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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Multiplying by $1$ adds a solution to an equation

I have a question, which is motivated by my book's solution to finding the inverse function of $f(x)=\frac{x}{1-x^2}$ with the domain of $f(x)$ restricted the open interval $(-1,1)$. The questions are ...
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summation of infinite terms contradiction

Consider the sum $$\sum_{i=0}^\infty (\tan^{-1}{(i+1)} - \tan^{-1}(i))$$ if we write $n^{th}$ terms ($T_n$) $$\begin{align} T_0 &= \tan^{-1}(1) - \tan^{-1}(0)\\ T_1 &= \tan^{-1}(2) - \tan^{-1}(...
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3 votes
1 answer
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Why are algebraic equations sometimes displayed as inverses instead of solving for the variable in question?

As the title suggests, why are some algebraic equations displayed as inverses instead of equalling the direct variable we're trying to solve? For example, consider the d-spacing formulas used in ...
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Question on term of derived formula for $\log\zeta(s)$

The derived formula $$\log\zeta(s)=-\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n\in\mathbb{P}} \left(2 \tanh ^{-1}\left(1-2 n^s\right)-i \pi\right)\right),\quad s>1\tag{1}$$ ...
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When is reversion just a conjugaison by $z \mapsto iz$ or by $z \mapsto -z$?

The origin of the question is the computation of two integrals $$F(x) = \int_0^x \frac{\mathrm{d}y}{\cos(y)} \textrm{ for } |x|<\pi/2,$$ $$\tilde F(x) = \int_0^x \frac{\mathrm{d}y}{\cosh(y)} \...
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-1 votes
1 answer
21 views

Local invertibility of

Please check my understanding. (a) Let $f:\mathbb{R^2}\to\mathbb{R^2}$ defined by $F(x,y)=(x^2-y^2, 2xy)$. Calculate derivative matrix of $F$ and show $F$ is locally invertible except possibly at the ...
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finding the inverse of a function from $M_2(R).$

I am trying to prove that the given map below is a diffeomorphism and it is pretty clear to me that it is a bijection but I do not know how to show that the inverse of the given map is smooth? in fact ...
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Is my proof of $\neg (f(f(x)) = f^{-1}(x)) \land (f(x) = x) $ correct?

Theorem: $$ f(f(x)) = f^{-1}(x) \implies f(x) = x, \ \ f : \Bbb R \to \Bbb R $$ Proof: $f(x)$ is a real, continuous function that satisfies $f(f(x)) = f^{-1}(x)$. That means that either $f : [x,f(x)] ...
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Inverse of $\psi(t) = \frac{\nu}{c^2}h(\frac{ct}{\nu})$ with $h(x) = 1 + x - \sqrt{1+2x}$

Given the inverse of $$h(x) = 1 + x - \sqrt{1+2x},$$ i.e., $$h^{-1}(y)=y+\sqrt{2y},$$ I am trying to compute the inverse of $$\psi(t) = \frac{\nu}{c^2}h(\frac{ct}{\nu}).$$ My first step was to define $...
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Inverting a cubed root equation with two cube roots

In a recent StackOverflow question, I came across this formula: $$up = (1 + p)^\frac{2}3 - (1 - p)^\frac{2}3 $$ and the question required the poster ...
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Efficient way of determining inverse of conformal complex function numerically

I am wondering if there is an efficient method to compute the inverse of a complex function numerically. More precisely I have a certain amount of points $x_1,\ldots,x_n$ in the upper complex half ...
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Chapter 12 Spivak Calculus: Spivak's comments about implicit differentiation

There is a passage in the Chapter 12 exercises of Spivak's Calculus (which is a book specific to real-valued functions) that reads as follows: In general, determining on what intervals a ...
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Solve the equations: $\cos\left(z\right)=2+4i, \text{Log}\left(z\right)=\left(1+i\right)\pi, \arctan\left(z\right)=1+i$

Solve the following equations: $$ \cos\left(z\right)=2+4i$$ $$ \text{Log}\left(z\right)=\left(1+i\right)\pi$$ $$ \arctan\left(z\right)=1+i$$ For the first one we know that $$cos(z)=\cos\left(z\right)...
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If the symbol '$f^{-1}$' shows up as an assumption, can I automatically assert that $f$ is a function?

In Chapter 12 of Spivak's Calculus, the definition of the inverse of function $f$ is provided as: For any function $f$, the inverse of $f$, denoted by $f^{-1}$, is the set of all pairs $(a,b)$ for ...
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Is this function composed of non-invertible functions invertible?

