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Questions tagged [inverse-function]

For questions regarding an inverse function as the dominant topic of the post, or for questions requesting guidance on finding the inverse function for a particular function.

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Will the domains of both these functions be equal?

Let f(x) = arcsec ($\frac{2x}{5x+2}$) and g(x) = arccosine ($\frac{5x+3}{2x}$) .So will their domains be equal? I had this doubt while finding the domain of f(x). I converted arcsec to arccosine, so ...
improvement dude's user avatar
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How do I derive the inverse of the function $f(x) = \frac{e^x -1}{x}$. [duplicate]

I'm trying to rearrange the function to isolate $x$, but I'm not sure what the first step is. $f(x) = \frac{e^x -1}{x}$ I suspect the Lambert W function crops up somewhere.
Elis's user avatar
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Are there Functions $f$ and $g$ such that $g$ is Bijective & is the Dirichlet Inverse of an Arithmetic Function $f$?

Let $f: \mathbb{N} \to \mathbb{N}$ be a function. I was wondering if, there is a function $g: \mathbb{N} \to \mathbb{N}$ such that $(f * g)(n)= \epsilon (n) = (g * f)(n)$ and $(f \circ g)(n) = id(n) = ...
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How to show these two functions are the inverses of each other?

The two functions in question are the $\text{dexp}_\Omega$ and $\text{dexp}_\Omega^{-1}$ operators, which are defined as follows: $$ \text{dexp}_\Omega(C) = \Phi(\text{adj}_\Omega)(C) \\ \text{dexp}^{...
Idieh's user avatar
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Computing inverse of sums of strictly monotanically function (e.g CDF of random variables)

I'm trying to compute the general case of inverse of CDF of $Y=X^2$, where $Y,X$ are random variables. Given that a CDF $F_X$ is a strictly increasing function, also has to be it's inverse. The CDF $...
Daniel Muñoz's user avatar
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Inverse higher order derivatives of a function with respect to changes in it's own gradient

I have an optimization problem operating on the implicit function $$F(x, y, z) = 0$$ and a vector $$(u, v, w)$$ where the solution to the problem is the point $(x_0, y_0, z_0)$ that satisfies the ...
sprw121's user avatar
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Inverse of equal functions

the function $f:\Re \to \Re : f(x)=e^x$ is one-one into and $g:\Re \to (0,\infty) : g(x)=e^x$ is a one-one onto function but both functions are same as the functions have the same domain and $f(x)=g(x)...
ca_100's user avatar
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2 answers
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Solve for $y$ in $ye^{\frac{1}{2}(y-\frac{1}{y})}=x$

I'd like to solve the following equation for $y$ in terms of $x$ $$ye^{\frac{1}{2}(y-\frac{1}{y})}=x$$ This equation is close in nature to the definition of the Lambert $W$ function, but different ...
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Inverse of a function in ordered pair form

Suppose a bijective function (say $f$) is in an ordered pair form. Then can we write its inverse as $$f^{-1}= \{(b,a):(a,b) \in f\} \quad ?$$ For example: Let $f=\{(4,5), (5,6), (3,4), (1,2)\}$. Then, ...
ca_100's user avatar
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Is the intersection of a function and its inverse on the line $y=x$ equivalent to the function being strictly increasing? [closed]

Is the following statement true? The functions $f$ and its inverse $f^{-1}$ intersect each other on the line $y=x$ if and only if $f$ is a strictly increasing function. I think this statement is not ...
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1 answer
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Inverse of smooth family of diffeomorphisms [closed]

Let $M$ be a smooth manifold, and let $I$ be an open interval. We say a map $\varphi:I \rightarrow \text{Diff}(M)$ is a smooth family of diffeomorphisms if the map $I \times M \rightarrow M$ given by $...
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Inverse function of a vectorial function with trigonometric functions

I have to find an expression of the inverse function of $\varphi:(-\pi,\pi)\times \mathbb{R} \to S\subseteq \mathbb{R}^3$ defined by $\varphi(u,v)=(\cos(u),\sin(u),v)$ where $S=Im(\varphi)$. My ...
Sigma Algebra's user avatar
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What is this kind of relationship among a set of given functions called?

