My attempt :
Given ##f(x)## and ##g(x)## for ## -1.6 < x < 1.6## we get ##0\leq f(x)<1.6##
Thus, for ##f(g(x))## we get ## -3 \leq g(f(x)) < -1.4##
Thus the required set should be the interval ##[-3, -1.4)##?
My Questions :
1. What have I missed since my answer does not match the given...
To check if it is injective :
##h'(x) = 3(x^2-1)##
##\implies h'(x) \geq 0## for ##x \in (-\infty, -1]##
Thus, ##f(x)## is increasing over the given domain and thus is one-one.
To check if it is surjective :
Range of ##f(x) = (0, e^4]## but co-domain is ##(0, e^5]## thus the function is into...
1. Homework Statement
Let ##f:X\rightarrow Y## with X = Y = ##\mathbb{R}^2## an euclidean topology.
## f(x_1,x_2) =( x^2_1+x_2*sin(x_1),x^3_2-sin(e^{x_1+x_2} ) )##
Is f continuous?
2. Homework Equations
f is continuous if for every open set U in Y, its pre-image ##f^{-1}(U)## is open in X...
1. Homework Statement
A clock runs irregularly but after 24 hours it has neither gained nor lost overall.
Find a way the clock can run irregularly such that there is no continuous 576 minutes during which the clock shows that 576 minutes have passed.
2. Homework Equations
24 hours = 1440...
1. Homework Statement
This is a solved problem, but I haven't understood a few things.
I've marked out sections of the solution in white for convenience. The markings are positioned where that particular section ends.
In part (1), how did they just assume
f1(0) = 2, f2(0) = 3, g1(0) = 3...
1. Homework Statement
find f(x) which satisfies f(x) = x + ##\frac{1}{\pi}## ##\int_{0}^{\pi} f(t) \sin^2{t} \ d(t)##
2. Homework Equations
3. The Attempt at a Solution
to solve f(x), I have to solve the integral which contains f(t). And f(t) is the f(x) with variable t? if yes, I will get...
Given a topological space ##(\chi, \tau)##, do mathematicians study the set of all metric functions ##d: \chi\times\chi \rightarrow [0,\infty)## that generate the topology ##\tau##? Maybe they would endow this set with additional structure too. Are there resources on this?
Thanks
1. Homework Statement
I need to draw this function:
however I don't get how?
I have the solution
but I don't understand how do I get that from the given function. Someone please try to explain? Thanks
What's the difference between f(x)=3 and f(x)=3x^0 ? and why Limit of the second function when x\rightarrow0 exists ? and is the second function continuous at x=0 ?
1. Homework Statement
cos(θ)^x+sin(θ)^x=1
find the number of real values of x satisfying this equation
2. Homework Equations
none
3. The Attempt at a Solution
the answer given is 1 which is 2.which is obvious but i don't know how to prove that it is the only solution available
Hi everyone,
I am currently preparing myself for my Bachelor thesis in local quantum field theory. I was encouraged by my advisor to read the books of M. Reed and Simon because of my lag of functional analysis experience but I have quite often problems understand the “obvious” conclusions.
For...
Say we have a function of the form f(x)=\bigg\lbrace\begin{matrix}c \qquad\quad\text{x=0}\\ 0\quad\text{elsewhere}\end{matrix} If we then integrate this over all space i.e. \int_{-\infty}^{\infty}f(x)dx Why does the result equal zero?
My wave function:
##\psi_2=N_2 (4y^2-1) e^{-y^2/2}.##
Definition of some parts in the wavefunction ##y=x/a##, ##a= \left( \frac{\hbar}{mk} \right)##, ##N_2 = \sqrt{\frac{1}{8a\sqrt{\pi}}}## and x has an arrange from ##\pm 20\cdot 10^{-12}##.
Here is my integral:
##<x^2> =...