# 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.

57 questions
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### Inverse of $y = x^3 + x$?

Can you help me find the inverse function for $y = x^3 + x$? This question was posed at the beginning of AP Calculus, so we can't use any math beyond precalc. Thanks!
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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 ...
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### Dirac delta function $\delta(f(x))$ of function $f$ with a higher-order zero

Dirac delta function have this property: $$\delta(f(x))=\textstyle \sum_i\frac{\delta(x-a_i)}{\lvert f^\prime(a_i)\rvert}.$$ And its derivation is: \begin{eqnarray} \int_{-\...
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### Sums of inverse trigonometric functions [closed]

I don't know how many of you will appreciate this one.But it will be really helpful to me if somebody can tell me how to remember these formulae with the given domains (from 6(a)) .Thanks in advance.
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### Compare $\arcsin (1)$ and $\tan (1)$

Which one is greater: $\arcsin (1)$ or $\tan (1)$? How to find without using graph or calculator? I tried using $\sin(\tan1)\leq1$, but how do I eliminate the possibility of an equality without ...
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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 ...
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### Inversion of Trigonometric Equations

I've been playing around with finding the domain-restricted inverses of trigonometric equations using the inverse trigonometric equations. One of the easier formulas I came up with was the formula for ...
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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 ...
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### A function satisfying $f \left ( \frac 1 {f(x)} \right ) = x$ [duplicate]

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 ...
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### How to prove the converse of Carathéodory's theorem

The following problem is an assignment of my complex analysis course, which seems to be a converse of the Carathéodory's theorem that a biholomorphism between two Jordan regions can be extended to a ...
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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 $\sin^{-1}(x)$,...
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### Inverse of a continuous function

Let $A$ be a closed bounded subset of $\mathbb{R}$. Suppose $f: A \rightarrow \mathbb{R}$ is a continuous injective function. Then $f^{-1} : f(A) \rightarrow A$ is also continuous. That $A$ is closed ...
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### Is there a method to find the inverse of an arbitrary function?

Is it possible to get inverse of all be functions? For example, can we calculate inverse of $y=x^3+x$?
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### Knowing which theorem of calculus to use to prove number/nature of solutions

How does one go about showing that the equation $$arctanx = x^2$$ has at least one solution and then in turn show that the equation has exactly one positive solution? I figured for part a) "show ...
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### Can someone prove that the inverse of $x^x$ is not an elementary function?

I want to prove that the inverse of $f(x)=x^x$ is not an elementary function. With elementary function I mean a function of one variable which is the composition of a finite number of arithmetic ...
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### Let $f(x) = 6\operatorname{arcsec}(2x)$. Find $f'(x)$.

Let $f(x) = 6\operatorname{arcsec}(2x)$. Find $f'(x)$. $$6 \cdot \frac 1{2x\sqrt{(2x)^2-1}} \cdot 2$$ $$\frac{12}{2x\sqrt{4x^2-1}}$$ $$\frac 6 {x\sqrt{4x^2-1}}$$ Why is that wrong?
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### (Why) Does $f(x,y)=\{x,G(x,y)\}$ having an inverse imply $D_{1}f(x,y)=0$?

My questions are: If the $\mathscr{C}^{1}$ mapping $f:\mathbb{R}^{m+n}\rightarrow\mathbb{R}^{m+n}$ defined by $f(\mathbf{x},\mathbf{y})=\{\mathbf{x},G(\mathbf{x},\mathbf{y})\}$ has an inverse such ...