Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...
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1answer
62 views

What does it mean when a matrix is to the (-1/2) power?

I'm reading a machine learning paper that uses a form of matrix normalization called symmetric divisive; given a matrix A and a diagonal matrix D derived from A, we define $$N=D^{-1/2}AD^{-1/2}$$ I am ...
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3answers
48 views

Yet another inverse function to calculate

Is it possible to evaluate the inverse of this function, in order to obtain for each $y\in\mathbb R^+$ an explicit value of $f^{-1}(y)$? Thanks in advance! ...
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1answer
44 views

Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
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1answer
49 views

Where exactly is the following process incorrect to yield an impossible answer

I was playing with my calculator and found some strange phenomena. $\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$ Verify here Now when we apply some inverses, then $\tan(\tan(\tan(\pi/4))) = ...
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Linear Algebra Review Questions

So I have a test on Monday and my professor posted a couple of non-graded review questions that she said we should look over. Anyhow, I have a couple of questions that I'd like answered if that's ...
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1answer
185 views

Determining the values of k for which the Matrix A has an inverse

I've been given this question in class, with the 3x3 matrix: 2 1 0 1 2 1 0 -3 k My job here is to find the values of k for which this matrix has an ...
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36 views

From concrete mathematics problem 4.35

From concrete mathematics problem 4.35. Let $I(m,n)$ be function that satisfies the relation $$I(m,n)m+I(n,m)n=\gcd(m,n)$$ when $m,n\in \Bbb Z^+$ with $m\neq n$. Thus, $I(m,n)=m′$ and $I(n,m)=n′$ ...
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Tangent line of the inverse function of $y = e^x + x$

I've been sitting on this problem for a while, hopefully you guys could give me a lead on what the hell is going on :) Let $f(x) = e^x + x$ Find the tangent line to $f^{-1}(y)$ (the inverse ...
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50 views

How to find inverse of…

How do you find the inverse of the equation in the form $y= b^{x-h} +K$ for example: $y=2^{x-4} +6$ I already know that the inverse of $b^x$ is $\log_bx$ but how do you find with the $H$ and $K$ ...
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39 views

Find the function $h(x) = g(2g^{-1}(x))$

Show that the function $g(x) = x^4 + x^3 + 1$ is one-to-one on [0, 2]. In addition, for the function $h(x) = g(2g^{-1}(x))$, find h′(3). For the first part, I manage to prove that g(x) is increasing ...
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1answer
41 views

an inverse of m with respect to n

From concrete mathematics problem 4.35. Let $I(m,n)$ be function that satisfies the relation $$ I(m,n)m + I(n,m)n = \gcd(m,n),$$ when $m,n \in \mathbb{Z}^+$ with $m ≠ n$. Thus, $I(m,n) = m'$ and ...
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2answers
68 views

How do I find $(f^{-1})'(a)$? [closed]

if $$f(x) = 3x^3 + 3x^2 + 6x + 9 $$ $$a = 9$$ and also $$f(x) = 2x^3 + 3\sin x + 3\cos x$$ $$a = 3$$ I know I have to find the inverse but I think I’m getting overly complicated answers and my ...
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1answer
117 views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
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0answers
21 views

Optimization that involves inverse operation.

$\newcommand{\diag}{\operatorname{diag}}$ I have the following optimization problem: \begin{align} \mathop{\arg\min}_\beta & \frac{1}{2} a' [ M + \diag( \beta ) \otimes I_d ]^{-1} a + ...
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1answer
18 views

Linear Growth Model

I have a problem where I have been given that $r(t)=at+b, 0 \leq t \leq \frac{100-b}{a}$. I have then been asked to find $t(r)$. Is this simply finding the inverse of $r(t)$?
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45 views

The rank of general inverse of $A$ times $A$?

