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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.

5
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5answers
62 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 "...
3
votes
1answer
96 views
+50

Inverse/Reverse of Number of Permutations and of Number of Combinations with Repetitions?

For an engineering application, I need the inverse functions of the computations of the number of combinations and permutations. In the thread How to reverse the $n$ choose $k$ formula? it shows how ...
0
votes
3answers
39 views

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$ So in order to show injectivity $f(a)=f(b) \Rightarrow a=b$ so $\frac{a}{1-|a|}=\frac{b}{1-|b|}$. But how do I prove that? ...
0
votes
0answers
24 views

Help with concepts with the inverse function theorem

I am solving the following problems: Be the transformation $T \in R ^ 2 $: by $T(u, v) = (u ^ 2-v ^ 2, uv) = (x, y)$. Calculate the Jacobian of $T$ and conclude whether the transformation is ...
0
votes
2answers
43 views

Number of solution of the equation $\cot^{-1}{\sqrt{4-x^2}+ \cos^{-1}{(x^2-5)}}=3π/2$

Number of solution of the equation $ \cot^{-1}{\sqrt{4-x^2} + \cos^{-1}{(x^2-5)}}=3\pi/2$ $$ \cot^{-1}{\sqrt{4-x^2}+ \cos^{-1}{(x^2-5)}}={3π/2}$$ Taking sine both side and solving this is I get $$1 +\...
0
votes
0answers
18 views

Finding the inverse of a complex function to derive information about it.

Im sorry if im butchering some of this concepts, i am a cs undergrad and have very little understanding of advanced mathematics. I found an intresting function i quite like, such function is : -y = ...
1
vote
0answers
32 views

Is there any formula for arctan x + arctan y + arctan z. ? ( For various case)

Is there any formula for $ arctan x + arctan y + arctan z $ ? One way to solve such type of question is to is two take two at a time and solve them. I know the general formula for $ tarctan x + ...
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.
0
votes
2answers
42 views

Why isn’t the integral of $x^2$ from $0$ to $5$ equal to the integral of $\sqrt x$ from $0$ to $25$? [duplicate]

My calculator says that the former is 41.67, and that the latter is 83.34. Why is that? Shouldn’t they be equal?
1
vote
1answer
37 views

Image of an interior point

Consider $\textbf{f}: U\subset \mathbb{R}^n\to \mathbb{R}^n$ and $\textbf{a} \in U$. Suppose that $\textbf{a}$ is an interior point of $U$ and $\textbf{f}$ is differentiable at $\textbf{a}$ with $\det(...
0
votes
0answers
12 views

Inverse of Legendre Dual

Reading the book "Concentration inequalities A nonasymptotic theory of independence" I came across the following results (Lemma 2.4): Let $f$ be a convex function such that $f(0)=f'(0)=0$ and such ...
1
vote
1answer
20 views

unexpected case of gradient of inverse function

if $y=x^3+3x+2$ is the original equation then, $$\frac{dy}{dx} = 3x^2+3$$ So the gradient of the inverse function atc $x=2$ should be $$\frac{dx}{dy}= \frac{1}{3x^2+3}$$ This gives me answer of $\...
1
vote
2answers
72 views

Proving $G(s,t)=(\cosh s \cdot \cos t, \sinh s \cdot \sin t)$ has global inverse in a certain region

Let $G(s,t)=(\cosh s \cdot \cos t, \sinh s \cdot \sin t)$, and let $A=\{(s,t)\in\mathbb{R^2}|s>0, 0<t<2\pi\}$. Prove that $G_{|A}$ has global inverse and give a graphical representation of $...
0
votes
4answers
25 views

If $x\in [-1,0)$, then what is the value of $\cos^{-1} (2x^2-1) - 2 \sin^{-1} x$?

If x belongs to $[-1,0)$ then what is $\cos^{-1} (2x^2-1) - 2 \sin^{-1} (x) $? This is a question in my book, but I have doubts about solving it. I tried putting $\sin a = x$ and got $\pi - 2a - 2a = \...
0
votes
0answers
30 views

How do I draw the graphs of $\sin^{-1}(\frac{2x}{1+x^2})$ and $\tan^{-1}(\frac{2x}{1-x^2})$?

