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.
2,146
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Inverse of composition of functions with intermediate change in dimension
Consider the simple function below:
$$y=f(x)=Ag(Bx)$$
where $x\in\mathbb{R}^n$, $y =f(x)\in\mathbb{R}^n$, $B\in\mathbb{R}^{m\times n}$, $A\in\mathbb{R}^{n\times m}$, and an invertible $g(\cdot): \...
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Deduce, the Taylor series expansion of $\arcsin x$ from the Taylor Series expansion of $e^{a\arcsin x}$
Find the Taylor Series expansion of $e^{a\arcsin x}$ and hence deduce, the Taylor series expansion of $\arcsin x$.
I could find the Taylor Series expansion of $e^{a\arcsin x}$ as $$1+ax+\frac{a^2x^2}{...
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1
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Find the Area Bounded by $y = \frac{2}{π} [|\cos^{-1}(\sin x)| - |\sin^{-1}(\cos x)|]$ and $x$-axis between $ \frac{3π}{2}≤x≤2π $.
Find the area bounded by $\displaystyle y = \dfrac{2}{π} \left[ \space \left|\cos^{-1}(\sin x)\right| - \left|\sin^{-1}(\cos x)\right| \space \right]$ and $x$-axis between $ \dfrac{3\pi}{2}≤x≤2\pi $.
...
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1
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Inverse function of a function involving $o(1)$
I am bumping into a problem when trying to write the inverse function of a function involving $o(1)$.
Here we have $t= \ln n(1+o(1))$ as $n\rightarrow \infty$
Now I want to write $n$ as a function of $...
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1
answer
52
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Intuitive explanation of why the area under the curve of a hyperbola (1/x) is infinite but not the area of a decreasing exponential?
In some of the videos I've watched on the Laplace transform, the authors say that if the exponential is decreasing, the area calculated by the transform is finite, and in control theory we can say ...
- 143
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1
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Is it possible to find the inverse of the following function explicitly?
I am interested in the following function
$$
f(x) = \sqrt{3x \left(1-\frac{x}{\theta} \right) \ln{\left[a \left(1-\frac{\theta}{x} \right) \right]}},
$$
with $ 0<a \leq 1$ and $\theta<x$. ...
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3
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what is the inverse function of 2[x]-x ,where[x] is the floor function? [closed]
$f(x) = 2[x]-x$,
$[x]$ is the floor function,
the domain of the function is $[-1,2]$,
its codomain is $\Bbb R$
$$
f(x) = \begin{cases}
-2-x & \text{if } -1 \leq x < 0 \\
-x & \text{if } 0 \...
- 13
2
votes
2
answers
49
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Proving the existence of multiple right inverses for surjective but not injective functions
Consider the function $f: \mathbb{R} \to \mathbb{R}^{\geq 0}$ defined by $f(x) = x^2$ for all $x \in \mathbb{R}$. $f$ is surjective, because for all $x \in \mathbb{R}^{\geq 0}$ we have $f(\sqrt{x}) = ...
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Domain and range of an inverse trigonometric composite function.
arcsin([x]/{x}) - what will the domain and range of such a function be?Here, [x] implies the greatest integer function while {x} is the fractional part function.
I tried using the fact that for arcsin(...
0
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1
answer
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What's the inverse function of the following function f(x)=2^x+3^x [closed]
I am a student and the question was raised in a channel that what is the inverse of this function and I thought a lot but I could not solve it and that is why I decided to express my question here so ...
- 19
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How to calculate inverse of modulo (remainder) operator?
I am running a C++ program with two arraysvec and nums which are related as:
...
-1
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28
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Finding the radius from a chord and arc, inverse sinc [duplicate]
This is based off some random YouTube comment. The person was asking about finding the radius of a circle when all you have is the length of an arc and the length of its chord. A respondent said it's ...
- 201
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Are there nice inverse functions for the diagonal indexing function $x + (x + y) (1 + x + y) / 2$?
I've been trying to work out nice inverse functions for the x and y coordinate parts of the diagonal indexing function
$$I(x,y) = x + \frac{(x + y) (1 + x + y)}{2},$$
which produces the table
$$\begin{...
- 363
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Is there an exact solution to $x \sinh\Big(\frac{1}{x}\Big) = a$?