I was learning about inverse functions, the book mentioned if $$y=f(x)$$ $$x=g(y)$$ then the functions $f$ and $g$ are said to be inverse of each other. My question is what is the following kind of ...
Aniket Harit's user avatar
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Nontriviality of generalised inverses of a nondecreasing function (corrected)

Given a function $f:[0,\infty[\,\rightarrow[0,\infty]$, its right inverse is defined as $$f^\top(t):=\sup\{s\geq0\mid f(s)\leq t\},$$ while its left inverse is defined as $$f^\bot(t):=\sup\{s\geq0\mid ...
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Find the inverse of $f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$ [duplicate]

This question is linked to Prove that the given function is invertible Find the inverse of $f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$. I am looking for an approach to find the inverse of such a 3d function, ...
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Prove that the given function is invertible

This question is linked with this question I have $$f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$$ and I need to prove that it is invertible at the point $(1,-1,2)$. I have read that proving such a quality ...
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Inverse of a vector function

Let $$\varphi:(-\pi, \pi)\times \mathbb{R} \to \mathbb{R}^3, \quad \varphi(u,v)=(\cos(u),\sin(u),v)$$ Prove that the map $\varphi$ is injective, $C^\infty$, and invertible with continuous inverse map ...
Sigma Algebra's user avatar
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1 answer
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Limit of function whose inverse has limit approaching infinity at finite point

Consider a continuous strictly monotone function $f:(-1,1)\to\mathbb{R}$, where $$\lim_{x\to1}f(x)=\infty, \lim_{x\to-1}f(x)=-\infty$$ I have proved that this function is bijective. I want to prove ...
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Limits coming from Lagrange Inversion Theorem

$y=e^x\implies x=\ln y=\ln(1+(y-1))\implies$ $$x=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}(y-1)^n\tag1$$ On the other hand, by Lagrange inversion theorem, with $y=f(x)=e^x$, $a=0$, we have $$x=g(y)=\sum_{...
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Elementary functions: Can an arithmetic operation be a composition?

The elementary functions are the functions of one complex variable that are generated by applying finite numbers of $\rm{exp}$, $\ln$ and/or unary or multiary $\mathbb{C}$-algebraic functions. Let $n\...
IV_'s user avatar
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If $f(g(x))=x$, does $g(f(x))=x$ hold?

If $f(x)$ and $g(x)$ are formal power series, i.e. $$f(x)=\sum_{n\ge 0} a_n x^n, g(x)=\sum_{n\ge 0} b_n x^n,$$ and $$f(g(x))=x,$$ can it be proved that always have $$g(f(x))=x?$$ It seems intuitive, ...
athos's user avatar
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1 answer
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Inverse Series of $x\sin x$

For some background before I get into my question: I am a Calculus 2 student who knows only some of the bare essentials to Complex Analysis, so bear with me. I was recently studying the transformation ...
Oiler's user avatar
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3 answers
95 views

Do any non-invertible functions satisfy $(f \circ g)(x)=x$

Is there a non one-to-one function $f$ such that , when composed with another function $g$ it gives back the identity function? I am not referring to inverse functions . Because from the definition , ...
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What is the inverse function of $x\mapsto\sum_{i=1}^n\frac{a_i e^{-a_i x}}{1+e^{-a_i x}}$?

I want to find the inverse expression of the following: $$\sum_{i=1}^n\frac{a_i e^{-a_i x}}{1+e^{-a_i x}} = T$$ $a_i , i \in\{1,2,...,n\} $ is known and $T$ is also known I want to find $x$ in terms ...
mohadeseh azari's user avatar
12 votes
1 answer
765 views

Taking the inverse (not the reciprocal) of both sides of an inequality

This is something I'm having a hard time finding online, but say we know that $f(x) > g(x)$ (for all inputs $x > a_{0}$ for some $a_{0}$), then would it always be true that $f^{-1}(x) < g^{-1}...
Bob Marley's user avatar
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Can I use function inversion to determine valid input range based on output range, for some vector valued non-linear function?

