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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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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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…
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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)…
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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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Does $f\colon x\mapsto 2x+3$ mean the same thing as $f(x)=2x+3$?

In my textbook there is a question like below: If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$ As a multiple choice question, it allows for the answers: A. $11$ B. $5$ C. $\frac{1}{11}$ D. $9$ If what I think is correct and I read the equation…
Soo Kyung Ahn
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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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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}…
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Solve functional equation $ h(y)+h^{-1}(y)=2y+y^2 $

I was trying to solve a certain physics problem, and encountered the functional equation that contains a function $h$ and its inverse $h^{-1}$: \begin{equation} h(y)+h^{-1}(y)=2y+y^2.\tag{1} \end{equation} Q: Does $(1)$ have a unique solution and…
Nemo
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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…
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True or False : If $f(x)$ and $f^{-1}(x)$ intersect at an even number of points , all points lie on $y=x$

Previously I have discussed about odd number of intersect points (See : If the graphs of $f(x)$ and $f^{-1}(x)$ intersect at an odd number of points, is at least one point on the line $y=x$?) Now , I want to know the even condition . For example…
S.H.W
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Asymptotic expansion of the inverse of $x\mapsto x+x^{\small\sqrt2}+x^2$ near zero

This is a follow-up to my previous question "Asymptotic expansion of the inverse of $x\mapsto x+x^\phi$ near zero". Consider a continuous real-valued monotone increasing function $f:\mathbb R^+\to\mathbb R^+$ satisfying…
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Prove that $\tan^{-1}\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}=\frac{\pi}{4}+\frac 12 \cos^{-1}x^2$

Let the above expression be equal to $\phi$ $$\frac{\tan \phi +1}{\tan \phi-1}=\sqrt{\frac{1+x^2}{1-x^2}}$$ $$\frac{1+\tan^2\phi +2\tan \phi}{1+\tan^2 \phi-2\tan \phi}=\frac{1+x^2}{1-x^2}$$ $$\frac{1+\tan^2\phi}{2\tan \phi }=\frac{1}{x^2}$$ $$\sin…
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A function satisfying $f \left ( \frac 1 {f(x)} \right ) = x$

Background. This question originates from the problem of finding a function $f$ such that its $n$-th iterate is equal to its $n$-th power, which I asked about here. Now I would like to focus on the related case $n = -1$, because it seems to be quite…
Luca Bressan
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