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.
1,036 questions
5
votes
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 "...
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
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
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
$$1 +\...
0
votes
0answers
18 views
Finding the inverse of a complex function to derive information about it.
Im sorry if im butchering some of this concepts, i am a cs undergrad and have very little understanding of advanced mathematics.
I found an intresting function i quite like, such function is : -y = ...
1
vote
0answers
32 views
Is there any formula for arctan x + arctan y + arctan z. ? ( For various case)
Is there any formula for $ arctan x + arctan y + arctan z $ ?
One way to solve such type of question is to is two take two at a time and solve them.
I know the general formula for
$ tarctan x + ...
0
votes
2answers
42 views
Why isn’t the integral of $x^2$ from $0$ to $5$ equal to the integral of $\sqrt x$ from $0$ to $25$? [duplicate]
My calculator says that the former is 41.67, and that the latter is 83.34. Why is that? Shouldn’t they be equal?
1
vote
1answer
37 views
Image of an interior point
Consider $\textbf{f}: U\subset \mathbb{R}^n\to \mathbb{R}^n$ and $\textbf{a} \in U$. Suppose that $\textbf{a}$ is an interior point of $U$ and $\textbf{f}$ is differentiable at $\textbf{a}$ with $\det(...
0
votes
0answers
12 views
Inverse of Legendre Dual
Reading the book "Concentration inequalities A nonasymptotic theory of independence" I came across the following results (Lemma 2.4):
Let $f$ be a convex function such that $f(0)=f'(0)=0$ and such ...
1
vote
1answer
20 views
unexpected case of gradient of inverse function
if $y=x^3+3x+2$ is the original equation then, $$\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}$$ This gives me answer of $\...
0
votes
0answers
30 views
How do I draw the graphs of $\sin^{-1}(\frac{2x}{1+x^2})$ and $\tan^{-1}(\frac{2x}{1-x^2})$?
How do I draw the graphs of $\sin^{-1}(\frac{2x}{1+x^2})$ and $\tan^{-1}(\frac{2x}{1-x^2})$? The solution of a question in my books how these graphs and draws the conclusion that from them it is seen $...
0
votes
4answers
25 views
If $x\in [-1,0)$, then what is the value of $\cos^{-1} (2x^2-1) - 2 \sin^{-1} x$?
If x belongs to $[-1,0)$ then what is $\cos^{-1} (2x^2-1) - 2 \sin^{-1} (x) $? This is a question in my book, but I have doubts about solving it. I tried putting $\sin a = x$ and got $\pi - 2a - 2a = \...
1
vote
4answers
61 views
Differentiating $\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$
Find the derivative with respect to $x$ of
$$\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
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
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
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 ...
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
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
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
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
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: $$F(x) = \begin{cases}
\frac{1}{2}e^{\...
1
vote
1answer
11 views
Derivate of inverse of composite function
I'm very confused, and this is probably a stupid question.
I want to calculate $ \frac{d}{dx} f^{-1}(g^{-1}(x))$. However, I get two seemingly different results taking two different approaches.
I. $\...
2
votes
1answer
101 views
Inverse of $\frac{\sin(x)}{x}$
How would one find the inverse of the function $y=\frac{\sin(x)}{x}$? Here are my steps:
$y=\frac{\sin(x)}{x}$,
$x=\frac{\sin(y)}{y}$,
$xy=\sin(y)$,
$\arcsin(xy)=y$,
After that step, I can’t find a ...
1
vote
2answers
49 views
Prove that $f(f^{-1}(V))=f^{-1}(f(V))$
Let $f: X\to Y$ be bijective, and $f^{-1}: Y\to X$ be it's inverse. If
$V\subseteq Y$, show that the forward image of $V$ under $f^{-1}$ is
the same set as the inverse image of $V$ under $f$.
I ...
2
votes
4answers
71 views
If $\tan 9\theta = 3/4$, then find the value of $3\csc 3\theta - 4\sec 3\theta$.
If $\tan9\theta=\dfrac{3}{4}$, where $0<\theta<\dfrac{\pi}{18}$, then find the value of $3\csc 3\theta - 4\sec 3\theta$.
My approach:-
$$\begin{align*} \tan9\theta &=\frac{3}{4} \\[6pt] \...
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
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 ...
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 ?
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 $f^{-1}(B)$ ...
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
$$\arcsin x + \arccos y = \frac{\pi}{2} \qquad \textrm{if and ...
0
votes
1answer
43 views
How to show $12^a \cdot 18^b$ is injective
We are told that $f: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$
I know one method is to prove that a inverse exists, but I'm not 100% sure how to do that in this case. so instead I decided ...
0
votes
0answers
45 views
Expected value of a non-linear function of a normal random variable
Consider a random variable $X\sim N\left(\mu,\sigma^2\right)$, and a monotonically increasing non-linear function of $X$, call it $Y=f\left(X\right)$, defined as: $$Y=f\left(X\right)=\Phi\Big(a-b\,\...
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?
-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, $\pi(p_n)=\frac{p_n}{\log{p_n}}$,
$\because \pi(p_n)=n$,
$\...
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
1answer
37 views
Inverse of a piecewise continuous function.
I am trying to find the inverse of a two dimensional map $f\left(\begin{bmatrix} x\\y\end{bmatrix}\right)$,
For example
$$
f\left(\begin{bmatrix}x\\y \end{bmatrix}\right) = \begin{cases} ax + by, &...
0
votes
1answer
19 views
digamma function inverse and special value
What the inverse of the digamma function?, and how can I write the x for
$$ ψ(x)=1$$ and $$1<x$$ $[x ≈ 3.20317146837693106929448152]$ as irrasional number [not a new one a familiar old one]
4
votes
1answer
117 views
Existence of a particular inverse transformation
Let $h : \mathbb{R}^D \rightarrow \mathbb{R}^d$, where $d < D$, be a differentiable function. I would like to find minimal conditions under which there exists a differentiable function $g : \mathbb{...
3
votes
1answer
96 views
+50
Inverse/Reverse of Number of Permutations and of Number of Combinations with Repetitions?
For an engineering application, I need the inverse functions of the computations of the number of combinations and permutations.
In the thread How to reverse the $n$ choose $k$ formula? it shows how ...
0
votes
2answers
42 views
An equivalence between function $f(x)$ and $f^{-1}(x)$.
Recently on this answer to one of my questions user farruhota replied that
Alternatively, note the property of inverse function:
$$f(f^{-1}(x))=f^{-1}(f(x))=x$$
Hence:
$$f(f(x))=x \iff f(x)...
0
votes
1answer
24 views
Prove that $(\pi_1)^{−1}(A) = A × Y$ and $(\pi_2)^{−1}(B) = X × B.$
Let $X$ and $Y$ be sets, let $A \subset X$ and $B \subset Y$ be subsets and let $\pi_1: X\times Y \to X$ and $\pi_2: X\times Y \to Y$ be projection maps. Prove that $(\pi_1)^{−1}(A) = A \times Y$ and $...
0
votes
0answers
31 views
How to invert this type of infinite series?
If have a function $f$ given by a series
$$
f(z) = \sum_{n,m = 0}^\infty u_{n,m} z^{n + m t}
$$
for some $t\in\mathbb{R}^+$.
Is there an straightforward way (something similar to the inversion ...
