# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Numerical inverse of a function

How to approximate the inverse of the function below? $$f(x) = \frac34 x - \frac 12\sin(2x) + \frac 1{16} \sin(4x)$$ The goal is to get $x$ values (range $[0, \pi]$) from values of $f$. The function ...
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### Groupoid element with multiple inverse elements?

Simply put, is there a groupoid whose element can have multiple inverse elements? I know how to prove that elements of a semigroup have unique inverses, but this is a bit diferent... If there is such ...
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### Uniqueness of right identity

I'm working on a problem which says the existence of a unique right identity and left inverse (which may not be unique) on a set with binary operation constructs a group. Of course I know that the set ...
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### Show the inverse of the One to One function

Is anyone able to guide me in the right direction for this question. This is for a beginner assembly language class. This is an online course so I am unable to ask the professor for guidance. Show ...
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### Finding selective inverses for functions: how?

So, I have a function $f(x) := \frac{3x}{7+x^2}$ and the question is that the domain is defined such as $x \geq a$, find the minimum value for $a$ for which there exists an inverse of the function. ...
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### Inverse of $\frac{1-e^{-x}}{x}$ on $(0,1)$

I am trying to invert (or to estimate the inverse of) $$y=\frac{1-e^{-x}}{x}$$ for $y\in(0,1)$. The function 'looks' monotonically decreasing between $x=0$ and $x=\infty$, but I have not been able to ...
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### Find injectivity/ prove one to one algebraically of $\frac{|x|x}{\sqrt{x^4-81}}$

I'm trying to find the range of this function: $$\frac{|x|x}{\sqrt{x^4-81}}.$$ However to find the range I have to first prove one to one, then find domain of inverse. I can't prove injectivity ...
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### Show $\int_0^a f(x)\,dx + \int_0^b f^{-1}(x)\,dx \ge ab$ for strictly increasing function $f(x)$

Let $f: [0, \infty) \to [0, \infty)$ be continuous and strictly increasing and with $f(0) = 0$. Prove that $$\int_0^a f(x)\,dx + \int_0^b f^{-1}(x)\,dx \ge ab$$ for any $a, b > 0$, and give a ...
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### Does the arccos itself contain all solutions or just one solution?

For the equation $$\cos(x)=\frac{1}{2}$$ All solutions are: $$x=\pm\frac{\pi}{3}+p2\pi,\quad p\in\mathbb Z\:.$$ To find these solutions, I use the inverse cosine ($\arccos$ or $\cos^{-1}$). Is the ...
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### Regarding an inverse trigonometric equation.

I tried to find the solutions of this equation $$\arctan\left(\frac{2x}{1-x^2}\right)+\text{arccot}\left(\frac{1-x^2}{2x}\right)=\frac{2\pi}{3}$$ I got solutions $\frac{1}{\sqrt{3}}$ and $-\sqrt3$. ...
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### f differentiable map of finite dimensional vector spaces, with derivative injective. Why is f injective?

Suppose A and B are finite dimensional vector spaces, $U\subseteq{A}$ is an open subset, $a\in U$ and $f:U\rightarrow B$ is $C^\infty$ with $(Df)_a$ injective. I need helping showing that there exists ...
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### Relation between $x,y,z$…Exponent problem…

The given equation is- $\sqrt[x]{75} = \sqrt[y]{45} =\sqrt[z]{15}$ Now,it is required to prove $x+y=3z$. I want the simplest possible solution.Thanks in advance.
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### is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
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### show multivarable functions are one-to-one, onto.

$F:\mathbb{R}^3 \rightarrow \mathbb{R}^3, F(x,y,z)=(2x,y,3z+y)$ My current method for these sort of questions is to try to find the matrix that represents this transformation and then see if i can ...
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### Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this. What is the difference ...
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### Composition and inverse mappings

Let $A\stackrel{\alpha} \rightarrow B \stackrel{\beta}\rightarrow A$ satisfy $\beta \alpha = 1_A$. If either $\alpha$ is onto or $\beta$ is one to one, show that each of them is invertible and that ...
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### Derivative of trace of inverse of a matrix function

I am trying to derive the derivative of the trace of inverse of a matrix function (of X), i.e. $$f(X)=Tr\left((HXH^{H}+I)^{-1}\right)$$ where $H\in R^{n\times m}, X\in R^{m\times m}$. So $HXH^{H}+I$...
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Let $f: x \mapsto (x,x^{2}) : \mathbb{R} \to \mathbb{R}^{2}$ and let $Y := f(\mathbb{R})$. Then $\mathbb{R}$ and $Y$ are in injection via $f$. Moreover, since $Y$ is the range of $f$, certainly $\... 2answers 55 views ###$(a,b) \mathbin\# (c,d)=(a+c,b+d)$and$(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse? Let$r\in \mathbb{R}$and let$0\neq s \in \mathbb{R}$. Define operations$\#$and$\&$on$\mathbb{R}$x$\mathbb{R}$by$(a,b) \mathbin\#(c,d)=(a+c,b+d)$and$(a,b) \mathbin\&(c,d)=(ac-bd(r^...
This is the homework I have done part a, b, but I don t have any idea how to do the rest $y = 5$ and $z = 12$ Those are the eigenvalues of matrix $A$ For part c, and d, I've tried to put some ...
Suppose we have symmetric positive definite $A$ with the size of $d \times d$, giving the SVD $A=V\Sigma U^T$ , if $D$ is an identity matrix, ie $D=I$, then \$(A^T A + \gamma I)^{-1}=U (\Sigma^2 + \...