## Let~~f(x)=h(x)+g(x) , where~~h(x)=10^{\sin x}~~and~~g(x)=10^{\csc x}##
##Then,~~D_f = {D_h}\cap {D_g}##
##Clearly,~~D_h=ℝ~~and~~D_g=ℝ-\{nπ|n∈ℤ\}##
##∴~~D_f =ℝ-\{nπ|n∈ℤ\}##
After considering the new domain, the range of ##\sin x## in ##10^{\sin x}## is ##[-1,1]-\{0\}##
Therefore, the range of...
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...
Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions."
Regarding the two main operators, X and D, each has...
Hi, I have a strange nonlinear operator which yields non-Hermitian solutions when treated in a simple ODE, ##H\Psi##=0. It appears from a paper by Dr Du in a different posting, that an operator can be non-self-adjoint in one domain, but be self-adjoint in another domain defined by the interval...
So, I'm a bit confused. The thing is, basically, all elementary functions are of the form ƒ:ℝ→ℝ. So the domain is ℝ and so is the codomain. However, if we have a function ƒ:ℝ→ℝ, given with f(x) = √x, it's domain is now x≥0. So, is the domain of this function ℝ or [0,+∞>?
Also, let's say we have...
Hi, I'm struggling to understand something. Does domain restriction work the same way for composition of inverse functions as it does for other composite functions? I would assume it does, but the end result seems counter-intuitive. For example:
If I have the function f(x) = 1/(1+x), with...
Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...
Hi,
I have a question about steel, for instance, silicon steel used in magnetic cores.
I was wondering about how Iron magnetises: can you align the grains/domains and heat treat it [or anything], so that it responds differently to flux one way then the other direction?
For example, say you...
1. Homework Statement
f(x,y) = 1/y^2-x
find the domain of f.
Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f.
2. Homework Equations
I know that the domain of the function is anywhere that the function is defined.
3. The Attempt at a Solution
in...