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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.

54
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4answers
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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}(...
30
votes
2answers
2k views

Functional equation: what function is its inverse's reciprocal? [duplicate]

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 ...
25
votes
4answers
1k views

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 ...
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
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4answers
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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 ...
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 ...
11
votes
1answer
1k views

Is a bijective smooth function a diffeomorphism almost everywhere?

Suppose I have $f: M \rightarrow N \in C^{\infty}$ a smooth bijection between $n$-dimensional smooth manifolds. Does it have to be a diffeomorphism except for a set of measure 0? I think the proof ...
10
votes
1answer
125 views

How to find the general solution to $\int f^{-1}(x){\rm d}x$ in terms of $\int f(x){\rm d}x$

I am trying to find a general proof of $\int f^{-1}(x)\,{\rm d}x$ in terms of $\int f(x)\,{\rm d}x$. The first step that I took was to piece apart what it means for a function to have an inverse. So I ...
10
votes
4answers
216 views

“Class” of functions whose inverse, where defined, is the same “class”

Please excuse the amateurish use of the term "class", I don't know what the exact term is for what I'm looking for. Anyway, details. I'm asking specifically about real-valued functions on the real ...
10
votes
1answer
196 views

How to find the inverse of $f(x)=x+ \frac{x^{3}}{1+x^{2}}$?

I know that given, $f(x)=x+ \frac{x^{3}}{1+x^{2}}$ I should set $y=x+ \frac{x^{3}}{1+x^{2}}$ and solve in terms of $x$, then just swap the $x$'s and $y$'s. I know that, since the derivative is ...
10
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0answers
226 views

Non-trivial solutions of a functional equation $(f \circ f \circ f)(x)=x$

I have come to this from the topic of self-inverse functions. Let's consider a more complicated case where: $$(f \circ f \circ f)(x)=x \tag{1}$$ Also assume that $f(x)$ is continous on some non-...
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 $\sin^{-1}(x)$,...
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 ...
9
votes
4answers
480 views

Why can't $y=xe^x$ be solved for $x$?

I apologize for my mathematical ignorance regarding this, but could someone help me understand why it isn't possible to (symbolically) find an inverse function for $f(x)=xe^x$? The most obvious (but ...
9
votes
1answer
183 views

Recovering $a+b+\cdots$ from $\exp(a)+\exp(b)+\cdots$ for $a,b,\ldots\in\mathbb N$

In a problem I working on, I have the following value $$ y = f(a_1,a_2,\ldots,a_n) = \varphi^{a_1} + \varphi^{a_2} + \cdots + \varphi^{a_n} \enspace, $$ for $(a_1,a_2,\ldots,a_n)\in\{0,1,\ldots\}$ and ...
8
votes
6answers
9k views

proof of $\log(xy) =\log (x) + \log (y)$ [closed]

$\log(xy) = \log (x) + \log ( y)$ and its division counter part were mentioned in an axiomatic way which I failed to proof. I noted the correlation with the exponential rules of adding powers in ...
8
votes
5answers
1k views

Derivative of $\arcsin(x)$

I was trying to find the derivative of $$\arcsin(x) = \sin^{-1}(x)$$ I thought that I could use the rule of inversion: $$({f^{-1}})'(x) = \dfrac{1}{f(x)'}$$ Therefor the derivative of $\arcsin(x)$ ...
8
votes
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!
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 ...
8
votes
2answers
577 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_{-\...
8
votes
1answer
167 views

If $n={k^2 \choose k}$, then what is $k$?

Given that: $$n={k^2 \choose k}$$ what is $k$ as a function of $n$? So far, I have found the following approximation: $$ n \approx (k^2)^k = (k^k)^2 $$ $$ \sqrt{n} \approx k^k $$ If we take this ...
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}...
7
votes
4answers
219 views

Integrate $\sin^{-1}\frac{2x}{1+x^2}$

Integrate $\sin^{-1}\frac{2x}{1+x^2}$ The solution is given in my reference as: $2x\tan^{-1}x-\log(1+x^2)+C$. But, is it a complete solution ? My Attempt $$ \int 2\tan^{-1}x \, dx=\int \tan^{-1}x \...
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 ...
6
votes
3answers
502 views

Are all square roots of diagonalizable matrices diagonalizable?

If a matrix is normal/unitarily diagonalizable, then its square roots are readily computed by taking the square roots of its eigenvalues (in the complex plane if needed). Any square root computed in ...
6
votes
1answer
158 views

Are Inverse Trig functions a different form of log?

When studying complex analysis, we realize that trigonometric functions are nothing but exponentials, and we can define real trigonometric functions in terms of complex exponentials. I was wondering ...
6
votes
5answers
172 views

Evaluate $\int_0^1x(\tan^{-1}x)^2~\textrm{d}x$

Evaluate $\int\limits_0^1x(\tan^{-1}x)^2~\textrm{d}x$ My Attempt Let, $\tan^{-1}x=y\implies x=\tan y\implies dx=\sec^2y.dy=(1+\tan^2y)dy$ $$ \begin{align} &\int\limits_0^1x(\tan^{-1}x)^2dx=\int\...
6
votes
4answers
4k views

Proving that $\cos(\arcsin(x))=\sqrt{1-x^2}$

I am asked to prove that $\cos(\arcsin(x)) = \sqrt{1-x^2}$ I have used the trig identity to show that $\cos^2(x) = 1 - x^2$ Therefore why isn't the answer denoted with the plus-or-minus sign? as in ...
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) ...
6
votes
2answers
115 views

