Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [inverse]

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.

An inverse is an operation that reverses the effect of another operation. This is a broad concept that arises in many areas of mathematics.

  • Multiplicative inverse: $2^{-1} = 1/2$
  • Inverse function: $\sin^{-1}x$ is the inverse of sine
  • Inverse matrix $A^{-1}$
  • Left and right inverses of group elements, of operators between linear spaces, etc.
4444 questions
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How to find the inverse function in explicit form?

For a function below: $$f(x)=a\cdot e^{-k_1 x}+b\cdot e^{-k_2 x}$$ How can I obtain its inverse function in explicit form?
LCFactorization
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From concrete mathematics problem 4.35

From Concrete Mathematics, problem 4.35. Let $I(m,n)$ be function that satisfies the relation $$I(m,n)m+I(n,m)n=\gcd(m,n)$$ when $m,n\in \Bbb N$ with $m\neq n$. Thus, $I(m,n)=m′$ and $I(n,m)=n′$ in (4.5). The value of $I(m,n)$ is an inverse of $m$…
meeponan
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Finding the correct angle from inverse cosine?

For my math homework, I have to find an angle of rotation, $\theta$, by cos $\theta$ = $-\sqrt3/2$. When I plug this into my calculator, I get 5$\pi$/6, but the correct answer is -5$\pi$/6. What is the procedure to find the correct angle.
Shankar Kumar
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Inverse Function of $ \frac{1}{2} ( e^x - e^{-x} ) $

Should find the inverse of: $$ f(x) = \frac{1}{2} ( e^x - e^{-x} ) $$ I tried a lot. But I don't know how to proceed on $$ 2x = \frac{(e^y)^2 - 1}{e^y} $$ Writing $ e^{-y} $ as $ \frac{1}{e^y} $ is right? I know somewhere I need to use the $ ln $…
loop
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Fixed point of an inverse?

According to a problem that I am working on a fixed point is defined as a point $x$ that satisfies $$f(x) = x.$$ A problem is asking me to find the fixed point of the inverse of $f(x)$, and the explanation tells me to find the inverse function…
hyg17
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Finding the all possible values of $x$ such that $\tan^{-1}(x+1) + \tan^{-1}(x) + \tan^{-1}(x-1) = \tan^{-1}(3)$

Find possible value of $x$ such that $$\tan^{-1}(x+1) + \tan^{-1}(x) + \tan^{-1}(x-1) = \tan^{-1}(3)$$ Progress: what I did was to consider a case when $x^2 -1 < 1$ $(xy < 1)$ and $3x>-1$ $(xy > -1)$ and then apply $\tan^{-1}(x) \pm tan^{-1}(y)$…
Orion_Pax
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Is there a symbol for inverse proportionality?

As the title said, is there a symbol for, or group of symbols for, Inverse Proportionality. I know that the symbol for proportionality is ∝, but I am having difficulty finding one for inverse proportionality.
Zoey
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Inverse of $\psi(t,s)=(e^t,se^{2t})$

I am trying to solve a problem involving finding the inverse of this map $$\psi:\mathbb{R}^2\to \mathbb{R}^2$$ $$\psi(t,s)=(e^t,se^{2t})$$ Here is what I am thinking Let $x=e^t$ and $y=se^{2t}$. Exchange $x,y$ and $s,t$. $t=e^x$ and…
Dima
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Is there an explicit formula for the inverse of $\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}$?

I apologize if this is trivial but I am stuck. Given the bijective function $f:(0,2\pi) \to (-2,2)$ with $$ f(x)=\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)} $$ where $\cot$ is the cotangent, how can I find an inverse $g:(-2,2)\to (0,2\pi)$? Is…
Ronald Bernard
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inverse of vandermonde matrix

I found a formula from that can simply derive the entries the of inverse of Vandermonde matrix. However, there is one notation that I couldn't understand which I hope someone can help me or give me a simple example. Link:…
ytj_banana
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The order in which to perform the transformation $y=f^{-1}(x+2)$ on the original point (10,8). Inverse first?

I'm having difficulty understanding this transformation. What is the correct order of these transformations? How do I find this correctly. I understand that $y=f^{-1}(x)$ can refer to switching the coordinates (the inverse) (appears as a reflection…
E.Yu
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Does there exist a basis function where its inverse function is in the same basis?

I'm interested in finding a basis function $\phi(x)$, which I can use to approximate some function $y(x) \approx \hat{y}(x) = \sum\limits_i c_i \phi_i((x - d_i)/s_i)$, where its inverse function, $\phi^{-1}(x)$, is in the same basis.…
user74223
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The scope of the multiplicative inverse?

First, Let $c$ denote the multiplicative inverse of $l~ (mod~ m)$, then $l\times c\equiv 1~(mod~ m)$ always hold. $l$ and $m$ are coprime. If we known $l$ and $m$, Extended Euclid's Algorithm can calculate $c$. I had known $l\in [a, b]$, $l>m$, and…
Yuansheng liu
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Is aI + bA invertible if rank(A) = n-1?

I am not able to prove this for sure by myself... To be more precise, $A$ is a $n \times n$ matrix of rank $n-1$ such that all diagonal elements of A are positive, off-diagonal elements can be positive or negative, and its last row is the weighted…
Tibo
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Inverse of $v \cdot v^\top$

Let's say I have a vector $v$. Now I want to calculate $(v\cdot v^\top)^{-1}$. Is there a known formula to solve this more directly than simply calculating it directly? Maybe something similar to the Sherman–Morrison formula…
Make42
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