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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72 views

Closed form of the inverse of a function

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$? Thanks in advance
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1answer
38 views

How to get the inverse function of this one?

Let's have function $$ \psi (x) = -\frac{1}{ax} - \frac{b}{a^2}\ln(x) + \text{const} + O(x). $$ I have read that the inverse function is written in a form $$ \psi^{-1}(t) = -\frac{1}{at} - ...
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37 views

Arc-Gamma Function.

Is there an arc-gamma function? Where gamma(x) = y... Arc-gamma(y) = x. I've searched and found something called DiGamma Function, but when I substituted it didn't seem to be "arc" but something ...
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3answers
96 views

What is the best way to find $g(x) = f^{-1}(x)$?

so the problem I have is if $f(x) = \sqrt{x+3} - 2$ and it asks to find the solution of $f(x) = f^{-1}(x)$. So i know to find the inverse, which I got as $f^{-1}(x) = (x+2)^2-3$. So to find the ...
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2answers
39 views

Getting inverse of polynoms with trigonometric functions

I'm trying to get the inverse of $$f(x) = \cos(x) + 3x$$ I tried it by definition of $\cos(x)$ with no luck: $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+...$$
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2answers
79 views

Showing that a matrix is invertible and finding its inverse

I'm incredibly rusty at linear algebra, and in preparation for my course I've been doing some review questions. I've been staring at this one for a half hour and still don't know how to approach it: ...
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1answer
34 views

Case Deletion Diagnostics

I have NO idea how to approach this problem. I don't see any connection between the corollary and the formula we need to prove. Does anyone have any hints? Corrolary: If $\mathbf{A}$ and ...
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0answers
27 views

Calculating the left pseudoinverse of a Matrix whose columns are Probablity Mass Functions

I have a matrix $A_{m\times n}$, where $A_j$ , a column of $A$ represents a probability mass function, and so the sum over the column is 1. This is true for all the columns of A, i.e. $\forall j \in ...
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3answers
36 views

Simple inverse function of $\frac{1-2x}{1+x}$

Just started learning about inverse functions, and got stuck on this one: $$f(x) = \frac{1-2x}{1+x}$$ So I tried multiplying by $(1+x)$ on both sides and got $y+yx = 1-2x$ but that doesn't seem to ...
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1answer
48 views

How to find the inverse of a function involving e with a coefficient?

I was wondering how I would find the inverse of the following function, since the e has a co-efficient: $\frac{e^x}{1+2e^x}=y$ I got as far as $\ln y+\ln(2e^x) = \ln e^x$, which would be changed ...
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47 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
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6answers
86 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
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1answer
45 views

Laplace transform, Inverse Laplace transform

Let $(\mathcal{L}f)(s)$ be the Laplace transform of a piecewise continuous function $f(t)$ defined for $t\geq 0$. If $(\mathcal{L}f)(s)\geq 0$ for all $s\in\mathbb{R^+}$ does this imply that $f(t)\geq ...
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1answer
378 views

Finding that values k that make this matrix invertible without using the determinant

The matrix in question is A = [(1,1,1),(1,2,k),(1,4,k^2)]. I know that I can row reduce the matrix to rref, which should in theory leave me with some k values in the matrix from which I can see what ...
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30 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
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1answer
40 views

inverse laplace transform of $s/(s^2+6s+13)$

Hi can anyone help with this inverse Laplace transform $$s/(s^2+6s+13) $$ I tried to do partial fraction $s+3/(s+3)^2+4 - 2/(s+3)^2+4$, but then I don't know what to do next...
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1answer
34 views

Do lines between determinants pass through the inverse?

Let A be a $2 \times 2$ matrix whose inverse also exists. If I was to draw a line from each of the 3 vertices (that are not the origin) of the determinant of A, to the 3 vertices of the determinant of ...
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1answer
70 views

To find the inverse of an implicit function

I have a function $t(f)$ here: $t(f) = T(sin(2\pi f/B)/2\pi + f/B) $ for $[-B/2 \le f \le B/2]$. $B$ and $T$ are constants. How to find the inverse of this function that is $f(t)$ using numerical ...
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2answers
62 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
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0answers
38 views

Finding inverse of a general linear transform

I'm not a mathematician, so I may abuse some notation here. Please comment for any clarification. Let's define a general linear transform as $$\int_XK(\mathbf{\omega},x)f(x)dx$$ where $X$ is some ...
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1answer
35 views

On matrices sharing the same smallest nonzero eigenvalue and related eigenvector

Suppose that $A$ and $B$ are square matrices with the proper size, then what kind of condition does $B$ have to satisfy such that $A$ and $AB$ share the same smallest(largest) nonzero eigenvalue and ...
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49 views

Canceling a Limit with 'Inverse Limit'

Is there an operator that cancels a limit (inverse limit)? For example $lim_{x\rightarrow0}e^{x}$: Can one take an 'inverse limit' to cancel with this limit and output $e^{x}$? Let A='inverse limit', ...
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55 views

What would be the inverse function for the following condition?

