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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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Find the inverse of the function (decoding)

Here is my problem: In cryptography, inverse functions are used by government agencies and other businesses to encode and decode information. If the encoding function is $f(x) = \sqrt{2x-3}+4$, ...
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42 views

Right inverse of the Poisson CDF

For an exercise, I have to find the right inverse of the Poisson CDF $$F_X = e^{-\lambda} \sum_{i=1}^{\lfloor k\rfloor }\frac{\lambda^i}{i!}$$ where the right inverse is: $$F_X^{-1}(p):=\text{inf}\{x ...
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5answers
72 views

Finding an inverse function (sum of non-integer powers)

I have a function: $$f(x)=x^{2.2} + (1-x)^{2.2}$$ It is defined on the interval $[0,1]$. Minimum: $x=0.5, y=2*0.5^{2.2} = 2^{-1.2}$. I want to find an inverse for it. Since the function has two "...
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0answers
24 views

Help with concepts with the inverse function theorem

I am solving the following problems: Be the transformation $T \in R ^ 2 $: by $T(u, v) = (u ^ 2-v ^ 2, uv) = (x, y)$. Calculate the Jacobian of $T$ and conclude whether the transformation is ...
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3answers
39 views

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$ So in order to show injectivity $f(a)=f(b) \Rightarrow a=b$ so $\frac{a}{1-|a|}=\frac{b}{1-|b|}$. But how do I prove that? ...
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2answers
43 views

Number of solution of the equation $\cot^{-1}{\sqrt{4-x^2}+ \cos^{-1}{(x^2-5)}}=3π/2$

Number of solution of the equation $ \cot^{-1}{\sqrt{4-x^2} + \cos^{-1}{(x^2-5)}}=3\pi/2$ $$ \cot^{-1}{\sqrt{4-x^2}+ \cos^{-1}{(x^2-5)}}={3π/2}$$ Taking sine both side and solving this is I get $$1 +\...
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0answers
18 views

Finding the inverse of a complex function to derive information about it.

Im sorry if im butchering some of this concepts, i am a cs undergrad and have very little understanding of advanced mathematics. I found an intresting function i quite like, such function is : -y = ...
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0answers
33 views

Is there any formula for arctan x + arctan y + arctan z. ? ( For various case)

Is there any formula for $ arctan x + arctan y + arctan z $ ? One way to solve such type of question is to is two take two at a time and solve them. I know the general formula for $ tarctan x + ...
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2answers
42 views

Why isn’t the integral of $x^2$ from $0$ to $5$ equal to the integral of $\sqrt x$ from $0$ to $25$? [duplicate]

My calculator says that the former is 41.67, and that the latter is 83.34. Why is that? Shouldn’t they be equal?
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1answer
37 views

Image of an interior point

Consider $\textbf{f}: U\subset \mathbb{R}^n\to \mathbb{R}^n$ and $\textbf{a} \in U$. Suppose that $\textbf{a}$ is an interior point of $U$ and $\textbf{f}$ is differentiable at $\textbf{a}$ with $\det(...
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0answers
12 views

Inverse of Legendre Dual

Reading the book "Concentration inequalities A nonasymptotic theory of independence" I came across the following results (Lemma 2.4): Let $f$ be a convex function such that $f(0)=f'(0)=0$ and such ...
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1answer
20 views

unexpected case of gradient of inverse function

if $y=x^3+3x+2$ is the original equation then, $$\frac{dy}{dx} = 3x^2+3$$ So the gradient of the inverse function atc $x=2$ should be $$\frac{dx}{dy}= \frac{1}{3x^2+3}$$ This gives me answer of $\...
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30 views

How do I draw the graphs of $\sin^{-1}(\frac{2x}{1+x^2})$ and $\tan^{-1}(\frac{2x}{1-x^2})$?

How do I draw the graphs of $\sin^{-1}(\frac{2x}{1+x^2})$ and $\tan^{-1}(\frac{2x}{1-x^2})$? The solution of a question in my books how these graphs and draws the conclusion that from them it is seen $...
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4answers
25 views

If $x\in [-1,0)$, then what is the value of $\cos^{-1} (2x^2-1) - 2 \sin^{-1} x$?

If x belongs to $[-1,0)$ then what is $\cos^{-1} (2x^2-1) - 2 \sin^{-1} (x) $? This is a question in my book, but I have doubts about solving it. I tried putting $\sin a = x$ and got $\pi - 2a - 2a = \...
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4answers
61 views

Differentiating $\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$

Find the derivative with respect to $x$ of $$\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$$ The partial solution to this problem is given as follows: $y=\left(\frac{x}{\sqrt{1-x^2}}\right)$ Then: ...
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2answers
33 views

Function inverses

From the definition of a inverse standpoint ($f^{-1}(f(x))=f(f^{-1}(x))=x$), why does interchanging variables ($x$ and $y$) work to find the inverse? It seems logical to me but I cannot come up with a ...
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1answer
26 views

Real analysis : Problem related to inverse

Let $f:[0,\infty)\rightarrow [0,\infty) $ be a continuous strictly increasing function and $g=f^{-1}$. Let $a,b>0$ and $a\neq b$. Then how $\displaystyle\int_{0}^af(x)dx+\int_0^bg(y)dy\geq ab.$ Any ...
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0answers
31 views