Is there an exact formula for solutions to the equation $x \sinh\Big(\frac{1}{x}\Big) = a$ where $a,x \in \mathbb{R}^+$? And if not, why?
I tried to rearrange to apply Lambert W somewhere to no avail.
...
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Inversion of the imcomplete elliptic integral of 2nd kind: $E(z|m)=x\;\Longleftrightarrow\; z=\text{bm}(x|m)$
I would like an opinion on this issue: similarly to the fact that
$$F(z|m)=x\Longleftrightarrow z=\text{am}(x|m)$$
I also tried to reverse the function $E(z|m)$ and got the following series (I ...
- 2,353
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Show that one bijection is the inverse of the other bijection if the two (non-identity) bijections commute [closed]
Suppose you have two bijections $\eta, \alpha: S \to S$. Both are not the identity maps on $S$, and that
$$\eta\alpha = \alpha\eta$$
Can we conclude that $\alpha = \eta^{-1}$?
Many thanks in advance!
- 3
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Building the inverse
If $\theta_1$ and $\theta_2$ are dependent, let $a_2$ be fixed at $\bar{a}_2$
$x_2= x_2(\bar{a}_2,\theta_2) \rightarrow \theta_2 =x_2^{-1}(\bar{a}_2,x_2)$
Why can you build the inverse dependend on $...
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How to solve $f(ix)=i f^{-1}(x)$
Is there some method to solve this equation?
$$f(ix)=i\cdot f^{-1}(x)$$
I found these solutions:
$x$
$c_1\arctan\left(\tanh\left(\frac{x}{c_1}\right)\right)$
$c_2\arcsin\left(\sinh\left(\frac{x}{c_2}\...
- 2,353
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Calculator doesn't show the steps to finding the inverse of this function.
I was recently given the function
$\sqrt{x}/(x-1)$
and was asked to evaluate $f^{-1}(1)$ find the inverse and draw the graph.
Firstly, in order for a function to be invertible, it must be bijective, ...
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433
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Why does deg(arcsin(sin(rad(degree)))) not produce degree?
I have an angle in degrees and I want it to be encoded between $-1, ..., 1$ to make it easier for using in a neural network.
I thought it might be a good idea to first convert the angle into radians ...
- 384
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1
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45
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Invertible function that takes four digits and generates one number
I'm trying to understand if there's a function (or a series of functions) that takes 4 digits (positive integers between 0 and 9) (a, ...
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2
answers
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Inverse function for $f(x)=x/(1-e^{-x})$
I'm looking for the inverse function for $f(x)=x/(1-e^{-x})$, over the domain $x>0$.
Wolfram says that the answer is $f^{-1}(x)=x+W(-xe^{-x})$, for $x>1$, where $W(x)$ is the Lambert W function. ...
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5
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110
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Show that $\arcsin(\frac{x-1}{x+1})=2\cdot\arctan(\sqrt{x})-\frac{\pi}{2}$
So I started by saying that $$y=\arcsin\left(\frac{x-1}{x+1}\right)$$
Then you could say that $$\sin(y)=\frac{x-1}{x+1}$$
Then calculating $\cos(y)$ with the trigonometric identity, I found the ...
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Inverse formula of "summing reciprocials of consecutive odd integers"
I'm looking for an explicit fomula for calculating the natural number $n$, as result value, from any (finite) given argument value $$\text{Sum} \left[ \frac{1}{2 k + 1}; \{ k, 0, n \} \right].$$
...
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1
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119
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Help me find the inverse of this function
While working on a recreational math problem I have come to the point were I need to find the inverse of one of the two following functions;
$$\frac{W_0(x)(W_{-1}(x)+1)}{(W_0(x)+1)W_{-1}(x)}$$
$$\frac{...
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Is there a name for a morphism which makes a left inverse act like a two-sided inverse?
$\newcommand{\Id}{\operatorname{Id}}$Consider
a morphism $f : A \rightarrow B$ which has a left inverse $g$, i.e. $g \circ f = \Id_A$. (That is, $f$ is a split monomorphism.) Of course, we don't ...
- 5,478
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Classification for involutory real infinite series
Out of curiosity I'm playing around with the concept of involutory functions. An involutory function (involution) is a function whose composition with itself is the identity function (i.e. ${f \circ f}...