How do I draw the graphs of $\sin^{-1}(\frac{2x}{1+x^2})$ and $\tan^{-1}(\frac{2x}{1-x^2})$? The solution of a question in my books how these graphs and draws the conclusion that from them it is seen $...
1
vote
4answers
61 views

Differentiating $\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$

Find the derivative with respect to $x$ of $$\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$$ The partial solution to this problem is given as follows: $y=\left(\frac{x}{\sqrt{1-x^2}}\right)$ Then: ...
0
votes
3answers
74 views

Solution of $\tan^{-1} \sqrt{x^2 + x} +\sin^{-1} \sqrt{x^2 + x + 1} = \pi/2$

My attempt at question $\tan^{-1} \sqrt{x^2 + x} +\sin^{-1} \sqrt{x^2 + x + 1} = \pi/2$ using the identity $\tan^{-1}(\sqrt{x^2 + x} + \sqrt{x^2+x+1/-x^2-x}/[1- \sqrt{x^2+x)(x^2+x+1/-x^2-x)}]$=π/2 $...
0
votes
2answers
33 views

Function inverses

From the definition of a inverse standpoint ($f^{-1}(f(x))=f(f^{-1}(x))=x$), why does interchanging variables ($x$ and $y$) work to find the inverse? It seems logical to me but I cannot come up with a ...
1
vote
1answer
26 views

Real analysis : Problem related to inverse

Let $f:[0,\infty)\rightarrow [0,\infty) $ be a continuous strictly increasing function and $g=f^{-1}$. Let $a,b>0$ and $a\neq b$. Then how $\displaystyle\int_{0}^af(x)dx+\int_0^bg(y)dy\geq ab.$ Any ...
-1
votes
0answers
31 views

Inverse function to the exponential integral

$Ei(-ln(f(x)) = c + ln(ln(x))$ where $Ei(x)= $ - $\int_{-x}^{\infty} \frac{e^{-t}}{t} dt $, where $x $ be a non zero real (called exponential integral ). Is there a way to express $f(x)$? If yes, ...
2
votes
2answers
72 views

Solve $2x^2-5x+2=$ $\frac{5-\sqrt{9+8x}}{4}$

Solve $2x^2-5x+2$= $\frac{5-\sqrt{9+8x}}{4}$ I simply do square both sides solve it and I get two value of x one is 2 and other is $\frac{3-√5}{2}$ but this approach it take more time so is there any ...
0
votes
2answers
39 views

How do I solve $y=x+B\sin(x+A)$ for $x$

I have a code that converts x into y using the formula: $y=x+B\sin(x+A)$ with $x, A$ and $B$ known values. $B$ is also very small so that $B\sin(x+A) < 0.035$. The problem is that in another ...
1
vote
1answer
44 views

Differentiation of inverse function

I know that first order differentiation of inverse of a function $f (x)$ is reciprocal of $f'(f^-1(x)) $. But I'm unable to evaluate the integration given in the question.
1
vote
1answer
33 views

Implicit equation $\ln(\frac{x}{y})-y=1$ to rectangular equation not in terms of $W(x)$

Backstory and Other Info I'm not sure if this is possible, I'm currently a precalculus student and have a very limited understanding of much of any of this. However, I do like to go on WolframAlpha ...
0
votes
2answers
25 views

Global inverse of $G(x,y)=(\ln (xy),\frac{1}{x^2+y^2})$

Let $G(x,y)=(\ln (xy),\frac{1}{x^2+y^2})$, with Dom$(G)=\{(x,y)\in\mathbb{R^2:0<y<x}\}$. I am asked to find the global inverse of $G$. I have tried to proceed in the following way: $$u=\ln(xy)=...
0
votes
1answer
34 views

Inverse Fourier transform of $F(k) = 1/(k^2+a^2), a>0$

I need help finding the Inverse Fourier transform of: $$F(k) = \frac1{ k^2 + a^2 },~ a>0$$ Here is what I have so far: Singular points at $k^2 = a^2$, namely, at $k = \pm ia$. The inverse ...
0
votes
2answers
33 views

If $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$ then what is the value of $g'(2)$?

I know that $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$. But how can I find the value of $g'(2)$?
0
votes
0answers
9 views

Preimage Structures: lists of all inverse images at a given depth (reference request)

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, the j-th such list would be ...
0
votes
1answer
41 views

Output response from closed loop transfer function using MATLAB

This transfer function is to control the position of Permanent Magnet DC motor. I was able to get the transfer function and now I need to analyze the output for the tuned closed loop for a given input ...
0
votes
0answers
27 views

Find $F ^{−1} (u)$, for $u ∈ (0, 1)$ by solving the equation $u = F(x)$ for $x$

I am working through a problem. I began with the pdf: $f(x)=\frac{1}{2}\alpha*e^{-\alpha|x|}$, where $\alpha > 0$ and $x ∈ (−∞,∞)$. I found the cdf to be: $$F(x) = \begin{cases} \frac{1}{2}e^{\...
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 ...
1
vote
1answer
11 views

Derivate of inverse of composite function

I'm very confused, and this is probably a stupid question. I want to calculate $ \frac{d}{dx} f^{-1}(g^{-1}(x))$. However, I get two seemingly different results taking two different approaches. I. $\...
2
votes
4answers
71 views

If $\tan 9\theta = 3/4$, then find the value of $3\csc 3\theta - 4\sec 3\theta$.