Inverse function to the exponential integral

$Ei(-ln(f(x)) = c + ln(ln(x))$ where $Ei(x)= $ - $\int_{-x}^{\infty} \frac{e^{-t}}{t} dt $, where $x $ be a non zero real (called exponential integral ). Is there a way to express $f(x)$? If yes, ...
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2answers
72 views

Solve $2x^2-5x+2=$ $\frac{5-\sqrt{9+8x}}{4}$

Solve $2x^2-5x+2$= $\frac{5-\sqrt{9+8x}}{4}$ I simply do square both sides solve it and I get two value of x one is 2 and other is $\frac{3-√5}{2}$ but this approach it take more time so is there any ...
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3answers
74 views

Solution of $\tan^{-1} \sqrt{x^2 + x} +\sin^{-1} \sqrt{x^2 + x + 1} = \pi/2$

My attempt at question $\tan^{-1} \sqrt{x^2 + x} +\sin^{-1} \sqrt{x^2 + x + 1} = \pi/2$ using the identity $\tan^{-1}(\sqrt{x^2 + x} + \sqrt{x^2+x+1/-x^2-x}/[1- \sqrt{x^2+x)(x^2+x+1/-x^2-x)}]$=π/2 $...
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2answers
39 views

How do I solve $y=x+B\sin(x+A)$ for $x$

I have a code that converts x into y using the formula: $y=x+B\sin(x+A)$ with $x, A$ and $B$ known values. $B$ is also very small so that $B\sin(x+A) < 0.035$. The problem is that in another ...
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1answer
44 views

Differentiation of inverse function

I know that first order differentiation of inverse of a function $f (x)$ is reciprocal of $f'(f^-1(x)) $. But I'm unable to evaluate the integration given in the question.
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1answer
33 views

Implicit equation $\ln(\frac{x}{y})-y=1$ to rectangular equation not in terms of $W(x)$

Backstory and Other Info I'm not sure if this is possible, I'm currently a precalculus student and have a very limited understanding of much of any of this. However, I do like to go on WolframAlpha ...
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2answers
72 views

Proving $G(s,t)=(\cosh s \cdot \cos t, \sinh s \cdot \sin t)$ has global inverse in a certain region

Let $G(s,t)=(\cosh s \cdot \cos t, \sinh s \cdot \sin t)$, and let $A=\{(s,t)\in\mathbb{R^2}|s>0, 0<t<2\pi\}$. Prove that $G_{|A}$ has global inverse and give a graphical representation of $...
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2answers
25 views

Global inverse of $G(x,y)=(\ln (xy),\frac{1}{x^2+y^2})$

Let $G(x,y)=(\ln (xy),\frac{1}{x^2+y^2})$, with Dom$(G)=\{(x,y)\in\mathbb{R^2:0<y<x}\}$. I am asked to find the global inverse of $G$. I have tried to proceed in the following way: $$u=\ln(xy)=...
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2answers
33 views

If $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$ then what is the value of $g'(2)$?

I know that $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$. But how can I find the value of $g'(2)$?
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0answers
9 views

Preimage Structures: lists of all inverse images at a given depth (reference request)

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, the j-th such list would be ...
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1answer
34 views

Inverse Fourier transform of $F(k) = 1/(k^2+a^2), a>0$

I need help finding the Inverse Fourier transform of: $$F(k) = \frac1{ k^2 + a^2 },~ a>0$$ Here is what I have so far: Singular points at $k^2 = a^2$, namely, at $k = \pm ia$. The inverse ...
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1answer
41 views

Output response from closed loop transfer function using MATLAB

This transfer function is to control the position of Permanent Magnet DC motor. I was able to get the transfer function and now I need to analyze the output for the tuned closed loop for a given input ...
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0answers
27 views

Find $F ^{−1} (u)$, for $u ∈ (0, 1)$ by solving the equation $u = F(x)$ for $x$

I am working through a problem. I began with the pdf: $f(x)=\frac{1}{2}\alpha*e^{-\alpha|x|}$, where $\alpha > 0$ and $x ∈ (−∞,∞)$. I found the cdf to be: $$F(x) = \begin{cases} \frac{1}{2}e^{\...
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1answer
11 views

Derivate of inverse of composite function

I'm very confused, and this is probably a stupid question. I want to calculate $ \frac{d}{dx} f^{-1}(g^{-1}(x))$. However, I get two seemingly different results taking two different approaches. I. $\...
2
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1answer
101 views

Inverse of $\frac{\sin(x)}{x}$

How would one find the inverse of the function $y=\frac{\sin(x)}{x}$? Here are my steps: $y=\frac{\sin(x)}{x}$, $x=\frac{\sin(y)}{y}$, $xy=\sin(y)$, $\arcsin(xy)=y$, After that step, I can’t find a ...
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2answers
50 views

Prove that $f(f^{-1}(V))=f^{-1}(f(V))$

Let $f: X\to Y$ be bijective, and $f^{-1}: Y\to X$ be it's inverse. If $V\subseteq Y$, show that the forward image of $V$ under $f^{-1}$ is the same set as the inverse image of $V$ under $f$. I ...
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4answers
71 views

If $\tan 9\theta = 3/4$, then find the value of $3\csc 3\theta - 4\sec 3\theta$.

