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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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
78 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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293 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
114 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
29 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
43 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 ...
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1answer
47 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
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1answer
37 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} \frac{...
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1answer
41 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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67 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
22 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
127 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
132 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\\ 1&1&0&-1&1&...
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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
45 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 & 1/...
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1answer
41 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) = \int_{...
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2answers
55 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
53 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: $ y^3-xy^2-x=...
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2answers
97 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 ...
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1answer
58 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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198 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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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 ...
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0answers
61 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 doesn'...
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3answers
28 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 $x$ ...
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3answers
40 views

Prove that the following function $f(x)=\log_{2}\left(x+{\sqrt{x^2+1}}\right)$ is invertible on the whole number line.

I think that it would help to show that this function is odd. But how can I show that the function $$\log_{2}\left(x+{\sqrt{x^2+1}}\right)$$ is invertible?
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486 views

Tricky trig question from GRE

Please evaluate \begin{align} 1-\sin^2\left(\arccos \frac{\pi}{12}\right) \end{align} What I've tried so far is to use Pythagorean identity and I got \begin{align} \cos^2\left(\arccos \frac{\...
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5answers
684 views

Will inverse functions, and functions always meet at the line $y=X$?

If I have a function, the inverse function, by definition will be a reflection of the original function in the line $y=X$, so if I wanted to find the point of intersection, instead of solving it with ...
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1answer
75 views

Inverse of Permutations

I'm having problems understanding inverting permutations, but I believe I understand how to do compositions. I may be second guessing myself, but some answers do not make sense to me. I've tried to ...
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0answers
17 views

Inverse z-transform of $z^4+1.827z^3+2.338z^2+1.827z+1$

I need to transform the following $H(z)$ back to time domain: $$ H(z)=(z-e^{j\frac{8}{15}\pi})(z-e^{-j\frac{8}{15}\pi})(z-e^{j\frac{12}{15}\pi})(z-e^{-j\frac{12}{15}\pi}) $$ I did the following steps ...
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0answers
18 views

Interpret the relationship of eigenmatrix of covariance matrix

I got a covariance matrix $C$ from a centered matrix $X$: $$C = X^TX/N$$where N is number of observations. Suppose $U$ is the eigenmatrix of $C$, obtained for example by SVD $$C=UDU^T$$ I understand ...
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1answer
155 views

Non-associative: set with a binary operation, but has inverses and identity

I've been thinking about an example of some set with a binary operation which would satisfy all axioms of groups except for associativity. I'm new to Group Theory, so I would appreciate your ...
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2answers
43 views

Finding the Inverse of $f(x) = \frac x{2^x}$

How does one find the inverse of the equation: $$ f(x) = \frac{x}{2^x} $$ for the least possible output value?
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4answers
58 views

Inverting $f(x)=\frac{a^x-1}{a^x +1}$

This is the problem: $$f(x)= \frac{a^x-1}{a^x+1}, \quad a > 0, \quad a \ne 1.$$ What I can get, but I don't think it is right: $$f^{-1}(x) = \frac{-x-1}{x \ln a-\ln a}.$$ So this is what I did:...
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1answer
83 views

Inverting an Equation Involving Floors?

How does one rewrite $$ l = \left\lfloor\frac{10\lfloor\frac{d}{7}\rfloor+\lfloor\frac{d}{10}\rfloor(\lfloor\frac{d}{10}\rfloor+1)}{2}\right\rfloor $$ in terms of $l$ such that the isolate variable,...
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0answers
26 views

How can I find an inverse of a Multivariate Function?

I have the following function h(X, Y) = [ h1(...); h2(...); h3(...) ] = Z Z is a 3*1 vector X is a 6*1 vector Y is a 3*1 vector I want to find this: h^-1(X, Z) = Y I know how to invert a simply ...
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1answer
45 views

Finding the inverse of a function using bisection method

It is said that we can find $f^{-1}(y)$ by solving the equation $y-f(x)=0$ using bisection method. But all sources that I can find use bisection to find roots, so I can't figure how and why. Could you ...
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1answer
30 views

Prove $f: S^1 \to S^1 \times S^1$ where $f(\theta) = (2\theta, 3\theta)$ is a homeomorphism

Prove $f: S^1 \to S^1 \times S^1$ where $f(\theta) = (2\theta, 3\theta)$ is a homeomorphism onto its image. We just need to prove continuity, one-to-one, and continuous inverse. For one-to-one, I ...
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215 views

First order differential equation involving inverses

My question is to find the solutions to the following $\frac{df(x)}{dx} = f^{-1} (x)$ where $f^{-1} (x)$ refers to the inverse of the function f. The domain really isn't important, though I am ...
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1answer
51 views

show that $(A'A + B'B)^{-1}$ is a g inverse of A'A

Notation: ' is for transpose, C() is for column space Here's what I've been given/figured out so far A is an n by m matrix rank(A) = rank(A'A) = r B is an s by m matrix rank(B) = rank(B'B) = m-r C(...
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1answer
113 views

Inverse Function of sum of exponential function

What is the inverse function for $$y=a^x+b^x+...+z^x$$ where $a, b, .. , z$ are positive constant and $x>0$ Thanks in advance!
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2answers
61 views

Pseudo-eigenvector times matrix inverse

Actually I don't know what should be a good title of my question. Here comes the simplified version of the question. Let's call it case 1. As we know, for a non-singular matrix $\textbf{A}$ with ...
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0answers
20 views

Transformation of a function; different methods, different values?

Given the function $ g(x) = 2f(2x + 2) - 3 $ for the point $ (1, 2) $. Now, we take 2 common and we get $ f(x) = 2f( 2(x + 1) ) - 3 $; the Horizontal stretch is $ \frac{1}{2} $. Solving for x: $2(...
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1answer
53 views

How to prove that $\,I-\gamma\, T\,\;\big(\,T\,$ is stochastic matrix, $\,0 \le \gamma \lt 1\,\big)\,$ is invertible

In case $\,T\,$ is right stochastic matrix (sum of each row is $1$) and $\,0 \le \gamma \lt 1$, Is there any way to prove that $\,I-\gamma \,T\,$ is invertible?
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1answer
37 views

How compute Inverse Fourier of this function $\mathcal{F}^{-1} \Big\{\frac{1}{iw - \alpha^2 k^2}\Big\}$

I need to compute this Inverse Fourier Transform to arrive at the given result $$ p(t) = \mathcal{F}^{-1} \Big\{\frac{1}{iw - k^2\alpha^2 }\Big\} = -\sqrt{2\pi}H(t)e^{-k^2\alpha^2t} $$ Where $H(t)$ ...
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4answers
66 views

Find inverse of $7x^{2}-112x+448$

Given the function $\; f(x) = 7x^{2}-112x+448, \;$ for $x\ge 8, \;$ find $\displaystyle \;$ $f^{-1}(x)$. To find inverse, I should just solve for x in terms of y: $$y = 7x^{2}-112x+448$$ I can ...
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0answers
53 views

another number group?

I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or ...
3
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2answers
61 views

Understanding the Bromwich Integral (Inverse Laplace Transform)

The formula for the Inverse Laplace Transform is (Bromwich Intergal): $$f_{(t)}=\frac{1}{2\pi i}\lim_{x\to\infty}\int_{\alpha-x i}^{\alpha+x i} \left(e^{st}\cdot F_{(s)}\right) \text{d}s$$ My ...
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3answers
35 views

Simplify Inverse Trigonometry Expressions

Help! I need to simplify this expression. I'm not even sure where to start. $$ \tan{(\arccos{(\frac{x}{4})})} $$