function analysis

1. Finding range of a function using inequalities

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...
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. Continuity of a function under Euclidean topology

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...
4. Irregular clock

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...
5. Parallelism of Time-varying Vectors

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...
6. Find f(x) which satisfies this integral function

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...
7. Rigor in Quantum physics -- Do I need to know Functional Analysis well?

Hello, I've a following question: Is necessary know well func. analysis, and all its theorems to handle well quantum physics...?
8. I Metrics which generate topologies

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
9. [basic help] Drawing functions

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
10. B Difference between these functions .

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 ?
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> =...