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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.

3
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3answers
366 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
votes
1answer
48 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....
1
vote
1answer
29 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
votes
2answers
64 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
27 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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0answers
23 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
25 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
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1answer
40 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
votes
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
65 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 ...
2
votes
2answers
37 views

Inverse function of $x-\lfloor x \rfloor $ and $(x-\lfloor x \rfloor)^2$

I need to find the inverse of these two functions if they exist: $$f_1 = x-⌊x⌋, 1\leq x<2, 0\leq f_1<1$$ and $$f_2 = (x-⌊x⌋)^2, 1\leq x<2, 0\leq f_2<1$$ I worked through it and I think ...
1
vote
1answer
27 views

If $z=\dfrac{(z_1+\bar{z}_2)z_1}{z_2\bar{z}_1}$ where $z_1=1+2i$ and $z_2=1-i$, then find $\arg(z)$

If $z=\dfrac{(z_1+\bar{z}_2)z_1}{z_2\bar{z}_1}$ where $z_1=1+2i$ and $z_2=1-i$, then find $\arg(z)$ My Attempt $$ z_1+\bar{z}_2=2+3i,\quad(z_1+\bar{z}_2)z_1=(2+3i)(1+2i)=-4+7i\\ z_2\bar{z}_1=(1-i)(1-...
1
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1answer
29 views

Find the domain of the trig function

Find the domain of the function. $f(x)=({2+\cos 2x-\sin2x*\tan x})$^1/2 $2+\cos2x-\sin2x*\tan x≥0$ Simplifying $-4\sin^2x+3≥0$ $\sin^2x≤3/4$ $-3/4≤\sin x≤3/4$ $\arcsin(-3/4)≤x≤\arcsin(3/4)$ My ...
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1answer
34 views

Need some help with trigonometry. Inverse functions.

Can someone help me to solve these examples? Evaluate: $$A= \text{arccot}\bigl(\tan\biggl[\frac{4\pi}{5}\biggr]\bigr)\;\; \text{and}\;\; B=\sin\bigl(\arctan\biggr[\frac{-3}{7}\biggr]\bigr)$$
1
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1answer
50 views

Double-precision algorithm for inverse log gamma or log factorial?

Question in a nutshell: Can anyone point me to an algorithm for computing to double-precision floating-point (roughly 16 digits) the inverse of either log gamma or log factorial? In other words, if ...
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0answers
17 views

Definition of the functional inverse

In the context of the calculus of variations, I have seen the following used to define $V^{-1}$ as the "functional inverse" of a functional $V$: $$ \delta(x-y) = \int V(x,t|f)V^{-1}(t,y|f) dt \tag{1} ,...
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2answers
43 views

About Theorem 4.17 on p.90 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

On p.90 in "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin. Theorem 4.17 Suppose $f$ is a continuous 1-1 mapping of a compact metric space $X$ onto a metric space $Y$. Then ...
1
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1answer
37 views

Looking for an explanation or insight on a form of the inverse of a restricted gamma function

I posted this yesterday asking about how to find an inverse of a restricted gamma function. To put it concisely, I was looking at the gamma function in the positive reals after restricting it to be ...
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2answers
36 views

Finding an Inverse of Restricted Gamma Function

I don't know/haven't used LaTeX yet but I'll do my best to keep it simple, I'm working on my undergrad senior project and I'm trying to find an inverse function for f(x)=(x-1)! just in the positive ...
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1answer
23 views

How do you find a and b in a ln function from (x1, y1) and (x2, y2)?

Like I was looking at one answer from JohnD https://math.stackexchange.com/users/52893/johnd and he answered that to find a and b of a ln function, you have to use a=y1−y2ln(x1/x2), b=exp(y2ln(x1)−...
0
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1answer
44 views

How to find the inverse of a function numerically

This is an extension of my previous question posted in here Inverse of a function of a 3rd order Now, I have another one which seems to be more complicated. I don't know how to solve them numerically....
0
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1answer
24 views

Calculating inverse - what about assumptions?