Supposing $X$ is the general inverse of $A$, that $AXA = A$. Then $XA$ is idempotent, that is $(XA)(XA) = XA$. Why is the rank of $XA$ equal to the rank of $A$ ? Thanks.
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1answer
46 views

Inversion of Matrix

What is the inverse of the following (n x n)-matrix? $$ \begin{bmatrix} 2 &-1 &0 &0 &... &0 &0 \\ -1 &2 &-1 &0 &... &0 &0 \\ 0 &-1 &2 &-1 ...
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2answers
30 views

How can I find the inverse of this function? [closed]

Can anyone help me find the inverse of this function? $$y=\frac{x}{2}-\frac{x^2}{16}$$
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0answers
11 views

Find the inverse function of a function relating to limited exponential sum

The function is given out as: $$y = 4x + {x^m} + {x^{ - m}},where{\text{ 0 < m}} \leqslant {\text{1, 0 < }}y < 6;$$ Closed form will be highly appreciate,but approximate results is also ...
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2answers
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Could someone please help me find the derivative of the inverse of $f$ at $0$?

The problem is: for $\displaystyle f(x)= \int_0^{\ln x} \frac{1}{\sqrt{4+\mathrm{e}^{t}}} \, \mathrm{d}t$, $x > 0$, find $(f^{-1})'(0)$. I know that I should use the fundamental theorem of ...
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1answer
87 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
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1answer
36 views

Find the inverse function about a exponential related function

Here is the function:$$y = 4x + {x^m},where{\text{ 0 < m}} \leqslant {\text{1;}}$$ Approximately results is acceptable.
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1answer
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Could someone please help me solve this calculus problem?

For f(x)= integral 1/sqrt(4+e^t) dt from 0 to lnx, with x>0, find (f^-1)'(0) (that is, the derivative of the inverse of f at 0)
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34 views

What is the space complexity of inverting a real valued sparse banded diagonal symmetric matrix?

Of course, when I say ``inverse'' what I really mean is solving a system of equations $Ax=b$ where $A$ is sparse, banded diagonal, symmetric, real valued $N \times N$ with a bandwidth of $k$. I know ...
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14 views

Easy Lagrange Interpolation (modulo)/multiplicative inverse

I am writing an examn in a few days and I just realized that Lagrange's polynomial interpolation is very fine and all, but pretty unfeasible if you are under examination stress and have no calculator ...
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53 views

Find the range of arcsin$((1-x^2)^{0.5})$

Title says it all, how do you get the answer to this? So far I only reach $0<1-x^2<pi/2$ but I get an invalid answer from here. the correct answer is $0<x<pi/2$. Any help is appreciated, ...
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1answer
41 views

When to use row operation or cofactor method to find matrix inverse?

I find two different answers by using these two methods in a same matrix. How can I decide to use row operation or cofactor method?
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1answer
15 views

Arc Tangents and Equation

For one of the problems in my book, it requires you to put the arc tangent into the 2piK equation and solve for the arc tangents and lie in [0,2pi]. For: arctan(117)+piK the answers are 1.5622 and ...
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1answer
81 views

Inverse Trig Functions with Double Angle Formulas

I am studying for a quiz tomorrow and one of the sections I am studying involves rewriting quantities as algebraic expressions of $x$. One of the problems I am having trouble with is: $$\sin ...
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Prove Inverse Function [closed]

Consider the function $f:\Bbb R\times\Bbb R→\Bbb R\times\Bbb R$ defined by $$f(x,y)=(x+y,x-y)$$ This function is invertible. Show that the inverse function is given by $$f^{-1} (a,b)=\left( ...
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1answer
56 views

Inverse of $a f(x)$ and inverse of $a f(x) + b$

Is there a general rule for the inverse of the function $ g(x) = a f(x) $, where $a$ is a constant, assuming $f^{-1}(x)$ is defined? Follow up: $g(x) = a f(x) + b$. Is the following correct, given ...
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19 views

Can we describe an original and inverse equation with one function?