Consider a nonlinear function $f:\mathbb{R}^3 \rightarrow \mathbb{R}^2$. Next define $g:\mathbb{R}^4 \rightarrow \mathbb{R}^4$ as per: \begin{align} (y_1, y_2, y_3, y_4) = g(x_1, x_2, x_3, x_4) = (f(...
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3 answers
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How to find the inverse function of the function $f$, defined as $f(x)=\frac{x}{1-x^2}$ for $-1 \lt x \lt 1$

I want to find the inverse function of the function $f$, defined as $f(x)=\frac{x}{1-x^2}$ for $-1 \lt x \lt 1$. Letting $y=f(x)$, we first note that $x=0 \iff y=0$. Next, we consider the case when $y ...
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$~\frac{\mathrm{d}}{\mathrm{dy}}\operatorname{arctg}\left(y\right)~$with integration where$~y~$is a function of$~x~$

$$A:=\int_{}^{}{1\over 4x^2+4x+2}\,\mathrm{dx}= \underbrace{{1\over 2}\operatorname{arctg}\left(2x+1\right)+\text{const}}_{\text{I want to derive this.} } \tag{1}$$ $$4x^2+4x+2=(2x+1)^2+1\tag{2}$$ $$\...
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Determining when a function intersects with another, $n$ times.

I would like to know if the following method of determining what missing variable of a function will yield to $n$ intersections of that function and another. #1 Demonstration of this method If $g(x)=\...
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6 votes
2 answers
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Why is the bizarre $f^{-1}(y)=y+\sum_{n=1}^\infty\frac{1}{n!}\frac{d^{n-1}}{dy^{n-1}}[y-f(y)]^n$ equivalent to the Lagrange inversion formula?

$\newcommand{\d}{\mathrm{d}}$EDIT: In the nontrivial example $f(x)=x-\frac{1}{4}x^3$ and using either of the two series to produce a result for $f^{-1}(3/4)$, I find that $(\ast)$ and $(1)$ produce ...
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Inverse Derivative: Error/Intuition

Problem is to find $[f^{-1}(4)]'$ given $f(x)=\frac{x^3+7}{2}$. Way One: Switch $x$ and $y$ to find inverse function. So, $x=\frac{y^3+7}{2}$. Therefore, $\sqrt[3]{2x-7}=y$ (i.e. our inverse function)....
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1 vote
1 answer
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Inverse of a rational complex function and order of zeroes

If we have a complex function $f(z)$ that can be written as $$ f(z) = \frac{P(z)}{Q(z)} $$ and $P(z)$ has a single zero $z = z_0$ of order $n > 1$, I've read in ''Conformal field theory'' by P. Di ...
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Prove that $``ff^{-1}(x)$ $=$ $x$ $=$ $f^{-1}f(x)"$ $\implies$ $``$the graph of $f$ and $f^{-1}$ are reflections of each other in the line $y = x"$.

According to the Cambridge International AS & A Level Pure Mathematics $1$ book $(2019$ edition, page $48)$, The graph of $f$ and $f^{-1}$ are reflections of each other in the line $y = x$. This ...
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Computing $\alpha$ such that $\mathbb{E}[(\tfrac{\alpha}{X + \alpha})^2] = t$, for $t \in (0, 1)$?

Suppose that $X$ is a Binomial random variable with parameters $n, p$. Let $$ F(\alpha) = \mathbb{E}\Big[\big(\frac{\alpha}{X + \alpha}\big)^2\Big], \quad \alpha > 0. $$ Clearly we have the limit ...
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1 vote
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Finding inverse of $g:\mathbb N × \mathbb N \to \mathbb N$

Let function $g:\mathbb N × \mathbb N \to \mathbb N$ $$g(x,y) = \frac{(x+y)(x+y+1)}{2}+y$$ I want to prove that $g$ is bijective. I tried to prove it is injective by contrapositive, but I had some ...
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2 Other Inverse Function of $f(x) = x^3+ax, \; a < 0$ for $x^2 \leq -\dfrac{4a^3}{27}$

For $a > 0$ the unique inverse of $f(x) = x^3+ax$ is $f^{-1}(x) = \dfrac{\sqrt[3]{9x+\sqrt{81x^2+12a^3}}+\sqrt[3]{9x-\sqrt{81x^2+12a^3}}}{\sqrt[3]{18}}$ But for $a < 0$ i know that this isn't ...
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0 votes
1 answer
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Inverse a cubic function

