domain

1. Find the domain and range of this function

$Let~~f(x)=h(x)+g(x) , where~~h(x)=10^{\sin x}~~and~~g(x)=10^{\csc x}$ $Then,~~D_f = {D_h}\cap {D_g}$ $Clearly,~~D_h=ℝ~~and~~D_g=ℝ-\{nπ|n∈ℤ\}$ $∴~~D_f =ℝ-\{nπ|n∈ℤ\}$ After considering the new domain, the range of $\sin x$ in $10^{\sin x}$ is $[-1,1]-\{0\}$ Therefore, the range of...
2. Finding the range of a function when checking if it is bijective

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...
3. I How to find admissible functions for a domain?

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...
4. A How to find the domain of functions of an operator

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...
5. B Domain and the codomain of a composite function

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...
6. I Domain of the identity function after inverse composition

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...
7. M

Transforming Piecewise Functions

1. Homework Statement The piece wise function is -x-2, x<-1 x^2-3x, -1≤ x ≤5 3x+5, x>5 The problem is to transform the function with...
8. I Must functions really have interval domains for derivatives?

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...
9. I Magnetic alignment of steel question

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...
10. Determining the domain and range of multi-variable function

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...