Can someone verify this and answer my questions? I've chosen simple function on purpose. I've also added my paper from an exam at the very end. Find inverse function $f^{-1}$ to function $f(x)= \frac{...
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0answers
47 views

How to find the inverse function

For this function: $f(x)=x\exp(ax)$ what is the inverse function $f^{-1}(\cdot)$?
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0answers
24 views

Relationship between inverse function, origin function and its domain - with an example

I am having problem with proper math-fashioned style of solving this task. It's task from my exam and apparently I didn't do well enough to get 2.5 out of 5 points. Find inverse function $f^{-1}$to ...
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0answers
18 views

Finding the inverse of an incomplete beta function

Is there a rigorous way of inverting $$\rho(r)=\frac{b_{0}}{1-q}B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$ where $B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$ is an ...
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0answers
14 views

Asymptotics of gauss hypergeometric function

My problem is focused on obtaining $r(\rho)$ which is the inverse of the $\rho$ given below. Along the way, I will be integrating a differential equation containing $r(\rho)$ in the regime $\rho>&...
0
votes
1answer
24 views

Inverse of power-2 rational function

I have a function $f(a,b) = \frac{ab}{(\frac{a+b}{2})^2}$, and (to me) it has some cool properties (e.g $f(a,b) = f(b,a)$, $f(x,0) = 0$, $f(x, x) = 1$, $0 \leq f \leq 1$, etc.). Now I wanted to know ...
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3answers
351 views

Algebra - find the formula for the inverse

I have the function $$y=\frac12\ln\left(\frac{1+x}{1-x}\right)$$ I have to find the inverse function. I know that $$e^{2y}=\frac{1+x}{1-x}$$ How do I solve this equation for $x$?
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2answers
63 views

Calculating exact values of “weird” functions like arcsin(sin 100)

This is pretty much the last thing I need to know for now. Tasks (calculate): $\arccos{(\cos{12})}$ $\arctan{(\tan{\sqrt{5}})}$ $\arcsin{(\sin{100})}$ Answers: $4\pi-12$ $\sqrt{5}-\pi$ $100-32\pi$ ...
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0answers
8 views

Inverting a function from asymptotic expansion

Can I invert the following functions to obtain $r(\rho)$? $\rho=r+a+b r^{q}$, where $q<0$ $\rho=cr+dr^{q}$, where $q>0$ $\rho=r+\frac{b_{0}}{2}\left(-1+\ln[2]-\ln[b_{0}/2]-\ln[1/r]\right)$ ...
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5answers
40 views

How to solve inverse trig. equations like sin(arctan 2)?

These are some of the tasks I am supposed to be prepared for. I have no idea where to begin when solving them. Below I present what I already know regarding the subject and what I have problems with. ...
0
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1answer
29 views

Generalized inverse function for $Y=\min(X, b)$

$X$ is a continuous and non-negative random valuable with CDF $F$, $b>0$, what is the generalized inverse function for $Y=\min(X, b)$
2
votes
1answer
29 views

Inverse of a function of a 3rd order

can someone help me how to find the inverse the following function? $$z(\zeta)=\frac{1}{\zeta}+m_1\zeta+m_2\zeta^2+m_3\zeta^3$$ In my case, $z$ is a complex number and cannot be zero. And $m_k$ is a ...
0
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1answer
25 views

Finding the inverse of $\phi_{\lambda}^2$ where $\phi_{\lambda}$ is the Mobius transform on $\mathbb{D}$

Let $\mathbb{D}$ denote the open unit disk. Fix $\lambda \in \mathbb{D}$. Define the Mobius transform $\phi_{\lambda}:\mathbb{D}\rightarrow\mathbb{D}$ by $$\phi_{\lambda}(z) = \frac{z-\lambda}{1-\...
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0answers
73 views

First Order Nonlinear PDE - Discrete Ray Method

In the application of the discrete ray method to the mixed differential-difference equation below , as $n \to \infty $: $G'_n(x) = e^{-\frac{x^{2}}{2\,n\,(n+1)}}\,G_{n-1}(x)$, $G_{0}(x)=1$ and ...
0
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1answer
31 views

Closed-form solution for $f(x)/x=y$ using $f^{-1}$

I'm programming a piece of math that requires solving an equation of a form $f(x)/x=y$. Now I already have $f^{-1}(z)$ coded (efficiently, and not by me) so I'd prefer using this implementation ...
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1answer
22 views

Inverse function theorem to prove onto

Let $f: \mathbb R^2 \to \mathbb R^2$ be $C^1$, $D_f(x)$ is invertible everywhere, and $\lim_{|x| \to \infty}|f(x)| = \infty$ Show that $\min_{x \in \mathbb R^2}|f(x)-a|$ exists, and that $f$ is onto. ...
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2answers
67 views