I've got this cubic function that I can't figure out how to calculate its inverse. $$f(x)=x^3+3x^2+3x, x\in \mathbb{R}$$ I've tried using online calculator to see if that would help, but none of them ...
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5 votes
1 answer
138 views

Inverse of a function of two variables with respect to one variable, and the curvature with respect to the other

Consider a function of two variables $f(x,y)$, where $x\in [0,\infty)$ and $y\in [0,1]$, which is a convex combination of two functions $f_a(x)$ and $f_b(x)$. Both, $f_a(x)$ and $f_b(x)$, are ...
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2 votes
1 answer
51 views

Tangents to a function and its inverse

I came across the following problem: Find all $x$ values such that the tangent to the function $f(x) = \frac{1}{x^2+1} + (1-2x)^{1/3}$ where $x \ge 0$ at that $x$ value is perpendicular to the tangent ...
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2 votes
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Finding a simple differential equation to define an inverse of $\,_2\text F_1(a,b;c;z)$ with respect to $z$ with the Gauss Hypergeometric function.

An “Inverse Gauss Hypergeometric function” with respect to $z$ in terms of a differential equation would define many special case inverse functions. Define: $$\,_2\text F_1(a,b;c;z)=\sum_{n=0}^\infty \...
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Do all roots of $f(x) = f^{-1}(x)$ have to lie on y=x and vice-versa? [duplicate]

One of my professors told me that all values of x that satisfy (1) $f(x)- x = 0$ are also solutions of (2) $f(x)- f^{-1}(x) = 0$ provided that the function is invertible. Thinking of graphs and noting ...
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0 votes
1 answer
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Left-inverse of linear function is linear?

Assume that $g\colon V\to W$ is linear. Show that if $f\colon W\to V$ is a left-inverse of $g$, then $f$ is linear. I am not sure how to prove that. I need to show that $f(ax+by)=af(x)+bf(y)$ for ...
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0 answers
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Trouble solving an exercise on differential geometry involving the arc length function and its inverse

Let $\alpha:I\to R^3$ be a regular parametrized curve (not necessarily by arc length) and let $\beta: J\to R^3$ be a reparametrization of $\alpha(I)$ by the arc length $s=s(t)$, measured from $t_0\in ...
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Inverse function of $ax + bx^{c}$ [closed]

I am struggling to find the inverse function of $f(x) = ax +bx^{c}$. The parameters $a$, $b$ and $c$ are all $\in \mathbb{R}^+$. Since $c$ is a real number, $x^c$ could be a root function and thus $f(...
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  • 41
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1 answer
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Is the empty function its own inverse

I suspect that the empty function $f: \emptyset \to \emptyset$ is self-invertible, i.e., $f^2 = \mathrm{id}$, but I don't know how I would go about proving it. I would need to show that for every $x \...
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4 votes
0 answers
68 views

Approximation of harmonic numbers and their analytical inverse.

In the same spirit as DeTemple–Wang for a series expansion of harmonic numbers, I tried to approach the problem as $$H_n\sim\frac 12 \log(n^2+n+a)+\gamma-\frac 1{b(n^2+n+a)+\Delta}\tag 1$$ hoping to ...
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0 votes
1 answer
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Necessary condition for locally invertible by inverse function theorem

If $f:R²$ to $R²$ such that f(x,y)=$(x³+3xy²-15x-12y,x+y)$. Let S={(x,y) such that f is locally invertible at (x,y)}. Then S is Answer: my approach, I had used the inverse function theroem and find ...
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2 answers
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The integral of $\frac{x}{16-x^4}$ [closed]

$$\int\frac{x}{16-x^4}\;dx$$ $\because$ $\frac{d}{dx}(\coth^{-1}\frac{x}{a}$)$=\frac{1}{a^2-x^2}$ ,And also $\frac{d}{dx}(\tanh^{-1}\frac{x}{a}$)$=\frac{1}{a^2-x^2}$ $\therefore$ If we say that $u=x^2$...
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-1 votes
2 answers
66 views

Why does the integral of $\frac{1}{\sqrt{a^2-x^2}}$ have 2 different answers for the same domain [closed]

$$\int\frac{1}{\sqrt{a^2-x^2}}dx=\sin^{-1}(\frac{x}{a})+C=-\cos^{-1}(\frac{x}{a})+C , \text{for} \space |x|<a$$ How are these 2 values equal ?
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1 vote
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Is q allowed to assume any real value in a trigonometric equation?

