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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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1answer
9 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. $\...
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2answers
72 views

Inverse of $y = 0.595x^2 + 0.387x^3$

[Edit] Simplify to: $$y = x^2 + x^3$$ Steps for solving an inverse would be to trade x and y (or f(x)) and then solve for y... can you do that here with x and y both still present in the equation? ...
2
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1answer
98 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
46 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
64 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
13 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
40 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
45 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
40 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
42 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
38 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
74 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
59 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
35 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
16 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]
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1answer
115 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{...
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0answers
37 views

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)...
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1answer
22 views

Prove that $(\pi_1)^{−1}(A) = A × Y$ and $(\pi_2)^{−1}(B) = X × B.$

Let $X$ and $Y$ be sets, let $A \subset X$ and $B \subset Y$ be subsets and let $\pi_1: X\times Y \to X$ and $\pi_2: X\times Y \to Y$ be projection maps. Prove that $(\pi_1)^{−1}(A) = A \times Y$ and $...
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0answers
29 views

How to invert this type of infinite series?

If have a function $f$ given by a series $$ f(z) = \sum_{n,m = 0}^\infty u_{n,m} z^{n + m t} $$ for some $t\in\mathbb{R}^+$. Is there an straightforward way (something similar to the inversion ...
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3answers
21 views

Inverse of a sum of functions

If $h(x) = f(x) + g(x)$, what is $h^{-1}(x)$ in terms of $f^{-1}(x)$ and $g^{-1}(x)$ ? Also, what are other useful inverse identities that you can give me? I know the basics like $(f(g(x)))^{-1} = g^{...
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1answer
56 views

How is the inverse of $y=4x^3 - 3x^4$ found?

I would like to calculate the inverse of $y = 4x^3 - 3x^4$ on the domain $x = [0,1]$. What would be the best way to tackle this? I'd preferably a general method, suitable for tackling other ...
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2answers
24 views

Similarity between graphs of sin and tan inverse

Why is it that the graphs of tan inverse and sin in the interval $$\left[-\frac \pi 2 , \frac \pi 2\right]$$ are so similar. Is it just some coincidence or something deeper?
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0answers
38 views

Equation of an Inverse of a function

I have a question about how to obtain an equation of an inverse function. The equation is $y=x^2+2x+3$ and I changed $x$ and $y$ to get $x=y^2+2y+3$ and it is only a matter of rearranging to have $y$ ...
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1answer
20 views

Invert a function containing integration term

Consider this simple equation $$ \tau(t)=\int\frac{dt}{a(t)},$$ where $\tau$ and $a$ are functions of $t$. Now, from this equation, how can I calculate $a(\tau)$ ?
3
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2answers
69 views

All definite integrals evaluate to 0 using periodic functions. [duplicate]

I know that my reasoning is incorrect, I just don't know where I went wrong. I did discuss this with my Maths teacher, and even she could not find what I did wrong. Let us begin by assuming a ...
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1answer
55 views

A few questions regarding the function $f(x) = x+\exp(x)\cdot \log(x)$

The function $f(x) = x+\exp(x)\log(x)$ occurs prominently at Lagarias inequality: $\sigma(n) \le H_n + \exp(H_n)\log(H_n)$ where $\sigma(n)$ is the sum of divisors, and $H_n$ is the n-th harmonic ...
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0answers
13 views

Factorization of Square-integrable random-variables and Generalized Inverses

Suppose that $X,Y,Z \in L^2(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^d)$, are $d$-dimensional random-vectors and there exists functions $f,g\in L^2(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),Law(X);\...
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0answers
36 views

Prove that there is a point $c\in(a,b)$ such that $f'(c)=0$.

Let $f:[a,b]\rightarrow\mathbb{R}$, $0<a<b$, a function that is differentiable and bijective such that $\int_{f(a)}^{f(b)}f^{-1}(x)dx=0$. Prove that there is a point $c\in(a,b)$ such that $f'(c)=...
1
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1answer
49 views

Finding the inverse of this isomorphic function.

I need to find the the inverse of the isomorphic function $f:\Bbb R^3 \rightarrow \Bbb R^3$ given by $\begin{pmatrix}a\\b\\c\end{pmatrix} \rightarrow \begin{pmatrix}3b-a\\3a+c\\3b-c\end{pmatrix}$ ...
0
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1answer
15 views

Showing inversion function for invertible matrices is differentiable

Consider the inversion function $f:GL_n( \mathbb{R}) \rightarrow GL_n (\mathbb{R})$ , $f(X)=X^{-1}.$ Where $GL_n( \mathbb{R})$ denotes the set of invertible $ n \times n$ matrices over the reals. The ...
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5answers
115 views

Showing that $ \int_0^{\pi/4} \tan x \;dx + \int_0^1\arctan x \;dx=\frac{\pi}{4} $

I would be glad if somebody pointed me in the right direction. The problem is stated as follows: Show that $$ \int_0^{\pi/4} \tan x \;dx + \int_0^1\arctan x \;dx=\frac{\pi}{4} $$
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0answers
43 views

