Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) =
Let f: R --> R be the function defined by f(x) = x2 + 3x - 4 for all x E R. a) Determine whether or not f is injective. b) Determine whether or not f is surjective. c) Find f^-1({0,75/4}). d) Find f([0,1]).
Let f: A ?> B and g: B ?> C be functions. a) Suppose that f and g are injective. Show that g o f is injective. b) Suppose that f and g are surjective. Show that g o f is surjective. c) Suppose that f and g are bijective. Then g o f is bijective by parts a) and b). Show that (gof)^-1 = f^-1 o g^-1. Please do number 3 in at
Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) =
1. If U.S. Treasury yields are as follows: 3 month 6.0% 6 month 6.3% 1 year 6.5% 2 year 6.6% 5 year 6.4% 10 year 7.5% 30 year 8.0% a. What is the expected yield on notes from year 1 to 2, assuming the PEH holds? b. What is the expected yield on notes from year 2 to 5, assuming th
(See attached file for full problem description with equations and diagrams) --- 1. Simplification of linear algebraic expressions and expressions with fractional coefficients and solve x; 2. Solving simple linear equations with fractional coefficient: 3. Solving inequalities with fractional coefficient: 4.
F(x) = 4sinx / (2sinx+4cosx) the equation of the tangent line to f(x) at a=0 is y=sx+b Find: y=sx+b.
The parabola y= x^2+4 has two tangents which pass through the point (0,-2). One is tangent to the parabola at (a, a^2+4) and the other at (-a, a^2+4) where a is a certain positive number. The question is a=?
Find the equation of the tangent line to the graph of the function f(x) = x + (4/x) at the point (4, 5).
F injective iff there exists g such that gf = 1 f surjective iff there exists h such that hf = 1.
Please explain how to sketch a graph of a differentiable function f such that f > 0 and f' < 0 for all real numbers x.
Please explain how to find a and b such that f(x) = {a(x^3), x is less than or equal to 2 {x^2 + b, x is greater than 2 is differentiable everywhere
Y=3x^4+4x^3-6x^2-12x What are the points where tangent line is horizontal? What is the relative maxima and minima? Intervals of decrease? Points of inflection? and interval(s) of concavity?
1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? 2. There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahrenheit Celsius Freezing point of water 32
See attached file for full problem description with equation. --- Find analytically the solution of this difference equation with the given initial values: Without computing the solution recursively, predict whether such a computation would be stable. (Note: A numerical process is unstable if small errors made at one
If x<-4, f(x) = {[-2(x^3) - 6(x^2) +14x+24] / (x+4)} if x>=-4, f(x) = 5(x^2) +5x+a What value must be chosen for a in order to make this function continuous at -4? Please note the value of a does not equal to 66.
Evaluate the sum (using generating functions) A) 0+3+12+...+3n2. B) 4x3x2x1+5x4x3x2+...+n(n-1)(n-2)(n-3)
A new fashion in clothes is introduced. It spreads slowly through the population at first but then speeds up as more people become aware of it. Eventually those willing to try the new fashions begin to dry up and while the number of people adopting the fashion continues to increase. It does so at a decreasing rate. Later the fas
1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? 2. There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahren
Please see the attached file for the fully formatted problems.
1).Let f(X) : R -> R be the following: f(x) = { 1 if x is in Q (rationals) , 0 if x is not in Q ( irrational)} Prove that f(x) is Borel measurable ( Borel functions).
(a) Suppose the graph in Figure 4.1.78 is that of a function g(x). Sketch the graph of the derivative g; (b) On the other hand, suppose the graph above is that of the derivative of a function f. For the interval ..., tell where the function f is (i) increasing; (ii) decreasing. (iii) Tell whether f has any extrema, and if so
For the function of f, given below in graph (a) Sketch (b) Where does change its sign (c) Where does have local minima and maxima Using the graph of write a brief description of complete sentences to describe the relationship between the following features of the function of: (a) the local maxima and minima o
You have been invited to present statistical information at a conference. To prepare, you must perform the following tasks: The following data was retrieved from www.cdc.gov. It represents the number of deaths in the United States due to heart Disease and cancer in each of the years; 1985, 1990, 1995, and 2002. The f
You have been invited to present statistical information at a conference. To prepare, you must perform the following tasks: 1. The following data was retrieved from www.cdc.gov. It represents the number of deaths in the United States due to heart Disease and cancer in each of the years; 1985, 1990, 1995, and 2002. Year Di
Identify the axis of symmetry, create a suitable table of values, then sketch the graph (including the axis of symmetry). y = -x2 + 3x - 3 Here, a = -1, b = 3, c = -3 The axis of symmetry is x = -b/2a = -3 (-2) = 1.5 See attached file for full problem description, equations, charts and diagrams.
Identify the axis of symmetry, create a suitable table of values, then sketch the graph (including the axis of symmetry). y = -x2 + 3x - 3
Consider an open-top box with a square base and a volume of 108 cubic inches. Let x be the length of a side of the base. a) Calculate the height h as a function of x. Is this function even, odd, or neither? b) What is the domain of the function above? (Note that there may be physical and/or mathematical restrictions.)
Problem: Find the vertex and intercepts for y = x^2 + x - 6
Prove : Let f1,f2.... be a sequence of continuous functions convergent uniformly on a bounded closed interval [a,b] and let c E[a,b] . For n = 1,2,...., define ..... Then the sequence g1,g2.... converges uniformly on [a,b]. Is the same true if [a,b] is replaced by ? Please see the attached file for the fully formatte