Suppose $f_q(x) = \sin^q x$. Next, we compute the inverse of $f_q(x)$: $$ \begin{aligned} y &= \sin^q x \\ y^{1/q} &= \sin x \\ \arcsin{y^{1/q}} &= x \\ \...
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  • 213
1 vote
1 answer
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Understanding the inverse function of a given function.

Here is the function I want to know its inverse: $\displaystyle\phi_{i}([x_{1},x_{2},...x_{n}])=(\frac{x_{1}}{x_{i}},...\hat{x_{i}},...\frac{x_{n}}{x_{i}})$ where the $\hat{}$ means we omit this ...
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1 vote
0 answers
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Question on inverse of implicit function

Let's consider the continuously differentiable functions $f:\mathbb{R}^n\to\mathbb{R}^n$ and $F: \mathbb{R}^{2n}\to\mathbb{R}^n$ where \begin{align*} F(x_1,x_2,\cdots ,x_n,y_1,y_2,\cdots ,y_n):=\...
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  • 3,061
4 votes
3 answers
417 views

Can you solve any mathematical function?

For any finite mathemathical function (consisting of addition, subtraction, division, multiplication, exponentiation, trigonometry) can you find $x$ in $f(x) = y$ where $y$ is a number you want? Is it ...
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2 votes
3 answers
68 views

Stuck on simplifying expressions involving trig and inverse trig functions

TL; DR Using Mathcad and Wolfram I can see that $$\sqrt{7}\cos\frac{\tan^{-1}\left(\frac{9\sqrt{3}}{10}\right)}{3}=2.5$$ The decimal value seems to be exact because Mathcad displays it like that with ...
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  • 163
0 votes
2 answers
45 views

To simplify the logarithmic expression and possibly find its inverse.

While solving an equation, I came across an unexpectedly symmetric solution. $$\frac{\log \left(\frac{a r+h}{\sqrt{(a r+h)^2-1}}+1\right)}{2 a}-\frac{\log \left(1-\frac{a r+h}{\sqrt{(a r+h)^2-1}}\...
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  • 279
8 votes
2 answers
111 views

Let $ f: ( 0, \infty) \rightarrow ( 0, \infty)$ bijective with $ f^{-1}(x) \cdot f(x) =1$.

Let $ f: ( 0, \infty) \rightarrow ( 0, \infty)$ bijective with $ f^{-1}(x) \cdot f(x) =1$.Prove that there is an interval $ I $ such that $f(I) $ is not an interval. I choose two elements $ x, y \...
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How to find an explicit equation for the inverse of an integral of a high degree polynomial with certain constraints?

I have a 6th degree polynomial function with decimal coefficients that I've rounded to make it easier: $$ f(x) = -0.1227 x^6 + 0.0227 x^5 +0.5221 x^4-0.2252 x^3 -0.5113x^2+ 0.3047x +0.1176$$ this ...
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0 answers
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Question on Inverse of a function

Let's consider $f:A\to B$. I thought that if we find another function $g$ such that $f\circ g=id$ and $g\circ f =id$ then $g$ is the inverse function of $f$ (and vice versa). However, our tutor said ...
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0 answers
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Find all increasing bijection $f$ of $\mathbb R$ onto itself that satisfy $f(x)+f^{-1}(x)=2x$ where $f^{-1}$ is inverse of $f$

Find all increasing bijection $f$ of $\mathbb R$ onto itself that satisfy $f(x)+f^{-1}(x)=2x$ (where $f^{-1}$ is inverse of $f$) My solution: Replace $x$ by $f(x)$ $k$ times we obtain $\underbrace{f\...
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  • 1,563
1 vote
0 answers
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Integral of (generalized) inverse functions with a twist

Consider a weakly increasing function $f : [0,1] \rightarrow [0,1]$ and define its generalized inverse (since an inverse does not necessariliy exist unless it increases strictly) as $$ f^{-1}(t):= \...
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Inverse function of $\sin: [-\pi,-\frac{\pi}{2}]\cup [\frac{\pi}{2},\pi]\to\mathbb{R}$?

What is the inverse function of $\sin: [-\pi,-\frac{\pi}{2}]\cup [\frac{\pi}{2},\pi]\to\mathbb{R}$? We know that this function is injective and restricting its codomain to $[-1,1]$ makes it bijective....
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