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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.

8
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4answers
24k views

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!
7
votes
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 ...
8
votes
2answers
576 views

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_{-\...
3
votes
3answers
132 views

Simplifying the integral $\int\frac{dx}{(3 + 2\sin x - \cos x)}$ by an easy approach

$$I=\displaystyle\int\frac{dx}{(3 + 2\sin x - \cos x)}$$ If $$\tan\left(\frac{x}{2}\right)=u$$ or $$x=2\cdot\tan^{-1}(u)$$ Then, $$\sin{x}=\dfrac{2u}{1+u^2}$$ $$\cos{x}=\dfrac{1-u^2}{1+u^2}$$ $$...
2
votes
4answers
229 views

What is exactly the inverse of the function $f(x)=\frac{x}{1-x^2}$?

How do we find the inverse of the function $f:(-1,1)\to \Bbb R$ by $f(x)=\frac{x}{1-x^2}$? The problem has been posted here and the answer below says "To show that $f$ has a continuous inverse, you ...
1
vote
2answers
2k views

$(\sin^{-1} x)+ (\cos^{-1} x)^3$

How do I find the least and maximum value of $(\sin^{-1} x)+ (\cos^{-1} x)^3$ ? I have tried the formula $(a+b)^3=a^3 + b^3 +3ab(a+b)$ , but seem to reach nowhere near ?
5
votes
1answer
2k views

Does the Gamma Function have an Inverse?

Does the Gamma Function have an Inverse? (Is there an "arc-gamma" function?) Where $\Gamma(x) = y... \Gamma^{-1}(y) = x\ (arc\Gamma(y)=x)$. I've searched and found something called DiGamma Function, ...
0
votes
1answer
200 views

How is the Inverse function theorem used to prove that the formulae in this question are the same?

I was informed in my last question that the Inverse function theorem: $$(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}\tag{I*}$$ was needed to show that $$\rho_x (x)=\rho_\alpha(\alpha)\left|\...
0
votes
2answers
19k views

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.
2
votes
2answers
70 views

Derivation of inverse sine, what is wrong with this reasoning?

I'm trying to find the derivative of $\sin^{-1}(x)$. I know the steps that lead to $\frac{1}{\sqrt{1-x^2}}$, however I don't understand why the following reasoning leads to a wrong answer. Because $$...
2
votes
1answer
415 views

Inverse function differentiability proof

My textbook states the proof as follows: Let $f$ be a continuous one-to-one function defined on an interval, and suppose that $f$ is differentiable at $f^{-1}(b)$, with derivative $f'(f^{-1}(b)) \not ...
54
votes
4answers
3k views

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}(...
9
votes
4answers
1k views

One-Way Inverse

My Algebra $2$ teacher stressed the fact that when you find the inverse $g$ of a function $f$, you must not only check that $$f \circ g=\operatorname{id}$$ but you must also check that $$g \circ f=\...
9
votes
7answers
651 views

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 ...
8
votes
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 ...
1
vote
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 ...
16
votes
6answers
3k views

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 ...
12
votes
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 ...
3
votes
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 ...
9
votes
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 $\sin^{-1}(x)$,...
7
votes
2answers
106 views

Determine a constant for a certain limit involving arctan function

Let us consider the sequence $(x_n)_{n \ge 0}$ such that $x_0\gt 0$ and defined as follows:$$x_{n+1}=\arctan x_n \ .$$ Find the numbers $\alpha \in \Bbb R$ such that $\lim _ {n \to \infty}n^{\alpha}...
6
votes
2answers
1k views

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) ...
3
votes
3answers
643 views

Is $f(x)=\frac{1}{x}$ invertible?

Let $f(x)=\frac{1}{x}$, then $f^{-1}(x)$ must equal $$y=\frac{1}{x}$$ and after swapping the variables $$x=\frac{1}{y}$$and rearranging to solve for $y$, $$\frac{1}{x}=y$$ that being said, can you ...
1
vote
2answers
98 views

Inverse derivative of a function

Problem We have function defined as: $$ f(x)=x^2\ln(x)$$ What is $$ (f^{-1})'(y_0) $$ on specific point which is $y_0=e^2$ Attempt to solve A derivative of a inverse function would be defined as:...
0
votes
1answer
72 views

What is the mathematical approach of inversing a function resulting in a piece-wise solution?

I've been trying to find the inverse of $$f(x) = e^{-\left(\displaystyle \frac{x}{\sqrt{1-x^2}}\right) \displaystyle \pi }$$ Here are my steps $$ \begin{align} x & = e^{-\left(\displaystyle \frac{...
4
votes
1answer
222 views

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 $y=x$?

I was reading about intersection points of $f(x)$ and $f^{-1}(x)$ in this site. (Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$) Then, I saw this statement ...
4
votes
3answers
324 views

Finding the inverse of $f(x) = x^3 + x$

How can one find the inverse of functions like $f(x) = x^3 + x$? I know how to do it for explicit quadratic functions; how do I express $x$ as a function of $y$ here?
2
votes
1answer
103 views

Composition of functions question

Am restricting this question to the elementary context of Riemann integrals and continuous functions $f,g.$ Because this came up in the context of another question, I would prefer to keep the examples ...
1
vote
4answers
160 views

$\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\frac{x+y+z-xyz}{1-xy-yz-zx}$ true for all $x$?