Finding $\phi^{-1}({id_{P(\mathbb{N})}})$, where $\phi:(\mathbb{N}\to\mathbb{N})\to(P(\mathbb{N})\to P(\mathbb{N}))$ and $\phi(f)(A) = f^{-1}(A)$

I have function $\varphi : (\mathbb{N} \to \mathbb{N}) \to (P(\mathbb{N}) \to P(\mathbb{N}))$ and $\varphi(f)(A) = f^{-1}(A)$. I have to find $\varphi^{-1}({id_{P(\mathbb{N})}})$, and I don't have ...
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2answers
62 views

How does one interpret $\dfrac{dx}{dy}$ for a function which isn't invertible?

I was just going through the proof of derivative of inverse functions. The statement reads: If $y= f(x)$ is a differentiable function of $x$ such that it's inverse $x=f^{-1}(y)$ exists, then $x$ ...
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1answer
32 views

on the inverse of trigonometric or/ and hyperbolic functions

If we want to find, say, the inverse $\tan$ function, $\tan^{-1}$, in terms of (complex) logarithm function we start with the equation $z=\tan w =\frac{\sin w}{\cos w}=\frac{1}{i}\frac{e^{iw}-e^{-iw}}{...
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1answer
26 views

How to find the inverse of the method to find the nth lexicographic logical permutation

I'm looking for a proper method to find the nth logical permutation of a particular sequence that is able to return the desired permutation. For example, for a sequence 1234567890, the 100,001st ...
0
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3answers
47 views

Can't figure out how $\sin\left(\tan^{-1}(x)\right)=\frac{x}{\sqrt {x^2+1}}$

Simplify the expression $$\sin\left(\tan^{-1}(x)\right)$$ Using a triangle with an angle $\theta$, opposite is x and adjacent is 1 meaning the hypo. is ${\sqrt {x^2+1}}$ Now because the problem has ...
0
votes
0answers
52 views

Correction terms in the asymptotics of hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ $($which is the inverse of $\rho$ below$)$, $$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\...
3
votes
1answer
44 views

If $g$ is the inverse of function $f$ and $f'(x)= \frac{1}{1+x^n}$, Find $g'(x)$

I tried the question and got an answer by the following steps: $f(g(x))=x$ Differentiating both sides w.r.t to $x$, we get $f'(g(x)).g'(x)=1$ And therefore, $g'(x)=1+\left[{g(x)} \right]^n$ Now ...
1
vote
1answer
33 views

Semi-Prime Number Originator Lookup For Identifying Prime Numbers

In one of BBC's documentaries on security, a mathematician stated that prime and semi prime numbers helped with the design of RSA encryption - a number like 91 can be easily calculated through ...
0
votes
3answers
76 views

Solve: $\arctan(2x)+\arctan(3x) = \frac{\pi}{4}$

Solve: $\arctan(2x)+\arctan(3x) = \frac{\pi}{4}$ I started by applying tan on both sides, $\frac{2x+3x}{1-2x\times3x}=\tan\frac{\pi}{4}$ This yields $x=\frac16,$ $x=-1$ But -1 doesn't satisfy the ...
0
votes
1answer
65 views

What is x in terms of p?

In this equation $$\frac x{\ln(x)} = p$$ How can I have $x$ as a function of $p$?
0
votes
1answer
28 views

Inverting the maximum of a function? Is this solvable analytically [= by hand?]

Define $$f_a(x) = ax - \log \left[ \frac{x}{5(1-x)} + 1\right], \ x < 1,$$ for some constant $a > 0$. Let $x'$ be the maximum-point of this function, and $f_a(x')$ the max-value. Clearly, ...
0
votes
0answers
17 views

Inverse of a trace based function?

I have the following linear operator: and I need to compute its inverse operator for y: vector of real values. The trace function is invertible as far as I know, so is this even possible? and if so,...
0
votes
1answer
45 views

Find the Range and Inverse of the function $f(x)=\sec x\tan x+\sec^2 x$

Find the Range and Inverse of the function $f(x)=\sec x\tan x+\sec^2 x$ My try: We have $$y=\sec x\left(\sec x+\tan x\right)$$ $$2y=\left(\sec x+\tan x+\sec x-\tan x\right)\left(\sec x+\tan x\...