Inverse Definition and 1000 Threads

Hi, I know from basic math courses that inverse trig functions are multi valued (e.g. arctan(c)=θ+n*2∏). Now, if I solve a partial differential equation and I get an inverse trig function as part of my solution, does that mean solutions to the pde are non-unique? For example, if...

Homework Statement Inverse Fourier of: [ jω+2 ] / [ (jω)2 +5jω+9 ] where j = sqrt(-1) I tried using partial fractions but the denominator can't be factored...I tried completing the square on the denominator but I get a sum of squares. What can I try? I am sure I don't have to use the formal...

Trying to find the inverse of f(x)=2x^3+3x+2 I've written it as x=2y^3+3y+2 and am trying to solve for y, but I can't seem to factor anything out correctly. Anyone see how this can be solved?

I think I found a procedure (other than the extended Euclidean algorithm) to compute the multiplicative inverse of a number; possibly a simpler method. It goes like this: ***EDIT: Ah, dang. I made a mistake. It will only work for a prime modulus. Suppose we need to find x satisfying ax \equiv...

Hello Everyone :) I have been facing a little difficulty when encountering such kind of problems . i have also written down my line of thinking and approach which i take to solve them. So, please try to give me the correct line of thinking while solving such problems: 1. If A is invertible...

Homework Statement Suppose f: A → B is a function. Show that f is surjective if and only if there exists g: B→A such that fog=iB, where i is the identity function.The Attempt at a Solution Well, I believe for a rigorous proof we need to use the axiom of choice, but because I have never worked...

I was reading <crackpot link removed> and was wondering if the inverse cube law for magnetic force still applied for situations where the object being attracted isn't another magnet itself? E.g. if there is an electromagnet attracting an iron nut is the rule still inverse cube and not inverse...

Homework Statement y=x+(1/x) at y=17/4 Homework Equations The Attempt at a Solution y^-1: x=y+(1/y) differentiate: 1=y'+ln(y)y' 1=y'(1+ln(y)) y'=1/(1+ln(y)) put that over 1: 1+ln(y) plug in y: 1+ln(17/4) =approximately 2.447

Consider a function f(x) and its inverse g(x). Then (f \circ g)(x) = x and (g \circ f)(x) = x Are both these statements separate requirements in order for the inverse to be defined? Is it possible that one of the above statements is true but not the other? If so, could I see an...

Homework Statement This is from a physics problem, but my question is more mathematically oriented. After working through the problem, I arrive at the last step. Sin(x)=.967 The question says that there are two possible angles for x. The Attempt at a Solution arcsin(.967)...

My analysis text defines inverse functions only for bijections. But y = e^{x} is not bijective, so according to my book it's inverse ( ln x ) wouldn't be defined? Am I missing something or is my textbook just plain wrong? I use the text by Bartle and Sherbert. Thanks! BiP

Homework Statement The Attempt at a Solution Suppose we take an arbitrary polynomial in P_2 (R), call this a_0 + a_1 x + a_2 x^2 T(a_0 + a_1 x + a_2 x^2) = (a_0, a_0 + a_1 + a_2, a_0 - a_1 + a_2) Now, I was under the impression that I could construct a matrix for T by showing what...

Homework Statement Find the inverse laplace transform of \frac{3s + 7}{s^{2} - 2s + 10}Homework Equations completing the square. e^{at}sin(bt) = \frac{b}{(s-a)^{2} + b^{2}} e^{at}cos(bt) = \frac{s-a}{(s-a)^{2} + b^{2}} The Attempt at a Solution F(s)= \frac{3s + 7}{s^{2} - 2s + 10} F(s) =...

Homework Statement Find the derivative of sec^{-1}(\frac{\sqrt{1+x^{2}}}{x}) Homework Equations sec^{-1}=\frac{U'}{U\sqrt{U^{2}-1}} The Attempt at a Solution U'=-\frac{1}{x^{2}\sqrt{1+x^{2}}} U\sqrt{U^{2}-1}= \frac{\sqrt{1+x^{2}}}{x^{2}} Therefore the derivative is...

Homework Statement Find derivative of tan^{-1}(\frac{3sinx}{4+5cosx}) Homework Equations deriviative of tan^{-1}=\frac{U'}{1+U^{2}} The Attempt at a Solution I found U'= \frac{12cosx+15}{(4+5cosx)^{2}} 1+U^{2}=1+\frac{9sin^{2}x}{(4+5cosx)^{2}} I think my components are correct but my...

Homework Statement Find the derivative of: sqrt(x^2-4)-2tan^-1{.5*sqrt(x^2-4)} Homework Equations U'/1+U^2 U'=x/2sqrt(x^2-4) 1+U^2=x^2 The Attempt at a Solution I combined the above components but my answer is incorrect. I feel that I might have the wrong answer for...