If $\tan9\theta=\dfrac{3}{4}$, where $0<\theta<\dfrac{\pi}{18}$, then find the value of $3\csc 3\theta - 4\sec 3\theta$. My approach:- $$\begin{align*} \tan9\theta &=\frac{3}{4} \\[6pt] \...
2
votes
1answer
101 views

Inverse of $\frac{\sin(x)}{x}$

How would one find the inverse of the function $y=\frac{\sin(x)}{x}$? Here are my steps: $y=\frac{\sin(x)}{x}$, $x=\frac{\sin(y)}{y}$, $xy=\sin(y)$, $\arcsin(xy)=y$, After that step, I can’t find a ...
1
vote
2answers
49 views

Prove that $f(f^{-1}(V))=f^{-1}(f(V))$

Let $f: X\to Y$ be bijective, and $f^{-1}: Y\to X$ be it's inverse. If $V\subseteq Y$, show that the forward image of $V$ under $f^{-1}$ is the same set as the inverse image of $V$ under $f$. I ...
0
votes
1answer
361 views

Use the complex definition of $\sin z$ to find an expression for $\sin^{-1} z$

Using $$\sin z = \frac{e^{iz}-e^{-iz}}{2i}$$ Prove $$\sin^{-1} z =\frac{1}{i}\ln(iz+\sqrt{1-z^2}) $$ Attempted solution: Let $\sin z = u$ and $e^{iz} = v$. \begin{align*}& 2iu = v - \frac{1}{v}\...
0
votes
0answers
15 views

continuity of isomorphism of unit circle

I try to show $\mathbb{S}^1\cong[0,1)$, by the map $f(x) = (\cos2\pi x,\sin2\pi x)$, for $x\in[0,1)$. It's clear that $f$ is continuous and bijective. But I don't know how to show the inverse map $f^{-...
1
vote
1answer
49 views

is it possible to solve it for variable r?

Here is the annuitet payment formula: $$p = s \cdot \left(r + \frac{r}{(r+1)^t-1}\right)$$ Is it possible to solve it for rate ?
1
vote
1answer
41 views

Inverse function of $ax + bx^3$

I am trying to find the inverse of the function $y = f(x) = ax + bx^3$, i.e. $x = f^{-1}(y)$. (The equation arises in the modeling of a certain type of transmission used in robots) Looking at the ...
0
votes
2answers
124 views

Inverse function Khan Academy [closed]

I'm doing an exercise on inverse functions on Khan Academy and cannot get with the first exercise. Where to find number $93$? There is no such number in the table...
0
votes
0answers
60 views

Inverse of $f(x) = x^{3}-x^{2}$

Can anybody find the inverse of $f:(-1,0) \to \mathbb{R}$ such that $f(x) = x^{3} - x^{2}$ ?
0
votes
0answers
23 views

Continuity of a function on a metric space and its consequences

Let $f : X → Y$ be a given function, and suppose that $f^{-1}(C)$ is an open subset of $X$ whenever C is an open subset of $Y$ . (a) Prove that $f$ is continuous on $X$. (b) Prove that $f^{-1}(B)$ ...
0
votes
1answer
30 views

Continuity and Uniform Continuity of inverse functions

Let $f : X → Y$ be a given function, and suppose that $f^{-1}(C)$ is an open subset of $X$ whenever C is an open subset of $Y$ . (a) Prove that f is continuous on $X$. (b) Prove that $f^{-1}(B)$ is ...
1
vote
4answers
48 views

Showing that $\arcsin x + \arccos y = \frac{\pi}{2}$ if and only if $x = y$

I was just wondering about this identity: $$\arcsin x + \arccos x = \frac{\pi}{2} .$$ That a thought came to my mind that in general $$\arcsin x + \arccos y = \frac{\pi}{2} \qquad \textrm{if and ...
0
votes
1answer
43 views

How to show $12^a \cdot 18^b$ is injective

We are told that $f: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ I know one method is to prove that a inverse exists, but I'm not 100% sure how to do that in this case. so instead I decided ...
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
1answer
117 views

Existence of a particular inverse transformation

Let $h : \mathbb{R}^D \rightarrow \mathbb{R}^d$, where $d < D$, be a differentiable function. I would like to find minimal conditions under which there exists a differentiable function $g : \mathbb{...
2
votes
1answer
75 views

An additional assumption to the inverse function theorem.

The theorem is given below: And here is the question: Could anyone give me a hint on how to prove the required in the question please?
0
votes
0answers
45 views

Expected value of a non-linear function of a normal random variable

Consider a random variable $X\sim N\left(\mu,\sigma^2\right)$, and a monotonically increasing non-linear function of $X$, call it $Y=f\left(X\right)$, defined as: $$Y=f\left(X\right)=\Phi\Big(a-b\,\...
-1
votes
3answers
64 views

inverse of $y=\frac{x}{\log{x}}$?

By Prime Number theorem $\pi(x)=\frac{x}{\log{x}}$ for large x Putting $x=p_n$ where $p_n$ denotes $n^{th}$ prime number, We have, $\pi(p_n)=\frac{p_n}{\log{p_n}}$, $\because \pi(p_n)=n$, $\...