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Sum regularization/ use of generalized functions in Ramanujan master theorem applied to inverse function of $\frac{\sin(x)}x$
$\def\Si{\operatorname{Si}}$
The converse Ramanujan master theorem finds Taylor series coefficients. The goal of our question is applying it on $h(x)=\cases{f^{-1}(x),x\ge1\\ 0,0\le x<1}$:
where $...
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Multivariate function inversion - Lagrange Inversion
If I have a (bijective) scalar function $f : \mathbb{R}^N \to \mathbb{R}$ and $\vec{x}\in \mathbb{R}^N$. Is it possible to obtain an inverse function $h:\mathbb{R} \to \mathbb{R}^N $ such that $\vec{h}...
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Composite functions and fixed point free property
Given 2 functions: $f: X \to X$ and $g: X \to X$,
$f$ and $g$ are one-to-one
$f$ and $g$ are fixed point free
composite functions $f \circ g$, $\; g \circ f$,
$\; f^2 \equiv f \circ f\;$ and $\;g^2 \...
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Can we compute $m_{yx}$ once we estimated $m_{xy}$ from $y=m_{xy}*x + 0 $?
Simply, Lets us say that we know $m_{xy}$ from the linear equation $y= m_{xy}*x + c$; And now we want to calculate $m_{yx}$ because we want to estimate the effect of x on y i.e $x = m_{yx} * y + c$ ...
- 53
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Geogebra struggles integrating the derivative of $\sec^{-1}(x)$
I'm currently using Geogebra 6.0.791.0 and I ran into a strange result while trying to integrate the derivative of the $\sec^{-1}(x) = \cos^{-1}(\frac{1}{x})$ function.
The first differentiation is ...
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What is the derivative of $\sin^{-1}(\frac{x+\sqrt{1-x^2}}{\sqrt{2}})$ [closed]
If we substitute $x=\sin \theta$ then we get the derivative to be $\frac{1}{\sqrt{1-x^2}}$ shown in the picture below
But instead of assuming $x=\sin \theta$ if we assume $x=\cos \theta$ we get the ...
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Mellin transform of Inverse function.
Mellin transform of $h(t)=f^{-1}(t)$ is:
$$M_s(h(t))=\int_0^\infty x^{s-1}h(t)dt\mathop=^{h(t)\to t}\int_0^\infty t f(t)^{s-1} df(t)=?$$
Could someone go through a step-by-step.
First part:
Using: $$h(...
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1
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75
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Find $dy/dx$ for $y =\operatorname{arcsec}(\tan x)$
If $$y =\operatorname{arcsec}(\tan x),$$ then find $\frac{{dy}}{{dx}}$.
Note: I was practising on finding derivatives for various types of questions and I stumbled upon the question mentioned above.
I ...
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Inverse logarithmic integral for x∈(0,1)
When looking for a suitable representation of the logarithmic integral, $li(x)=\int_0^x \frac{dx}{\log{x}}$, one can found many texts for $x>1$, which is understandable because of its relations to $...
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2
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Why codomain is more than the range in an Inverse function
While solving inverse function problems, I got confused in a part, like for any Inverse function to be defined, it must be one-one and onto, then in many questions why the codomain is given more than ...
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How are derivatives and integrals inverse functions when they're not injective transformations?
By using the kernel of both transformations, we can easily check that they're not injective functions (due to the fact that kernel does not only contain the zero vector by seeing both operators as ...
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2
answers
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Solving $\operatorname{arccsc}(\sqrt{37})+\operatorname{arcsin}\left(\frac{x}{\sqrt{4+x^2}}\right)=\frac{1}{2} \operatorname{arcsin}\frac{3}{5}$ [closed]
My question is
$$\operatorname{arccsc}(\sqrt{37})+\operatorname{arcsin}\left(\frac{x}{\sqrt{4+x^2}}\right)=\frac{1}{2} \operatorname{arcsin}\frac{3}{5}$$
Find value of $x$.
I have converted all the ...
user1198475
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3
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About derivatives of inverse functions [closed]
Suppose $f(x)$ is the inverse function of $g(x)$.
Does that also imply that $\frac{df(x)}{dx}$ is also the inverse function of $\frac{dg(x)}{dx}$?