Functional equation: $f(2x)=f(x)+f^{-1}(x)$

I have been working to find solutions to the functional equation $$f(2x)=f(x)+f^{-1}(x)$$ $$f:\mathbb R^+\to \mathbb R$$ So far I have found the trivial solution $$f(x)=x$$ and, by mere luck, I ...
5
votes
4answers
171 views

Difference between $\sin^{-1}(x)$ and $\frac1{\sin(x)}$? [closed]

How are arcsine and cosecant different mathematically if cosecant is $\frac{1}{\sin(x)}$ and arcsine is $\sin^{-1}(x)$ which is $\frac{1}{\sin(x)}$? I have tried to find an answer before but nobody ...
5
votes
4answers
182 views

why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$

How do i find the When $x$ is ranging from $-1$ to $1$? I want know why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$ I have already tried inverse-function. thanks.
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{...
5
votes
3answers
163 views

If $\sin^{-1} (x) + \sin^{-1} (y)+ \sin^{-1} (z)=\dfrac {\pi}{2}$, prove that $x^2+y^2+z^2+2xyz=1$

If $\sin^{-1} (x) + \sin^{-1} (y)+ \sin^{-1} (z)=\dfrac {\pi}{2}$, prove that: $$x^2+y^2+z^2+2xyz=1$$ My Attempt: $$\sin^{-1} (x) + \sin^{-1} (y) + \sin^{-1} (z)=\dfrac {\pi}{2}$$ $$\sin^{-1} (x\...
5
votes
2answers
199 views

Limit of a two variable function involving arcsin and arctan

I recently came across this problem of finding the limit of a function in two variables as we approach the origin, defined as follows: $$\lim_{(x,y)\to (0,0)} \frac{\arcsin(x+2y)}{\arctan(2x+4y)}$$ ...
5
votes
4answers
807 views

Inverse of a function - Set Theory

Please help me with this exercise... I already showed that the function is bijective, and I do not know how to find the inverse of the function... Be the function $f : \mathbb{N} \times \mathbb{N}  ...
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, ...
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
1answer
234 views

Find the roots of $e^x+e^{1/x} + a = 0$

Find the roots of this equation $e^x + e^{1/x} + a = 0$ where $a \in \Bbb R$ Is there any nice formula for this type of equation?
5
votes
5answers
72 views

Finding an inverse function (sum of non-integer powers)

I have a function: $$f(x)=x^{2.2} + (1-x)^{2.2}$$ It is defined on the interval $[0,1]$. Minimum: $x=0.5, y=2*0.5^{2.2} = 2^{-1.2}$. I want to find an inverse for it. Since the function has two "...
5
votes
1answer
147 views

Functional equation $3f(-3x) -f(x) = 3x^2$

I am trying to solve the following functional equation: $f(x)$ is a continuous function, satisfying (1) $f(f(x))=x$ (2) $ 3f(-3x)-f(x)= 3x^2 $ for $x>0$. From (1) & (2), I found that $...
5
votes
2answers
210 views

Prob. 2, Chap. 5 in Baby Rudin: How is the inverse function differentiable?

Here is Prob. 2, Chap. 5 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $f^\prime(x) > 0$ in $(a, b)$. Prove that $f$ is strictly increasing in $(a, b)$, ...
5
votes
1answer
205 views

Existence of a function whose derivative of inverse equals the inverse of the derivative

I've been thinking about the calculation of inverse functions through Taylor series expansions. My hypothesis was that if we had: $$\ f(x) =\sum_{n=0}^\infty \frac{(x-x_0)}{n!}f^{n}(x_0),$$ then $$\ ...
5
votes
0answers
98 views

Show that $f+\epsilon g\in{\rm Diff^1}(\Bbb R^m)$ for all $\epsilon\in(-\epsilon_0,\epsilon_0)$

This is an exercise on page 220 of Analysis II of Amann and Escher Here $\rm Diff^k$ means the set of diffeomorphisms where the $k$-th derivative is also an homeomorphism. My work below. The ...
5
votes
1answer
187 views

The function $\mathrm{Li}_2(x)=\int_2^x\frac{dt}{\log^2t}$, its inverse and summation

I am reading the more understandable mathematics in the section Preliminary Results of a paper in which the authors give a explanation of facts for the logarithmic integral and its inverse. In this ...
4
votes
4answers
117 views

$\cos(\arcsin(x)) = \cdots $

I've been asked to prove $$y=\frac{\sqrt{3}} 2 x+\frac 1 2 \sqrt{1-x^2}$$ given $x=\sin(t)$ & $y=\sin(t+\frac \pi 6)$ I did $t=\arcsin(x)$ and plugged that into the $y$ equation. Used the $\sin(...
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?
4
votes
3answers
388 views

Limit involving inverse functions

When I am given the limit $$\lim\limits_{x \rightarrow \infty}\frac{x\arctan\sqrt{x^2 +1}}{\sqrt{x^2+1}}$$ would it be possible to evaluate it giving some substitution? L'Hospital's rule seemed an ...
4
votes
6answers
92 views

Solving for $x$ in $\sin^{-1}(2x) + \sin^{-1}(3x) = \frac \pi 4$

Given an equation: $$\sin^{-1}(2x) + \sin^{-1}(3x) = \frac \pi 4$$ How do I find $x$? I tried solving by differentiating both sides, but I get $x=0$. How do you solve it, purely using ...