# 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
4answers
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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!
2answers
3k views

### 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 ...
2answers
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### Dirac delta function $\delta(f(x))$ of function $f$ with a higher-order zero

Dirac delta function have this property: \begin{equation} \delta(f(x))=\textstyle \sum_i\frac{\delta(x-a_i)}{\lvert f^\prime(a_i)\rvert}. \end{equation} And its derivation is: \begin{eqnarray} \int_{-\...
3answers
132 views

2answers
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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.
2answers
70 views

7answers
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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 ...
2answers
2k views

### 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 ...
1answer
272 views

### 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 ...
6answers
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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 ...
2answers
183 views

### 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 ...
1answer
155 views

### 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 ...
5answers
2k views

### 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 $\sin^{-1}(x)$,...
2answers
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1answer
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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 ...
2answers
307 views

### 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 $y=x^3+x$?
5answers
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4answers
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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 ...
1answer
148 views

### 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 ...
3answers
62 views

### 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?
2answers
58 views

### (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 ...
3answers
93 views

1answer
331 views