$\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\dfrac{x+y+z-xyz}{1-xy-yz-zx}$ true for all $x$ ? This expression is found without mentioning the domain of $x,y,z$, but I don't think its true for all $x,y,...
0
votes
1answer
378 views

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 ...
5
votes
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 $y=x^3+x$?
5
votes
5answers
4k views

Find the greatest and least values of $(\sin^{-1}x)^2+(\cos^{-1}x)^2$

Find the upper and lower limit of $$ (\sin^{-1}x)^2+(\cos^{-1}x)^2 $$ My Attempt: $$ \frac{-\pi}{2}\leq\sin^{-1}x\leq \frac{\pi}{2}\quad\&\quad0\leq\cos^{-1}x\leq\pi\\(\sin^{-1}x)^2\leq\frac{...
4
votes
0answers
472 views

Inverse Function Theorem and Injectivity

I have the following problem (which involved the Inverse Function Theorem and Injectivity): Let $f(x_1,x_2,x_3)=(u(x_1,x_2,x_3),v(x_1,x_2,x_3),w(x_1,x_2,x_3))$ be the mapping of $\mathbb{R}^3$ and $\...
3
votes
3answers
2k views

How to calculate the inverse of a known optical distortion function?

Assume I have the following lens distortion function: $$ x' = x (1 + k_1 r^2 + k_2 r^4) \\ y' = y (1 + k_1 r^2 + k_2 r^4) $$ where $r^2 = x^2 + y^2$. Given coefficients $k_1$ and $k_2$, I need to ...
3
votes
1answer
188 views

Powers of Möbius transformations equal to identity?

I'm looking at "Mobius transformations" where $a,b,c,d\in\mathbb R$. I want to know for which $n$ there exists $a,b,c,d$ such that for $f(x) = \dfrac{ax+b}{cx+d}$, $$f^n(x) = f(f(...(f(x)))) = x$$ ...
3
votes
1answer
55 views

Computing the inverse of $Ax+Bx\log x$

I'm trying to inverse the function $f:x\mapsto Ax+Bx\log x$. I know from Wolfram Alpha that the result is: $f^{-1}(x)=\frac{A}{BW(\frac{Ae^{x/B}}{B})}$ where $W$ is the W-Lambert (or product ...
3
votes
4answers
847 views

Why is the inverse tangent function not equivalent to the reciprocal of the tangent function?

I know that $$ {\tan}^2\theta = {\tan}\theta \cdot {\tan}\theta $$ So I guess the superscript on a trigonometric function is just like a normal superscript: $$ {\tan}^x\theta = {({\tan}\theta)}^{x} ...
2
votes
4answers
96 views

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 ...
2
votes
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 ...
1
vote
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?
1
vote
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 ...
1
vote
3answers
93 views

Find $\arctan(\sqrt{2})-\arctan\left(\frac{1}{\sqrt{2}}\right)$

I did it as follows: $$\arctan(\sqrt{2})-\arctan\left(\frac{1}{\sqrt{2}}\right)=\tan\Bigg(\arctan(\sqrt{2})-\arctan\left(\frac{1}{\sqrt{2}}\right)\Bigg)=\frac{\sqrt{2}-\frac{1}{\sqrt{2}}}{1+\frac{1}{\...
1
vote
2answers
2k views

Evaluation of $\sum^{\infty} _{n=1} \arctan\left(\frac{4n}{n^4-2n^2+2}\right)$

Evaluate $\displaystyle\sum^{\infty} _{n=1} \arctan\left(\frac{4n}{n^4-2n^2+2}\right).$ I know we know to convert it in the of $\arctan\left(\frac{a-b}{1+ab}\right)$ but I am not able to do so here. ...
1
vote
1answer
227 views

Evaluate a limit involving the inverse of a function [duplicate]

Let $f : [1, \infty) \to [1, \infty)$ be a function such that $f(x) = x(1 + \ln x)$. Prove that $f$ is bijective and then calculate: $$\lim_{x \to \infty} \frac{f^{-1}(x) \ln x}{x}$$ I have no ...
1
vote
1answer
58 views

Expression for $\cos^{-1}x\pm\cos^{-1}y$

As mentioned in Proof for the formula of sum of arcsine functions $\arcsin x+\arcsin y$ for $\sin^{-1}x+\sin^{-1}y$ $$ \sin^{-1}x+\sin^{-1}y= \begin{cases} \sin^{-1}( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;...
1
vote
1answer
331 views

How can we show that parallel transport is invertible?

We have that the map $\Pi^{pq}_{\gamma}: T_pS \rightarrow T_qS$ that takes $v_0 \in T_pS$ to $v_1 \in T_qS$ is called parallel transport from $p$ to $q$ along $\gamma$. $\Pi^{pq}_{\gamma}: T_pS \...
1
vote
1answer
230 views

Derivative of double integral with direct and inverse functions

I found a couple of similar question, but I am struggling applying their logic to my example. Derivative of double integral with respect to upper limits Differentiation under the double integral ...
1
vote
3answers
71 views

Find angle given ratio between sine and angle

I know an angle is between 0 and $\pi$ (180 degrees). I know the ratio between its sine and the angle itself. Specifically it's $\frac{15}{16}$, but I am more interested in the general case. Since ...
1
vote
2answers
2k views

Derive the conditions $xy<1$ for $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}$ and $xy>-1$ for $\tan^{-1}x-\tan^{-1}y=\tan^{-1}\frac{x-y}{1+xy}$

$$ \tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy} \text{, }xy<1\\ \tan^{-1}x-\tan^{-1}y=\tan^{-1}\frac{x-y}{1+xy} \text{, }xy>-1 $$ But, How do I reach the conditions $xy<1$ for the first ...
0
votes
1answer
109 views

Inverse of the asymptotic expansion of Gauss Hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below). Basically I want to series expand $\rho$ for large $r$ (i.e. as $r\to \infty$) and then ...