Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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54 views

How to Find the Domain of The Inverse of (4(e^x)-5)/(25(e^x)+12)

These are the steps I have taken so far: In order to find the inverse of the function, I did the following steps: ...
7
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1answer
544 views

Is there a way to calculate the definite integral of inverse of a 5th degree polynomial?

I want to calculate the definite integral of inverse of a 5th degree polynomial. The problem is that the inverse of the polynomial cannot be calculated (by using Matlab). However without calculating ...
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1answer
55 views

Inverse Laplace transform of $\frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}}$ (MGF of noncentral chi-squared distribution)

I am trying to use the countour integral to calculate the inverse Laplace transform of the function $$F(s) = \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} \hspace{1cm}\mathrm{for} \hspace{1cm} ...
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1answer
104 views

Inverse Laplace transform of one complicated function

I want to ask the inverse Laplace transform of the following function: $$F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
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1answer
18 views

Calculate the inverse of a multi-variable function

How would I calculate the inverse of $F(x,y)=\left( \textrm{arctan} \left(\frac{ay}{x}\right),\frac{x^2+a^2y^2}{2a}\right)$ $a$ is a constant.
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2answers
41 views

If $X$ is a $n$ by $k$ real matrix, and we know that $X^{T}X$ is invertible, is $X$ invertible as well?

If $X$ is a $n$ by $k$ real matrix, and we know that $X^{T}X$ is invertible, we know that $rank(X^{T}X) = n$ by $n$. Can we say that $X$ is invertible as well and hence has rank $n$? thanks!
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1answer
26 views

Effective way to calculate the inverse (A+kB)^-1 with k changing and A, B fixed

I have a Simulink modell where I need to calculate $(A+c_k B)^{-1}$ in every time step with $c_k$ changing each iteration. Does someone know any more effective way to do it, instead of calculating a ...
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1answer
90 views

Derivative of Inverse Function from AP

An AP question asks: The function f(x) = x^5 + 3x - 2 passes through the point (1,2). Let f^-1 denote the inverse of f. Then (f^-1)(2) equals? The inverse of this function should be y^5 + 3y - 2. ...
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2answers
85 views

Inverse Laplace Transfrom of $s^{-1}e^{-a\sqrt{s} + b/s}$

I am trying to find the inverse Laplace transform for following function and it seems almost impossible for me to find the answer. Can anyone help me please with final answer and also the way to get ...
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4answers
37 views

Find the interval for which $2\arctan x + \arcsin \frac{2x}{1+x^{2}}$ is an independent of x?

I used formula and simplified the expression to $\Rightarrow 4\tan^{-1} x$
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1answer
98 views

How to find the inverse of an upper triangular matrix

I want to find the inverse of an upper triangular matrix in an efficient way. I googled a lot, but all the articles discussed about a lower triangular matrix. Is it possible to edit the matlab code ...
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1answer
32 views

Calculation of inverse function

My question is physics related but the problem itself is just math. I have an expression for a refractive index depending on the wave length $\lambda$: $$ ...
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1answer
49 views

Inverse of $x\log(x)$ for $x>1$

Let $y(x)=x\log(x)$ for $x>1$. Is it possible to write down the inverse function explicitly? Has this inverse function been named? (For example, the Bessel functions are "named" but cannot be ...
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3answers
87 views

Inverse of a matrix!

Let $I_n$ be the $n$ by $n$ identity matrix and $b$ and $c$ be two vectors in ${\mathbb R}^n$ such that $b^Tc\ne 0$. Then one can easily see that the $n+1$ by $n+1$ matrix $X$ defined as following $$ ...
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2answers
42 views

Derivative of the inverse of a matrix

I've seen in a scientific paper this equation: $\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$ where K is a $n\times n$ matrix which depends on $p$. In my calculations I ...
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2answers
62 views

If $y=ax^be^{-cx}$ then $x=g(y)$, find $g$

I have this function $$y=0.384394\cdot x^{0.341429}\cdot e^{-0.004749 x}$$ Based on this function I would like to know how I can I get $x=g(y)$.
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1answer
31 views

Inverting functions containing ceiling

I have a recurrence relation for a countably infinite sequence that contains the integers divisible by 5 but not by 7. The relation I came up with is: $5((n-1) + \lceil \frac{n}{6} \rceil)$ The ...
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0answers
38 views

Order of the sum of elements of the inverse of a matrix

For each $T$, let $A_T$ be a $T\times T$ matrix of real numbers. let $e_T$ be the $T\times 1$ vector of ones. Assume that the sum of all entries of the matrix $A_T$ divided by $T^2$ is limited as $T$ ...
2
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0answers
30 views

