Mathematics Stack Exchange
Mathematics Stack Exchange

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.

In mathematics, an inverse function or $f^{-1}$ is a function that "reverses" another function. That is, if $f$ is a function mapping $x$ to $y$, then the inverse function of $f$ maps $y$ back to $x$.

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Can there be an injective function whose derivative is equivalent to its inverse function?

Let's say $f:D\to R$ is an injective function on some domain where it is also differentiable. For a real function, i.e. $D\subset\mathbb R, R\subset\mathbb R$, is it possible that $f'(x)\equiv f^{-1}(x)$? Intuitively speaking, I suspect that this is…
polfosol
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If $f(x)-f^{-1}(x)=e^{x}-1$, what is $f(x)$?

$f(x)$ is an increasing, differentiable function satisfying $f(x)-f^{-1}(x)=e^{x}-1$ for every real number $x$ I couldn't figure it out whether such function $f(x)$ exists or not. And if it exists, I want to know the method to find what $f(x)$…
Gastly
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A function with a non-zero derivative, with an inverse function that has no derivative.

While studying calculus, I encountered the following statement: "Given a function $f(x)$ with $f'(x_0)\neq 0$, such that $f$ has an inverse in some neighborhood of $x_0$, and such that $f$ is continuous on said neighborhood, then $f^{-1}$ has a…
Ran Kiri
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Functional equation: what function is its inverse's reciprocal?

The fact that so many students confuse functional inverse notation $$f^{-1}(x)$$ with multiplicative inverse notation $$[f(x)]^{-1}$$ got me to thinking... does there exist a function whose inverse is its inverse? That is, is there a function…
Franklin Pezzuti Dyer
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Can we invert this transform?

Consider the following transform $g\mapsto f$, where $$ f(x) = \int_{0}^{\infty} \exp\left\{-\int_0^t\int_{x-\tau}^{x+\tau} g(y) \, dy \, d\tau\right\} \, dt $$ Assume $f,g>0$ are $C^\infty(\mathbb{R})$, and $f,g\in L^2(\mathbb{R})$. Assume also…
sam wolfe
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What is $\arctan(x) + \arctan(y)$

I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$ which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by which I mean, has different definitions for different…
Max Payne
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Deriving the closed form for $\sum_{n=-\infty}^{\infty} \tan^{-1} (an+b) $

I saw this amazing identity elsewhere : $$ \bbox[5px,border:2px solid #C0A000]{\sum_{n\in\mathbb Z} \tan^{-1} (an+b) =\lim_{N\to \infty} \sum_{-N}^N \tan^{-1} (an+b) = \tan^{-1} \left( \tan\frac{b\pi}{a} \cdot \coth \frac{\pi}{a} \right)…
Vishu
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Is there an explanation why the reflection of $f(x)$ through $y = x$ is its inverse?

e.g. The function $e^x$ reflected through $y=x$ is $\ln x$. Is this always true OR just in some cases?
RHS
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Inverse function of the Exponential Integral $\mathrm{Ei^{-1}}(x)$

The Exponential integral is defined by $$ \mathrm{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm dt, $$ and has the following expansion $$ \mathrm{Ei}(x) = \log x + \gamma + \sum_{k=1}^\infty \frac{x^k}{k \cdot k!}. $$ Someone recently made a…
Nolord
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Differentiable bijection $f:\mathbb{R} \to \mathbb{R}$ with nonzero derivative whose inverse is not differentiable

I had an exam today, and I was asked about the inverse function theorem, and the exact conditions and statement (as stated in Mathematical Analysis by VA Zorich): Let $X, Y \subset \mathbb{R}$ be open sets and let the functions $f: X \to Y$ and…
Matija Sreckovic
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Alternative notation for inverse function

We all known the problems that presents the notation of inverse/reverse/anti functions as $f^{-1}(x)$, being the most important one the confusion with ${f(x)}^{-1}$, as in the classical $\sin^{-1}(x)$, or the strange cases such as…
pasaba por aqui
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Inverse function of $y=\frac{\ln(x+1)}{\ln x}$

I've been wondering for a while if it's possible to find the inverse function of $y=\frac{\ln(x+1)}{\ln x}$ over the reals. This is the same as finding the positive real root of $x^y-x-1$. I realize that it's impossible with elementary functions,…
B H
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How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition)

I'm doing exercise on discrete mathematics and I'm stuck with question: If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f \circ\ g)$ is given by $(f \circ\ g) ^{-1} = g^{-1}…
idonno
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Inverse of the polylogarithm

The polylogarithm can be defined using the power series $$ \operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}. $$ Contiguous polylogs have the ladder operators $$ \operatorname{Li}_{s+1}(z) = \int_0^z \frac…
Simon
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An idea for this difficult integral: $\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$.

I am being stuck in caculating this integral: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$$ I tried to change to another variable: $x = - t$ then $dx = - dt$, also I got:…
Lê Trung Kiên
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