Let us say we have two real values, 1 and x. I want to determine the absolute value of the difference or their sum between 1 and x without specifying whether I am dealing with 1 - x or 1 + x For ...
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2answers
22 views

Inverting this equation as a function of X

I'm trying to inverse this equation as a function of x $$z = x + \frac{x^2}{2}$$ but couldn't wrap up my head around it. If anyone can provide a step by step solution to this it will be really ...
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0answers
28 views

Inverse of a non-singular linear transformation

The question is about showing that if A is a non-singular linear transformation of an n-dimensional linear space to itself, then there must be some polynomial $c_0 + c_1 z + ... + c_k z^k$ such that ...
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1answer
110 views

How find this matrix the inverse $A^{-1}$

Let $a,b>0$,and the matrix $A_{n\times n}$ and such $$A=\begin{bmatrix} a&b&0&\cdots&0&0\\ b&a&b&\cdots&0&0\\ 0&b&a&\cdots&0&0\\ ...
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5answers
88 views

Derivative of $ h(t)= \sin (\cos^{-1}t$)?

Can someone please explain the steps/rules I need to preform to find the derivative of $h(t)= \sin (\cos^{-1}t)$? I tried to used the product rule, and realized it was obviously a failure. Using ...
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2answers
46 views

Prove $ X = \left(\begin{array} &A & C \\ 0 & B \end{array} \right) $ is invertible iff A and B both are.

Suppose $A$ is a $n \times n$ matrix, $B$ is a $m \times m$ matrix, and $C$ is a $n \times m$ matrix. Prove $ X = \left(\begin{array} &A & C \\ 0 & B \end{array} \right) $ is invertible ...
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1answer
54 views

Cholesky decomposition of the inverse of a matrix

I have the cholesky decomposition of a matrix M. However, I need the cholesky decomposition of the inverse of the matrix, invM. Is there a fast way of getting this, without first inverting the matrix ...
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3answers
106 views

Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...
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36 views

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = ...
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3answers
39 views

Please solve this in details inverse problem i am using complement angle formula

For any $x \in [-1,0) \cup (0,1]$, how can I prove that: $$\sin^{-1}(2x\sqrt{1-x^2})=2\cos^{-1}x$$ Also, can someone explain to me how to understand the graphs of $sin$ and $cos$ functions?
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123 views

Proof of the Inverse of a Scalar times a Matrix

How would I prove that given a square matrix $A$ and non-zero scalar $c$ that $$(cA)^{-1}=c^{-1}A^{-1}$$
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How do I calculate the inverse function of this function?

I have this function: $$ f(x)=\frac{1+\ln(x)}{1-\ln(x)} $$ And i should calculate $f^{-1}(x)$ I am not really sure how to proceed but I think that the first step would be to have x alone, how do I ...
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1answer
53 views

Laplace Transform Damp Harmonic Motion

http://gyazo.com/19d18f085731c6dbc304fefdaece4f3c.png I'm currently on (a) where so far I have gotten; $ y'' + 2y' + 5y = f(t) $ Using Laplace transforms, I get; $ Y(s)$ = $ F(s) + s+2\over(s^2 ...
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2answers
123 views

Power series for a matrix inverse

Is there a power series expansion for a matrix inverse of the form $$\left(\frac{1}{m}I+A\right)^{-1} \mbox{ where $m$ is a scalar?}$$ $A$ is not invertible but the expression above is defined. I ...
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1answer
20 views

Inverse integral manipulation

Why is the following true: If $\rho = B^{-1}(t)$ where $$ B(t) = \int_0^t \frac{1}{g(\gamma(s))}ds $$ then $$ t = \int_0^\rho \frac{1}{g(\gamma(s))}ds $$ I know it must be something fundamental, ...
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43 views

Moore-Penrose inverse multiplication

I really need help proving that when $AB=0$ then $B^+A^+=0$ and also the other way: when $B^+A^+=0$ then $AB=0$. Where $B^+$ and $A^+$ are Moore-Penrose Pseudo-inverse of B and A.
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42 views

Show that $ (f^{-1})^{-1}=f $

$$ f:X\to Y $$ $f$ is invertible, show that $(f^{-1})^{-1}=f$ Here it is not given that how the function is defined, so I think that making equations and solving them will not help me. So I have ...
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45 views

Find the inverse of a function.

$$ g:[-1,1] \to \mathbb R\\ g(x)={\frac{x}{x+2}} $$ $f:[-1,1] \to$ range of f. Find the inverse of $ f.$ $\forall y\in \text{range of }g$ there exist some ${\frac{2y}{1-y}}\in [-1,1]$ such that ...