Differential Equation for Inverse of a Function given the ODE for the function

Assume we have and ODE for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ in the form of \begin{equation} \mathbf{J}f \times F(x)=D(\lambda_i)\times f(x) \end{equation} where $\mathbf{J}$ denotes ...
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1answer
69 views

How to estimate the size of the neighborhoods in the Inverse Function Theorem

Given a function $f:U \subset V\to W$ such that $\textbf{D}f(x_0)\neq 0$ for some $x_0$. How to estimate the neighborhood for which it's invertible? Assuming the second derivative exists and is ...
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1answer
155 views

Which graphs do have invertible adjacency matrices?

I would like to know if there is any class of graphs for which the adjacency matrices are invertible. At this moment I am aware of only the class of graphs $n K_2$ which is the disjoint union of $n$ ...
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0answers
19 views

Inverse function of a function of Laplacian Matrix

I have a function $f:\mathbb R^n \rightarrow \mathbb R^n$ defined by $f(\mathbf{X})=L(\mathbf{X})\mathbf{X}$ where $L(\mathbf{X})$ is a (nonlinear) Laplacian matrix of an undirected but of-course ...
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2answers
79 views

Find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$

From the formula $\sin\left(\frac{π}{2}−x\right)=\cos x$, find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$. I have figured out that the domain of $x$ is ...
2
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1answer
68 views

Inverse of a function's integral

The function $g$ is strictly positive. Let the function $f$ be defined as $$f(x) = \int_0^x g(u) du$$ Is there a way to express $f^{-1}(x)$ in terms of $g$?
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1answer
99 views

Help with Inverse Function and Composition of Functions?

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to Relations and Functions. To be specific, I'm stuck on ...
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2answers
76 views

Find the inverse of the $n\times n$ matrix whose entries are given by $a_{ij} = \max (i,j)$

The actual question on the past papers is "Let $n\ge 1$ be an integer and consider the $n\times n$ matrix $A$ whose entries are given by $a_{ij} = \max(i,j)$ for all $1\le i,j\le n$. Show that $A$ is ...
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2answers
241 views

Show that f(x)=e^x from set of reals to set of reals is not invertible…

Yes, this is my question... How can you prove this? That $f(x)=e^x$ from the set of reals to the set of reals is not invertible, but if the codomain is restricted to the set of positive real numbers, ...
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5answers
112 views

Prove that if $f$ is increasing then so is $f^{-1}$

Prove that if $f$ is increasing then so is $f^{-1}$, when $f$ is a one-to-one function. I'm having trouble figuring out how to get started with this question. I'm assuming it has something to do with ...
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1answer
27 views

Find inverse $z$-transform of $\frac{5}{z^{2}-z-6}$

How can I find inverse z transform of $$X(z)=\frac{5}{z^{2}-z-6}$$ What I did: first i factored denominator and i got (z+2)(z-3), now we get A(-2^{n}) + b(3^{n}). To get A and B i used Partial ...
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1answer
40 views

Cofactor expansion to check if matrices is invertible.

I gave question regarding a co-factor expansion question. I understand that an easy way to check if a matrices is invertible is to do co-factor expansion and if $A \ne 0$ then its invertible. I'm ...
0
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1answer
46 views

Find inverse $z$-transform of $\dfrac{(z-1)^2}{z^3}$

How can I find inverse z transform of $$X(z)=\frac{(z-1)^{2}}{z^{3}}$$ What I did: I am thinking to do Partial Fraction Decomposition or long division. Is there another method ?
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1answer
22 views

Write $(h \circ f)^{-1}$ when $h(x)= x \ln(3 + x)$ and $f(x) = x^2 − x$

I have arrived up to a point but haven't solved it yet: $$(h \circ f)^{-1} = y= (x ^2 − x )· \ln(3 + x^ 2 − x)$$ $$ x = (y^ 2 − y )\cdot \ln(3 + y^ 2 − y)$$ Any suggestions? Thank you
0
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1answer
36 views

Inverse Matrices. Unsure how to solve it.

Find the inverse of matrix $A$. Can some please show me how to do this question.I've been attempting this question for quite awhile now, although don't know how to proceed. $$A=\begin{bmatrix} ...
0
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1answer
40 views

Inverse of a Function with Complex Variables

I'm taking Abstract Algebra, and we're currently covering isometries of the Real and Complex plane. I'm going through and studying for our first midterm, and I'm working on a problem that asks to show ...
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0answers
64 views

Inverse function theorem - good proof

I am looking for a reference which give a full demonstration of the inverse function theorem (let say in Banach spaces) where we can have estimates of the bounds of the neighbourhoods that we build to ...
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0answers
21 views

Proof of the surjectivity of $f$ given it has a right inverse gives $f$ not-a-function?