I'm trying to determine the range of valid inputs for a given function based on its output range. I'm considering the possibility of using function inversion as a method to achieve this. Specifically, ...
desert_ranger's user avatar
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1 answer
44 views

given a positive and decreasing function, will its inverse be decreasing? [closed]

Given a positive function $f(x)>0$ such that $f'(x)<0$. consider the domain $[0,1]$ and the range $[0,\infty)$ is there any proof that the inverse function $y = f^{-1}(x)$ is DECREASING? That ...
Beatrice F-Weber's user avatar
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0 answers
71 views

Does the inverse of this function exsist?

For a personal project i derived the function, $$f(x) = x^{\left(\frac{c+1}{x^b+d}+1\right)}$$ where $b,c,d \in \mathbb{R} |b,c,d > 0$ (I think that is how you write it, i am still learningD:) I ...
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When can I just take complex inverses in absolute value inequalities?

For example, consider the inequality $\left|\sin z\right| < a$. Why is it correct to take $\left|z\right| < \arcsin a$ (restricting $-\pi < |z| < \pi$)? Is it a property of complex $\sin$, ...
GingerBadger's user avatar
2 votes
0 answers
170 views

Is this multivalued inverse logarithmic integral $\operatorname{li}(x)<0$ power series valid?

$\DeclareMathOperator\li{li}$To derive the multivalued inverse of the logarithmic integral $y=\li(x)$$;y<0,x>1$, denoted $\li_-^{-1}(x)$ and for $0\le x<1$, denoted $\li_+^{-1}(x)$, we use: ...
Тyma Gaidash's user avatar
1 vote
1 answer
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Inverse Z-Transform of $\frac{1}{(1+z)^2}$ with ROC : |z|<1

I'm trying to Find the inverse Z-Transform of $\frac{1}{(1+z)^2}$ My steps are as such : $\frac{1}{(1+z)^2}$ = $\frac{z^{-2}}{z^{-2}\cdot(1+z)^2}$ = $\frac{z^{-1}\cdot z^{-1}}{(1+z^{-1})^2}$ = $\frac{...
Losh_EE's user avatar
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0 answers
57 views

Ways to invert complicated matrix formulas

I have two somewhat complicated matrix formulas that convert the mean vector and covariance matrix for a certain variable, $\mu \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{n \times n}$, into the ...
dherrera's user avatar
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5 votes
1 answer
136 views

General formula for reversing double integral bounds

The double integral over the region: $$ R = \left\{ \left( x,\: y \right) : a \leqslant x \leqslant b,\: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right\} $$ is expressed as $$ \...
LightninBolt74's user avatar
17 votes
1 answer
377 views

Inverse function of the Exponential Integral $\mathrm{Ei^{-1}}(x)$

The Exponential integral is defined by $$ \mathrm{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm dt, $$ and has the following expansion $$ \mathrm{Ei}(x) = \log x + \gamma + \sum_{k=1}^\infty \frac{x^...
Nolord's user avatar
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0 votes
1 answer
52 views

Taking the inverse of normal CDF inverse after an additive operation

$\Phi\left(x\right)$ is the CDF of a normal distribution with parameters $m$ and $\sigma$. Is there a way to solve for $\rho$ here? $\Phi\left(x\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\frac{x-m}{\...
Victor Yerz's user avatar
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0 answers
40 views

Existence and continuity of inverse operator

Let $E$ and $F$ be normed spaces. Let $T:E \to F$ is a linear operator and suppose that exists $c>0$ such that $$\|T(x)\| \geq c \|x\|, \quad \forall x \in E.$$ Then it's easy to see that $\...
Guilherme Costa's user avatar
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0 answers
23 views

A function with multiple extremes from a function with a single extremum through reversible variable replacement

I recently studied normalizing flows, a machine learning method. I was wondering if it is possible to reproduce the illustrations that I often see in articles using this method. Example The theory of ...
illigarium's user avatar
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38 views

Find an explicity formula for the inverse of the function $\textbf{f}$ given by $f_1=e^x\cos y$ and $f_2=e^x\sin y$ [duplicate]

Let $\textbf{f}$ be given by $f_1=e^x\cos y$ and $f_2=e^x\sin y$, and put $\textbf{a}=(0,\pi/3)$, $\textbf{b}=\textbf f(a)$. Find an explicity formula for the inverse of $\textbf{f}$, denoted $\textbf{...
Superunknown's user avatar
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0 votes
1 answer
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erfcinv is giving incorrect results in MATLAB

I am working on a Communication Theory problem. To summarize the problem, I have to code the following: Plot the ideal bit error probability for BPSK, which is given by $Q(\sqrt{\frac{2E_{b}}{N_{0}}})...
AverageStudent123's user avatar
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0 answers
36 views

Roche Geometry Visualization

I have a task in hand to create a Roche Geometry Visualization. All I am is a programmer and despite I have basic knowledge of math and algebra I am not match for what is awaiting me. I have a fairly ...
MSH's user avatar
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0 answers
36 views

Faster way to calculate compound function of trigonometric and inverse trigonometric function.