If $\tan9\theta=\dfrac{3}{4}$, where $0<\theta<\dfrac{\pi}{18}$, then find the value of $3\csc 3\theta - 4\sec 3\theta$. My approach:- $$\begin{align*} \tan9\theta &=\frac{3}{4} \\[6pt] \...
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0answers
15 views

continuity of isomorphism of unit circle

I try to show $\mathbb{S}^1\cong[0,1)$, by the map $f(x) = (\cos2\pi x,\sin2\pi x)$, for $x\in[0,1)$. It's clear that $f$ is continuous and bijective. But I don't know how to show the inverse map $f^{-...
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1answer
41 views

Inverse function of $ax + bx^3$

I am trying to find the inverse of the function $y = f(x) = ax + bx^3$, i.e. $x = f^{-1}(y)$. (The equation arises in the modeling of a certain type of transmission used in robots) Looking at the ...
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1answer
49 views

is it possible to solve it for variable r?

Here is the annuitet payment formula: $$p = s \cdot \left(r + \frac{r}{(r+1)^t-1}\right)$$ Is it possible to solve it for rate ?
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0answers
60 views

Inverse of $f(x) = x^{3}-x^{2}$

Can anybody find the inverse of $f:(-1,0) \to \mathbb{R}$ such that $f(x) = x^{3} - x^{2}$ ?
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0answers
23 views

Continuity of a function on a metric space and its consequences

Let $f : X → Y$ be a given function, and suppose that $f^{-1}(C)$ is an open subset of $X$ whenever C is an open subset of $Y$ . (a) Prove that $f$ is continuous on $X$. (b) Prove that $f^{-1}(B)$ ...
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4answers
48 views

Showing that $\arcsin x + \arccos y = \frac{\pi}{2}$ if and only if $x = y$

I was just wondering about this identity: $$\arcsin x + \arccos x = \frac{\pi}{2} .$$ That a thought came to my mind that in general $$\arcsin x + \arccos y = \frac{\pi}{2} \qquad \textrm{if and ...
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1answer
43 views

How to show $12^a \cdot 18^b$ is injective

We are told that $f: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ I know one method is to prove that a inverse exists, but I'm not 100% sure how to do that in this case. so instead I decided ...
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0answers
45 views

Expected value of a non-linear function of a normal random variable

Consider a random variable $X\sim N\left(\mu,\sigma^2\right)$, and a monotonically increasing non-linear function of $X$, call it $Y=f\left(X\right)$, defined as: $$Y=f\left(X\right)=\Phi\Big(a-b\,\...
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1answer
75 views

An additional assumption to the inverse function theorem.

The theorem is given below: And here is the question: Could anyone give me a hint on how to prove the required in the question please?
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3answers
64 views

inverse of $y=\frac{x}{\log{x}}$?

By Prime Number theorem $\pi(x)=\frac{x}{\log{x}}$ for large x Putting $x=p_n$ where $p_n$ denotes $n^{th}$ prime number, We have, $\pi(p_n)=\frac{p_n}{\log{p_n}}$, $\because \pi(p_n)=n$, $\...
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1answer
30 views

Continuity and Uniform Continuity of inverse functions

Let $f : X → Y$ be a given function, and suppose that $f^{-1}(C)$ is an open subset of $X$ whenever C is an open subset of $Y$ . (a) Prove that f is continuous on $X$. (b) Prove that $f^{-1}(B)$ is ...
1
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1answer
37 views

Inverse of a piecewise continuous function.

I am trying to find the inverse of a two dimensional map $f\left(\begin{bmatrix} x\\y\end{bmatrix}\right)$, For example $$ f\left(\begin{bmatrix}x\\y \end{bmatrix}\right) = \begin{cases} ax + by, &...
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1answer
19 views

digamma function inverse and special value

What the inverse of the digamma function?, and how can I write the x for $$ ψ(x)=1$$ and $$1<x$$ $[x ≈ 3.20317146837693106929448152]$ as irrasional number [not a new one a familiar old one]
4
votes
1answer
117 views

Existence of a particular inverse transformation

Let $h : \mathbb{R}^D \rightarrow \mathbb{R}^d$, where $d < D$, be a differentiable function. I would like to find minimal conditions under which there exists a differentiable function $g : \mathbb{...
3
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2answers
110 views
+50

Inverse/Reverse of Number of Permutations and of Number of Combinations with Repetitions?

For an engineering application, I need the inverse functions of the computations of the number of combinations and permutations. In the thread How to reverse the $n$ choose $k$ formula? it shows how ...
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2answers
42 views

An equivalence between function $f(x)$ and $f^{-1}(x)$.

Recently on this answer to one of my questions user farruhota replied that Alternatively, note the property of inverse function: $$f(f^{-1}(x))=f^{-1}(f(x))=x$$ Hence: $$f(f(x))=x \iff f(x)...