I am working on a proof of the surjectivity of $f:X\rightarrow Y$ given that it has a right iverse $g_R:Y\rightarrow X$ such that $f(g_R(y))=y\,\,\, \forall y\in Y$. My question stems from this. We ...
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2answers
41 views

Given $f(x)$ with inverse $g$, find $g'(2)$

Problem Given the function $$f(x) = \ln^3(x) - 2\ln^2(x) + \ln(x)$$ defined for $$x\in[e, e^3]$$ show that the function has an inverse $g$ on the given interval, and find $g'(2)$ Progress I have ...
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2answers
117 views

Find a formula relating $\arcsin (x)$ and $\arccos (x)$ [duplicate]

From the formula $\sin(\frac{\pi}{2}-x)=\cos x$, find a formula relating $\arcsin (x)$ and $\arccos (x)$. I have no idea where to start.
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1answer
117 views

Inverse of this $6 \times 6$ matrix?

Let $A= \begin{pmatrix} 1 & 1 & 1&0&0&1 \\ 1 & -1&1&-1&0&1\\ 1&-1&-1&0&-1&1\\ 1&-1&0&1&1&0\\ ...
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0answers
20 views

Finding points on an inverse function

How do you verify if a point is on an inverse function of a graph if a point on the original function is given?
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1answer
43 views

find $\det(\det(A)B[\det(B)A^{-1}])$

$A$ and $B$ are matrices, let $A= \begin{bmatrix}2 & 0 & 3\\-1 & -2 & 1\\2 & 0 & 1\end{bmatrix}$ and $B=\begin{bmatrix}1 & -1 & 0\\1 & 0 & 1\\-1 & 1 & ...
0
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1answer
39 views

How to Invert a Monotonic Function that Contains an Integral?

Consider $y=b(c)$ where the function $b$ is continuous, strictly increasing in its argument. So that there exists an inverse. However, $b(c)$ is quite complicated and has an integral: $$ b(c) = ...
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2answers
51 views

Determinant of $2\times 2$ matrix over $\mathbb{Z}/2\mathbb{Z}$

I have to prove that for any square matrix that is in $M_2(\mathbb{Z}/2\mathbb{Z})$ it is invertible if and only if its determinant is not $0$. Here are my thoughts: Since all entries are modulo $2$, ...
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1answer
52 views

Find the inverse of given functions

Suppose $ g(x)=\frac{x^3}{x^2+1} $ is the inverse of f(x). Find the inverse functions of the function f(x+1), and 4f(x). I tried replacing all x’s with y and all y’s with x and had this: $ ...
4
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2answers
96 views

Prove that if $AA^T=A$ then $A^3=A$

The approach I'd like to use to prove this particular property necessitates that $A$ be invertible, but I don't wish to assume this (though it would certainly make the task simpler). Is there some ...
2
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1answer
53 views

What are the relations between the columns, rows and rank of a matrix A, in order to show that A has an inverse?

Consider an m×n matrix A (with m rows, n columns) of rank r. What relations between m, n and r are necessary and sufficient for the existence of: 1) a right inverse B such that AB=I 2) a left ...
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2answers
140 views

How is the determinant related to the inverse of matrix?

Whenever I needed to find the inverse of a matrix, I was told to check if its determinant is not zero. However, once I directly applied the Gauss-Jordan's method for finding the inverse of matrix ...
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4answers
1k views

Does the inverse of the matrix always rely on the determinant of a matrix?

I always thought that if the determinant of a matrix $A$ is $0$ then it has no inverse, $(A^{-1})$, until I saw an exercise in Contemporary Abstract Algebra by Gallian. This asks me to prove that the ...
0
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0answers
60 views

Finding matrix inverse using Gauss method

I have been trying to find the inverse of a matrix using Gauss method and I want to know suppose what happens if I don't get the "1" in reduced matrix on the left? Does it mean that the inverse ...
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3answers
27 views

inverse of $\arcsin (\frac{x}{x-1})$

determine the inverse of a) $y=\arcsin \left(\dfrac{x}{x-1}\right)$ b) $y= \dfrac{1-2e^{-x}}{4}$ I learned you the steps for finding the inverse are 1) get it in a form of $x= \dots$ 2) change $x$ ...