Homework Statement Hi there I'm trying to solve this question: Homework Equations The Attempt at a Solution I figured i should just multiply them together and show that you get the identity matrix, but I'm having trouble cancelling out some of the terms. I'm not sure if I...

Using the laplace transform, find the solution to the differential equation: y'' + y' + y = 0 , y(0)=0, y'(0)=1 Using the laplace transform and its properties I end up with: f(s) = 1/(s2+s+1) How can I find the inverse of this/ does anyone know the inverse of it? Setting y=eax I got a...

Homework Statement Hi. I found in the answears that the inverse of function f(x)=3-\sqrt{x-2} is f^{-1}(x)=(3-x)^{2}+2 only if we restrict it to {x:x\leq3}. I understand that the restriction is needed because the found inverse is a parabola (and thus not one-to-one function). My general...

Derivative of inverse function at x=0 [SOLVED] Homework Statement Let f(x) = x + Ln(x+1), x > -1 Find \frac{d}{dx} f^{-1} |_{x=0}; Note that f(0) = 0 Homework Equations (f^{-1})'(x) = \frac{1}{f^{'}(f^{-1}(x))} (or) \frac{dx}{dy} = 1/\frac{dy}{dx} The Attempt at a Solution...

Find the inverse Laplace transform of $\displaystyle \frac{2s+7-e^{-2s}}{(s+1)^2}$.

Suppose that $\displaystyle f(x)=\frac{ax+b}{cx+d}$. What conditions on $\displaystyle a,\ b,\ c,\ d$ are necessary and sufficient in order that $\displaystyle f(x)$ coincide with its inverse function. My attempt: $\displaystyle...

Homework Statement The Attempt at a Solution So I observed: T(B) = λB T-1(B) = λ'B Also, T-1(T(B)) = λ'λB = B This implies, λ'λ = 1 And so, there should be a relation λ = \frac{1}{λ'}. Is that right?

Homework Statement How do I find the inverse of these functions step by step? y= e^-x^3 y= sin(1/x) I know the solutions but I don't know how to work with these two functions. Does anyone know the steps to finding the inverse of these?

Hi all I have been working on some unique solutions to advection-diffusion type problems. One inverse Laplace transform that I seem to continue to encounter is the following: Inverse Laplace[F(s)] where F(s)=[(1/(((s-α)^2)+β)*exp(-x*sqrt(s/D))] In their classic 1959 text, Carslaw...

Why are derivatives of inverse functions important? My students are giving me questions like: When would using the theorem be useful? Can't you just find the inverse function and take its derivative? I'm sure many of you know the type of question: "Who cares?" My answers are that the theorem...

Homework Statement What is the inverse of matrix A? A= (1) (2) (1) (2) (-1) (2) (1) (2) (1) Homework Equations The Attempt at a Solution The determinant works out to be 0 The inverse is 1/determinant x adj(A) Therefore the inverse is 1/0 x adj(A) So is the...

Let T: V-->W be a surjective linear transformation and let X be a subspace of W. Assume tat Ker(T) and X are finite dimensonal. Prove that T^-1 (X) = {v | T(v) is in X} is a subspace of V. Ok I absolutely suck at showing things are a subspace of something...I don't even know where to...

Hello everyone, I've been trying to figure out how to obtain the multiplicative inverse of an integer in Zn in Mathematica but I haven't found a way. Is there a way to do this anyone can help me with?

Homework Statement This is an integration of an inverse trig function. I don't see how they go from 1/2 to 1/4. I understand how they get the 1/2, du = 2dx, divide both sides by 2, but where does the 1/4 come from?

How does one use IFT to find the inverse of a function? I thought it was something like \int \frac{dx}{df(x)}dx. But that doesn't work with f(x)=x^2:\int \frac{dx}{2x}=\frac{1}{2} \log{x} \neq f^{-1}(x).

Assume that there is a original wall which does not have any defect inside. After an excitation of heat at one side of the wall, Ti is 37°C and To is 30° C. Thermal capacitance is assume 1. So the heat transfer is 7. For another condition, i want to find out if the wall has air pocket inside...

Homework Statement tan-1[2x/(1-x2)]+cot-1[(1-x2)/2x]=2π/3 2. The attempt at a solution tan-1[2x/(1-x2)]+cot-1[(1-x2)/2x]=2π/3 tan-1[2x/(1-x2)]=2π/6 take tan on both sides 2x/(1-x2) =sqrt(3) quadratic equation so it should have 2 solutions(sqrt(3) and sqrt(1/3)).But this question...

Hey everyone. Sorry to post another topic, but i thought this would be easy to find on google, to help me with , but i can't find it. All I am really lookin for is someone to let me know if my answers are right, and if not, then how i can fix them :) Anyways, i was asked to graph...

f(x) \ = \ \dfrac{1 - \sqrt{x}}{1 + \sqrt{x}}Edit: \ \ I \ sent \ a \ PM \ to \ a \ mentor.

