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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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
147 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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23 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
19 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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1answer
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
49 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
31 views

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
91 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
49 views

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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0answers
35 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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0answers
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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2answers
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
43 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
16 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
105 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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4answers
27 views

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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0answers
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
24 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
30 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
116 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
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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
62 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
118 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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37 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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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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2answers
152 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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2answers
46 views

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
55 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
151 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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1answer
46 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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2answers
43 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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2answers
46 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 ...
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1answer
81 views

Given its pseudo-inverse, is there a fast way to measure the degree of full-rankness of a nonsquare matrix?

update: I realized the core of question is about ill-conditioning of the matrix (aka Multicollinearity). In a computer, with floating point arithmetic, it is impossible to talk about full-rankness. We ...
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4answers
71 views

$f:\mathbb R \to (0,\infty)$ defined by $f(x)=e^x$. Describe its inverse.

How do I go about describing it? Well first is the inverse $e^{-x}$ or $\ln(x)$? Additionally, since I have no clue how to solve these problems as I am probably overthinking them... $f:\mathbb R\to ...
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1answer
43 views

What is the same as the inverse of a logarithm?

I am trying to simplify $f(n) = \frac{n}{\log(n)}$ into a more easily understandable function. Up until now, I got as far as $n\cdot(\left(\log(n)\right)^{-1})$. Is there any way I can further ...
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47 views

Primes and Inverses of an integer

I have the following question which I do not understand. Here it is: Consider the primes $5$, $7$ and $11$ as n. For each integer from $1$ through $n - 1$, calculate its inverse. I do not ...
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303 views

Proof of Vandermonde Matrix Inverse Formula

I'm working through Exercise 40 from section 1.2.3 of Knuth's The Art of Computer Programming volume 1, but am finding myself unable to produce a rigorous proof, and the one here is suspect and not ...
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1answer
42 views

If A = BC and B is invertible, then how does reducing “B to I” also reduce “A to C”?

If $A = B*C$, where $B$ is an inverse, use row-ops to reduces "$B$ to $I$" also shows that it will reduce "$A$ .. $C$". Big-Hint: Represent the row operations by a sequence of elementary matrices.
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31 views

Inverse trig functions [duplicate]

What would be the inverse function of $f(x) =x \cot \frac{\pi }{x}$ on the interval $ x\geq 3$? I don't know what to do about that pesky x in the front of the problem. Otherwise, the problem would be ...
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2answers
275 views

Show that A is invertible and that it is Lower Triangular.

Does anybody have a solution to the given word problem below? Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is invertible and and that A-inverse is lower ...
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126 views

Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
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55 views

How to calculate the inverse of a known optical distortion function?

Assume I have the following lens distortion function: $$ x' = x(1 + k_1r^2 + k_2r^4) \\ y' = x(1 + k_1r^2 + k_2r^4) $$ where $r^2=x^2 + y^2$. Given the coefficients $k_1$ and $k_2$ I would need to ...
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2answers
221 views

convexity of inverse function

I have a question on the reverse of a convex function. Let $f(x)$ be a convex function. Is the reverse function, say $g(x)=f(x)^{-1}$, is necessarily a concave function ? Considering that such ...
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1answer
40 views

Factoring a matrix out of linear matrix equation

I'm having a bit of trouble following a solution in a textbook, one step in particular. I have the equation $(Z + tV)^{-1}$ where $Z$, $V$ are matrices and $t$ is a scalar. $Z$ is positive definite, ...