the U.S. dollar during the period ($0.50 to $0.45). After transferring the


Ask 20 people how long they workout at the gym, ten men and ten women: ten men said they workout at less 2 hrs a day, five women said one hr, three said 2 hrs, two, one hr and half. What is the mean, median, and mode?

Cost function, input price and production function.

A firm uses one input, L, to generate output, q, according to the production function q=16L^2. The input price is w and the fixed costs are c_0 > 0. i) Show that the firm's cost function is given by C(q)=c_0 + w(square root)q/4. ii) Show that dq/dL increases with L while dC/Dq decreases with q. See the attached file for th

Z x Z Functions and Range

See the attached file. 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.

trigonometric functions of any angle

For an arc length s, area of sector A, and central angle theta of radius r, find the indicated quantity for the given value. A + 61.4 mi ^2, r = 61.4 mi Theta = ? Theta equal ? radian Do not round until the final answer. Then round to three decimal places as needed.

Proof about Functions

Let f: X-->Y be a function from one set X to another set Y, let S be a subset of X, and let U be a subset of Y. What in general can one say about f^-1(f(S)) and S? What about f(f^-1(U)) and U?

Function and Inverse Function

If the functions f(x) and g(x) are defined by the table below and if z(x)=g(f(x)), then what is the value of z^-1 (1)? (Use table below) x 1 2 3 4 5 6 7 8 9 10 f(x) 10 1 8 2 6 3 4 9 5 7 g(x) 7 8 9 10 1 2 3 4 5 6.

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


Bob owns a watch repair shop. He has found that the cost of operating his shop is given by C(x) = 4x2-296x+85 , where c is cost and x is the number of watches repaired. How many watches must he repair to have the lowest cost?

using quantifiers sequences

Please show complete solutions. By definition, 〖lim〗_(n→∞ ) a_n=L if for every ε>0, the re exists a positive number N such that if n is an integer with n>N, then |a_n-L|<ε. By taking the negation of this definition, write out the meaning of 〖lim〗_(n→∞) a_n≠L using quantifiers. Then write out the meaning o

Notation for Intervals

Suppose n is a positive integer, and let a1, a2, ... , an be real numbers such that a1 < a2 < ... < an. ** Please see the attachment for complete question description ** Please show complete solution with steps.

Expressing domain in interval notation

1) Given that f(x)= sqrt(4x+6) and g(x) = (x-1)/x, find the following functions and express their domains in interval notation. a. (f + g)(x) = _________, its domain is ________ b. (f/g)(x) = __________, its domain is________ c. (g/f)(x) = __________, its domain is __________ d. (f [of] g)(x) = _________, its domain is ___

Truth Equivalencies

1. Determine which, if any, of the three statements are equivalent. I) If the house is not brick, then the house is small. II) Either the house is brick or the house is small. III) Either the house is not brick or the house is not small. 2. Determine which, if any, of the three statements are equivalent. I) If tom

Solving for density when volume and mass are known

If the mass of an object = 32 g +/- 2g and its volume is 11cm cubed +/- 1cm cubed, what is the density in g/cm cubed? Select from the choices below. 2.90 +/- 3% 0.34 +/- 3% 2.9 +/- 15% 2.9 +/- 27% 3 +/- 2%

average price for a Big Mac

Do not send in PDF format. Sec. 2.1 # 38. The set of countries in which the average price for a Big Mac is between $2.00 and $2.99 Switzerland $5.46 Denmark $4.97 Sweden $4.46 United Kingdom $3.61 Germany $3.58 New Zealand $3.16 United States $3.00 Turkey $2.80 Peru $2.74 Canada $2.60 Chile $2.56 Japan $2.50

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)

second order equation

1. Write the given second order equation as its equivalent system of first order equations. Use to represent the "velocity function", i.e. . Use and for the two functions, rather than and . (The latter confuses webwork. Functions like are ok.) 2. Write the given second order equation as its equivalent s

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