IVP Laplace Transform Problem -- Tricky Inverse Laplace Transform Homework Statement Solve x"+x'+x=1, given x(0)=x'(0)=0 Homework Equations The Attempt at a Solution Plugged in transforms: s2*Y(s)-s*y(0)-y'(0)+s*Y(s)-y(0)+Y(s)=1/s Plugged in initial value points, simplified...

Homework Statement Hello all, Having difficulty with this one question that involves complex roots. Here it is: F(s)=\frac{s+3}{s^3+3s^2+6s+4} I tried two different ways to tackle it. First method I divided it right away: F(s)=\frac{s+3}{s^3+3s^2+6s+4}\rightarrow{s^2+6-\frac{14}{s+3}} Is there...

Why are Newtons law of universal gravitation, F=G\frac{m_{1}m_{2}}{r^{2}} and Coulombs law, F = K_{e}\frac{q_{1}q_{2}}{r^{2}} inverse square laws? I understand why they are inverse because the force decreases with distance but why is the distance, r, squared? Thanks AL

I feel kind of lame, but here's my situation: We start with the operator g_{\mu \nu} \Box - \partial_{\mu}\partial_{\nu} and convert to momentum space to get -g_{\mu \nu} k^{2} - k_{\mu}k_{\nu}. Apparently it's easy to see that this has no inverse? I'm told that if it *did* it would be...

I am reading the proof of the Inverse Function Theorem in baby Rudin and I have a question about it. How does associating a function phi(x) (equation 48) with each point y tell us anything about if f(x) is one-to-one? I'll show the proof below. Also, if f'(a) = A, and f(x)=x2, what would A-1 be?

Homework Statement find the derivative of the function f(x)=arcsec(4x) Homework Equations I think this is a Relevant equations. d/dx[arcsecu]=u'/(|u|(√u2-1) The Attempt at a Solution f'(x)=4/(|4|(√42-1) =1/√15 I keep getting wrong in my online homework why? :confused:

Homework Statement arctan(8x-8)=-1 Homework Equations I'm sure what this part wants. The Attempt at a Solution tan[arctan(8x-8)]=tan(-1) 8x-8=tan(-1) x=(1/8)(tan(-1)+8) x=(1/8)(-(tan(1)+8) x= I am stuck on the last part since the homework says Simplify the above equation...

Homework Statement Find the derivative of the compositional inverse of f(x) = sin(1/x) restricted to (1,∞). You may use without proof that sin(x) is differentiable with derivative cos(x). Homework Equations (f^{-1})'(y_0) = \frac{1}{f'(f^{-1}(y_0))} The Attempt at a Solution...

Homework Statement Find (f^{-1})'(a) of: f(x)=\sqrt{x^{3}+x^{2}+x+22} ; a=5. Homework Equations (f^{-1})'(a)=\frac{1}{f'((f^{-1})(a))} The Attempt at a Solution Well, I know to find an inverse: I need to set the equation equal to y, solve for x, then swap x and y. But I don't...

Homework Statement The function f:x→ 4-x2 for the domain x≤0. Find the inverse of is denoted by f-1 and state the domain and range of f-1. Homework Equations Set equation to 0 and solve for x to find inverse. The D and R is going to be switched for the inverse..? The Attempt at a...

I am having problem with the inverse transformation of a Fourier transformed function which should give the function itself. Let f=f(x) and let f be Fourier transformable (whatever that implies) Let \tilde{f}(k)=∫^{\infty}_{-\infty}dx e^{-ikx}f(x) (1) then we should have...

Homework Statement For α > 0, determine u(x) by the inverse Fourier transform u(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ \frac{e^{ikx}}{ik+\alpha}\ dk Homework Equations The Attempt at a Solution This seemed like a relatively simple residue problem. You just note that...

Homework Statement ∫ (x+2)dx/√(4x-x2) Homework Equations why was the -2 in -2(x-2) was ignored? The Attempt at a Solution so first i let u= 4x-x2 then, du=4-2x = -2(x-2) so to get (x+2) i equate it to (x-2)+4 so ... ∫ (x+2)dx/√(4x-x2) = ∫(x-2)+4dx/√(4x-x2) = ∫...

This is insane, I am trying to revise the inverse of matrices and this one element is being really stubborn, please help. Here is the matrix 3 -1 7 2 0 1 5 -2 6 I have transposed it 3 2 5 -1 0 -2 7 1 6 Now as for replacing the element of...

Homework Statement The function y=x^{2}+4x-6 has two inverses. What are they and which domains lead to these inverses? Homework Equations The Attempt at a Solution y=x^{2}+4x-6 x=y^{2}+4y-6 y(y+4)=x+6 Not really sure where to go from here.