We know that $\arcsin(\sin x) = x$ holds true only for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ for if $x$ does not lie in that interval, we need to add or subtract some multiple of $\pi$, or even need ...
Ansh's user avatar
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0 votes
2 answers
67 views

inverse of a rational / irrational defined function

Going through Spivak's chapter on inverse functions, there is an interesting problem: Find $f^{-1}$ if: $$ f(x) = \begin{cases} x & \text{if } x \text{ is rational} \\ -x & \text{if } x \text{...
nz_'s user avatar
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0 votes
1 answer
25 views

If $\psi(u,v) = (u, u^2 + v^2, v)$ and $\alpha(t)=(ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha$ in $\text{img}(\psi)$, then $a + b + c$ is

If $\psi(u,v) = (u, u^2 + v^2, v)$ with $(u,v) \in \mathbb{R}^2$ and $\alpha(t) = (ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha \subset \text{img}(\psi)$, then $a + b + c$ is .... Is anyone ...
Angelo's user avatar
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5 votes
1 answer
291 views

Question regarding inverse of exponential function

This question is in the context of the following problem Find the inverse of the function $$f:(-\infty, 1] \rightarrow \Biggr[\frac{1}{2}, \infty\Biggr], \text{ where } f(x) = 2^{x(x-2)}$$ I proceeded ...
koiboi's user avatar
  • 316
2 votes
2 answers
52 views

$\int^{0.75}_{0.25} (f \circ g^{-1})(x) dx$

Let $f(x)= \frac{2 \cdot 10^x +10^{2x}}{1+10^x}$ and $g(x)=\frac{10^x}{1+10^x}$ What is the value of $\int^{0.75}_{0.25} (f \circ g^{-1})(x) dx$ How I solved it: I first found $g^{-1}(x)$ Let $y=g(x) $...
Antony Theo.'s user avatar
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1 vote
1 answer
20 views

Finding point $S$

I have $f: x\mapsto 2x\ln(\frac{x}{2})$ in $\mathbb{R}^+$. Let $E (x_E \mid y_E)$ be the extremal point. The restriction of $f$ to the interval $]0; x_E]$ has the inverse function $g_1$, the ...
Zrste's user avatar
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2 votes
0 answers
31 views

Inverse of a piecewise non-continuous function

Let $$f(x) = \begin{cases} 2x-2, & \text{if $x \leq1$} \\ \frac{1}{x^2-1}, & \text{if $x>1$} \end{cases}$$ For $x\leq1$ $f'(x)=2>0 \implies f$ is increasing at $(- \infty,1]$ the range ...
Antony Theo.'s user avatar
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6 votes
3 answers
140 views

If $f$ is continuous at $a$, is $f^{-1}$ continuous at $f(a)$?

Let $I\subseteq\mathbb{R}$ be an open interval, and $f:I\to\mathbb{R}$ an injective function. Let $a\in I$, and suppose that $f$ is continuous at $a$. Does it follow that $f^{-1}$ is continuous at $f(...
ashpool's user avatar
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2 votes
0 answers
86 views

Coefficients of the Inverse Modified Bessel Function

I am curious if there is a closed form that represents the coefficients of the inverse of the modified Bessel function of the first kind $I_{0}(x)$. I can find the series representation using ...
Kyler Rusin's user avatar
-1 votes
1 answer
84 views

Upper bound for series with $\arcsin$, $\arccos$ and $\arctan$

Just a only doubt. Supposing that I have these three series: $$\sum_{n=1}^\infty\left[\arcsin(p(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arccos(q(x))\right]^n, \quad \sum_{n=1}^\...
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