And can I thus use $g'(x)= \frac{1}{f'(g(x))}$ for ...
1
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0
answers
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Inverse of a Lipschitz function
If it is given that $\hat{\rho}(s)=\tilde{\rho}(s)s$. How do I prove that $s=(\tilde{\rho} \circ \hat{\rho}^{-1}(s))\hat{\rho}^{-1}(s)$ where $s \in \mathbb{R}$ and $\tilde{\rho} : \mathbb{R} \to \...
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0
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364
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Surprising approximation of exponential series?
Consider the following expression
$$
y_j= \sum_{k=0}^{L} \frac{e^{-\sum_{i=-k}^k(k-|i|)x_{j+i}}-e^{-\sum_{i=-k}^k(k+1-|i|)x_{j+i}}}{\sum_{i=-k}^k x_{j+i}}\tag{1}
$$
for $1\leq j \leq L$. Given smooth ...
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119
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Does $f(x)=x$ and $f(f(x))=x$ have exactly the same set of solutions when $f(x)$ is monotone?
Does $f(x)=x$ and $f(f(x))=x$ have exactly the same set of solutions when $f$ is a monotone function?
My try:
It is pretty obvious that every solution of $f(x)=x$ is also the solution of $f(f(x))=x$, ...
- 12.8k
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4
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Why is not $\,f(x)=\arcsin\left(2x\sqrt{1 - x^2}\right)$ a one to one function?
Why is not $\,f(x)=\arcsin\left(2x\sqrt{1 - x^2}\right)$ a one to one function?
I've always knew inverse trig functions to be one to one on a specific range. But from the graph of the above inverse ...
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1
answer
47
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Let $f(x)=x^2+3x-3,x\gt0.$ If $n$ points $x_1,x_2,...,x_n$ are chosen on the x-axis, evaluate $\frac{x_1+x_2+...+x_n}n$
Question:
Let $f(x)=x^2+3x-3,x\gt0.$ If $n$ points $x_1,x_2,...,x_n$ are so chosen on the x-axis such that
(i) $\frac1n\sum_{i=1}^nf^{-1}(x_i)=f(\frac1n\sum_{i=1}^nx_i)$
(ii) $\sum_{i=1}^nf^{-1}(x_i)=\...
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Implicit Inverse function Theorems on $\mathbb R^2$ and injective functions
I have the following question on the applications of Implicit Inverse function Theorems on $\mathbb R^2$ and injective functions:
Let $U\subset\mathbb{R}^{2}$ be an open set containing $(0,0)$ and ...
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Is there a single-valued function linear in $\sin(x)$ and $\cos(x)$ that is invertible on $[-\pi, \pi)$ or $[0,2\pi)$?
What about $\sin(x/2)$ and $\cos(x/2)$? My intuition is no, for both, because $f = \sin(x)+\cos(x)$, $f = \sin(x)-\cos(x)$, $f = \sin(x/2)+\cos(x/2)$, and $f = \sin(x/2)-\cos(x/2)$ all don't do the ...
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0
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Inverting the sum of two inverses of integrals of strictly positive functions
I have a function $M: [0,1] \rightarrow R_{\geq 0}$ defined as
$M(p)=\frac{F_1^{-1}(p) +F_2^{-1}(p)}{2}$,
with $F_1(t)=\int_0^t f_1(\zeta) d\zeta$ for some continuous $f_1: R_{\geq 0} \rightarrow [0,1]...
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0
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46
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Are these two series of the arctangent function the same?
I came across the series axpansion of the arctangent function shown here:
$$
\tag{1}\arctan(x)=\sum_{k=0}^\infty\frac{4^kx^{2k+1}}{(2k+1)\binom{2k}{k}\left(1+x^2\right)^{k+1}}.
$$
Is it possible to ...
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2
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Proving $\arctan(\sinh(x))= 2\arctan(e^x) - \frac{\pi}{2}$ [closed]
I’m attempting to prove this:
$$
f(x) = \arctan(e^x) \\
g(x) = \arctan(\sinh(x)) \\
\forall x \in \mathbb{R} \\
g(x) = 2f(x) - \frac{\pi}{2}
$$
Taking the tan of both sides, yields good results. ...
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