Inverse Fourier Transform involving Gamma

My Question: How do I complete the inverse Fourier Transform of: $\displaystyle \int_{-\infty}^\infty F(\omega)e^{-k\omega^2t}e^{-\gamma t}e^{-i\omega x}\,d\omega$ I cant figure out quite how to ...
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0answers
27 views

Problem inverting the Callendar-Van Dusen equation

I am currently using the Callendar-Van Dusen Equation to solve for R(Resistance) when I input T(Temperature, in °C). $$R(T) = R(0)(1 + AT + BT^2 + (T-100)CT^3)$$ I want to invert this equation so I ...
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0answers
16 views

How to find the inverse or a tight bound on a series

If $ f(x)=1-\frac{4}{\pi}\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}e^{-\frac{\pi^2 (2k+1)^2}{8 x^2}}$, find $\min\{x:f(x)\geq 1-y \}$. The function $f(x)$ is increasing and its output falls in $[0,1]$...
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0answers
13 views

Conditions for inverse to be continuous in parameters

Let $f(x,\theta,\omega):X \ \times \ \Theta \ \times \ \Omega \to \mathbb{R}$, where $X\subset\mathbb{R}, \ \Theta\subset\mathbb{R}, \text{and} \ \Omega\subset\mathbb{R}_{++}$ are intervals, be ...
1
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1answer
54 views

Limits of the inverse function

Consider a real function $f$ which is defined, continuous and strictly monotonic on a given interval $I$, yielding an inverse $f^{-1}:f(I) \rightarrow I$. Can one assert that for any $a$ and $b$ in ...
4
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3answers
386 views

Limit involving inverse functions

When I am given the limit $$\lim\limits_{x \rightarrow \infty}\frac{x\arctan\sqrt{x^2 +1}}{\sqrt{x^2+1}}$$ would it be possible to evaluate it giving some substitution? L'Hospital's rule seemed an ...
3
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1answer
67 views

Approximate $\int_0^x e^{f^{-1}(t)} \; dt$ from an approximation of $f$

I am wondering if it is possible to have an approximation of this integral $\int_0^x e^{f^{-1}(t)} \; dt$ I have only an approximation of $f$: $f(\frac{i}{n}),\; i=0, \dots, n$? Many thanks, Peter....
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2answers
43 views

Inverse trigonometric functions without calculator, arcsin, arctan

How can I evaluate this expression without the use of a calculator and only assuming i know the standard angles ($\pi/3, \pi/4, \pi/6$)? \begin{align*} \arcsin\left(\frac{5\sqrt{3}}{14}\right)-\...
2
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2answers
72 views

What is the inverse operation of a gradient?

I notice that the function $$f(x,y,x;a,b,c) = ke^{-a/x-b/y-c/z}$$ has partial derivatives $$\nabla f = \begin{bmatrix} \partial f / \partial x \\ \partial f / \partial y \\ \partial f / \partial ...
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1answer
28 views

inverse of the function whose exponential value is different

$f(x)=x^2+2 $ $f^{-1}(x)= \sqrt {x-2} $ $g(x)=x^7+x^3+2$ $g^{-1}(x)= ?$ and for more $x^y$? for example: $f(x)=x^{11}+x^8+x^7+x^4+x^3+x+7$
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1answer
70 views

Are there any equation that could produce a monotonic smooth step function by parameter

I want to write a mathematic formular that, given any number of monotonic arbitrary point, it will produce a monotonic smooth step function Such as a figure below, I give it 2 point (the intersect ...
0
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0answers
26 views

If P(x)=Q(R(x)) and $P(x)=(\log_3 x)^2$ what is R(x)?

The full problem is here. I am seriously lost on how I should approach this question. Thanks in advance.
2
votes
1answer
41 views

Write the Taylor expansion of order $2$ at $x=0$ of $h(x)=g^{-1}(x+\sin(x))$, for $g(x)=x\ln(2+x^2)$

Can anyone tell me whether I carried out properly this exercise and where are mistakes? Thank you. Let be $g: \mathbb{R} \to \mathbb{R}$ the function defined by: $$g(x)\,=\,x\ln(2+x^2)$$ Show that $...
0
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2answers
72 views

finding the inverse function $f(x) = x^5+ x^3 + x$, then find what is $f^{-1}(3)$

if $f(x) = x^5+ x^3 + x$, then $f^{-1}(3) = ?$ I TRIED to do it, and I got this answer : 3/91 . I don't know if it is correct or not? I tried to do this work like this: $$ y = x^5+x^3+x\\ y = x(x^4+...
4
votes
2answers
80 views

How do I know whether the inverse function has a closed form?

I am interested in the function $$ y(x) := \left( x +\frac{3\pi}{2} \right) \sin(x) + \cos(x). $$ Over the range $ x \in \left[ -\frac{\pi}{2} ,\frac{\pi}{2} \right]$, this function grows ...