What would be the inverse function condition for the above question.
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0answers
41 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
3
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0answers
70 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
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2answers
25 views

Prove that $B$ is invertible:$B = A_{11} - A_{12}A_{22}^{-1}A_{21}$ if…

Let $$A = \begin{bmatrix}A_{11}&A_{12} \\ A_{21}&A_{22} \\ \end{bmatrix}$$ Prove that $B$ is invertible:$$B = A_{11} - A_{12}A_{22}^{-1}A_{21}$$ $$$$ in case of $A$ and $A_{22}$ being ...
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0answers
36 views

Formal inverse of a matrix ressembling Fourier's matrix

What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$? When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and ...
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1answer
28 views

Complex Field - Proving $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$

Like the title states, I'm trying to prove that $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$ where z is a complex number and z* is its conjugate. I keep ...
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1answer
106 views

Is there a closed form for the inverse of $y=x^{x^x}$?

It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any ...
3
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1answer
47 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
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8answers
204 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
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1answer
86 views

How to find $10^{-1}$ in $\{0, 2, 4, 6, 8, 10, 12\} \subseteq \mathbb Z_{14}$?

Here is my question: Consider the subset $S = \{0, 2, 4, 6, 8, 10, 12\}$ in $\mathbb Z_{14}$, with the operations of addition and multiplication in $\mathbb Z_{14}$. (a) Show that $S$ has ...
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1answer
323 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
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268 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
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1answer
67 views

Solving least-squares: why ever use iterative descent methods over pseudoinverse?

I recall doing an assignment in machine learning where we ran regression tests on a data set, both using our own implemented gradient descent program, and then using the (right) pseudoinverse ...
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130 views

The inverse of a Moment generating function

The moment generating function of $X$ is $M_X(t) = \mathbb{E}[e^{tX}] = \int e^{tu}f_X(u)du$ where t is a complex variable and $f_X$ is the density of X. The cumulant generating funtion of $X$ is ...
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2answers
129 views

Evaluating an inverse function by sketching a unit circle

Problem I'm working on: "Evaluate the inverse function by sketching a unit circle, locating the correct angle and evaluate the ordered pair on the circle." The function I got was $\cos^{-1}(0)$. ...
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50 views

Local inversion theorem (théorème d'inversion local)

I don't understand how to use the local inversion theorem to prove that a nondegerate critical point of a function $f\in C^2(U,\mathbb{R})$ is isolated Thank you.
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Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [closed]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
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2answers
39 views

Inverse of $x(x+2)$ given $x\ge -1$

Consider the function: $y=x(x+2)$ . Consider its domain to be $x \geq -1$ . Graphically it makes sense that the inverse of this function is $-1 + \sqrt{x+1}$. But how to compute it analytically? ...
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2answers
46 views

How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$?

Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$ This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$ I say it is surjective because for all really $x$ there is all real $y$, $x \in ...
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4answers
352 views

How to find a modular multiplicative inverse when GCD is not 1

I am working on a problem that requires finding a multiplicative inverse of two numbers, but my algorithm is failing for a very simple reason: the GCD of the two numbers isn't 1. I figured I must've ...
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0answers
32 views

Looking for reference for the criterion of inveribility of a difference of two invertible matrices

It is pretty easy to show that $A-B$ is invertible if either $AB^{-1}$ or $BA^{-1}$ have all eigenvalues of absolute value less than $1$. But I am specifically looking for a handy reference of this ...
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2answers
63 views

If a one-to-one function's inverse is the same what must be true of the graph of f?

As a followup to this question. I'm trying to determine what must be true of the graph of $f$ in these cases. I've examined the two functions $f(x)= x$ and $f(x)= \frac{1}{x}$ and I'm not seeing any ...
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1answer
77 views

Can an Elementary Matrix's Inverse's Determinant = 0?

Can someone explain to me why an elementary matrix's inverse determinant cannot equal 0? Or can it? Is there some theorem to elementary matrix inverses? THANKS FOR YOUR INSIGHT! :)
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1answer
44 views

Finding the area bounded by two curves when in terms of $x = y^2$?

I can't seem to figure this problem out. Find the area bounded by the curves $x=2y-y^2$ and $x=4-y^2$, in the first quadrant. I am having difficulties with graphing the equations and coming up ...
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1answer
55 views

Solving a set of non-linear matrix equations

Consider the following set of equations $$\begin{cases}PAQ^{-1}&=T \\ QBR^{-1}&=T\\ RCP^{-1}&=T, \end{cases} $$ where A,B,C and T are known real-valued $3\times3$ matrices and P, Q, R are ...
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1answer
28 views

Show that a square matrix with linear transformation T(M) = A·M is bijective when A is inversible

Suppose $K$ is a body (a field), $n ≥ 1$ and $A ∈ M_n(K)$ a fixed matrix. Consider the linear transformation $T : M_n(K) → M_n(K)$ defined by $T(M) = A · M$ for $M ∈ M_n(K)$ The mark scheme says ...
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2answers
67 views

Why does the square root of an inverse function turn negative?

For example, $$f(x)=x^2$$ $$y=x^2$$ $$-\sqrt{x} = f^{-1}$$ Why does $\sqrt{x}$ become negative? Edit: Sorry for all the confusion, I will state the problem on my textbook and the solution. ...
3
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3answers
244 views

Determine the greatest interval where the function is invertible

The assingment is to determine the greatest interval around $x=0$ where the function: $$f(x)=x^5-5x+3$$ is invertible. After that, determine $(f^{-1})'(3)$ I have totally forgotten all about ...