Let P be the power set of {A, B} and let S be the set of all binary strings of length 2. A function f: P -> S is defined as follows: For A in P, f(A) has a 1 in the high-order bit position (left end of string) if and only if a is in A. f(A) has a 1 in the low-order bit position (right end of string) if and only if b is in A. Is f one-to-one? Prove or disprove. Is f onto? Prove or disprove.

Solution Summary

This is a proof regarding one-to-one and onto functions.

1. If f(x)=2x-3 and g(x)=x2+1, find each of the following:
a) f(g(2))
b) g(f(3))
2. Let f(x)=x2+2 and g(x)=square root of 1-x2.
a) Find the domain and range of f and g.
b) Are the functions of f g and g f defined?
3. Given F(x)=cubed root of x+5, find functions f and g such that F=f g. Explain the answer.

Differentiate the given functions. For this question, you may use the following differentiation rules and formulas:
If r is any rational number, then
d/dx (x^r) = rx^(r-1)
If f and g are differentiable functions of x, and if ? [alpha] and ? [beta] are constants, then
d/dx ( ?f(x) + ?g(x)) = ? d/dx (f(x)) + ? d

Prove, from the definition of function (using ordered pairs), that composition of functions is associative. (i.e. prove that f * (g*h) = (f* g) *h) for suitable functions f, g, h
I would like to know how to use ordered pairs to proof the associative of composited functions.

1. Find the derivatives for the following functions ("^" means "to the power of", sorry I can't do double exponents on my keyboard) :
a. f(X) = 100e10X
b. f(X) = e(10X-5)
c. f(X) = e^X3
d. f(X) = 2X2e^(1- X2)
e. f(X) = 5Xe(12- 2X)
f. f(X) = 100e^(X3 + X4)
g. f(X) = e^(200X - X2 + X100)
2. Fi

I need some help with how to answer this question:
Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverse of each other:
f(x)=8x+8 and g(x)= x-8/8
a) f(g(x))= (simplify)
b) g(f(x))= (simplify)
c) f and g are inverse or are not inverse of each other

I'm looking for help in solving question 3 which is attached as a PDF document to this description.
The question asks for the Green Function, GF of: d(^2)y/dt(^2) +5*dy/dt + 6*y and show
that's its generic sol is y(t)= e^-2t int e^2z f(z) dz - e^-3t int e^3z f(z) dz.

Refer to the graph given below and identify the graph that represents the corresponding function. Justify your answer.
y = 4^x
y = log4x
Plot the graphs of the following functions. Scan the graphs and post them to the Facilitator along with your response.
1. f(x) = 5x
2. f(x) = 4x+2
3. f(x) = (1/3)x
4. f(x

Answer the following question:
For the exponential function ex and logarithmic function log x, graphically show the effect if x is halved. Include a table of values for all four functions.
Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Rank in order from fastest to slowest the rate at whi