0
votes
0answers
10 views

The inverse of a Moment generating function

The moment generating function of $X$ is $M_X(t) = \mathbb{E}[e^{tX}] = \int e^{tu}f_X(u)du$ where t is a complex variable and $f_X$ is the density of X. The cumulant generating funtion of $X$ is ...
0
votes
2answers
37 views

How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$?

Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$ This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$ I say it is surjective because for all really $x$ there is all real $y$, $x \in ...
1
vote
1answer
42 views

Finding the area bounded by two curves when in terms of $x = y^2$?

I can't seem to figure this problem out. Find the area bounded by the curves $x=2y-y^2$ and $x=4-y^2$, in the first quadrant. I am having difficulties with graphing the equations and coming up ...
4
votes
3answers
130 views

Determine the greatest interval where the function is invertible

The assingment is to determine the greatest interval around $x=0$ where the function: $$f(x)=x^5-5x+3$$ is invertible. After that, determine $(f^{-1})'(3)$ I have totally forgotten all about ...
1
vote
4answers
63 views

If a function $f$ is decreasing on its domain then would its inverse be increasing or decreasing?

I have a question concerned the inverse of a function $f$ and the sign of its derivative. If we are given a function $f$ that is decreasing on its domain, would its inverse $f^{-1}$ be increasing or ...
0
votes
2answers
58 views

How to find Inverse function value at given point? [closed]

How to solve this, If $f(x)=x^5+x^3+x$, then find $f^{-1}(3)$
0
votes
1answer
49 views

Show that a linear mapping is invertible over all $\Bbb R^{2}$

Show that (under appropriate assumptions) a general linear mapping $F(x,y) = (ax+by,cx+dy)$ is invertible over all of $\Bbb R^2$ (i.e. there is a single inverse for all of $\Bbb R^2$). What ...
2
votes
1answer
80 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
1
vote
2answers
49 views

How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
1
vote
1answer
31 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
1
vote
2answers
70 views

I need help finding the derivative of the inverse function.

So $$f(x)=\frac{x+1}{2x-1}$$ and $$g(x)$$ is an inverse of $$f(x)$$ I have the points on $f(x)$ of (2,1). So I know that $f(2)=1$, $g(1)=2$ and $g'(1)=\frac{1}{f'[g(1)]}$ so $g'(1)=\frac{1}{f'(2)}$ ...
0
votes
1answer
23 views

Inverse of Continuous Function on Closed Bounded Part of R. Why Bounded?

Consider the following proposition: Let $A$ be a closed bounded part of $\Bbb R$. Assume $f: A\rightarrow \Bbb R$ is a continuous injective function. Then $f^{-1}: f(A) \rightarrow A$ is also ...
0
votes
0answers
24 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
1
vote
3answers
48 views

Why rotating a function around line $y=x$ gives an inverse of this function?

So I'm trying to read through a book on calculus on my own and there is a statement that if we have a graph of some function $y=f(x)$ and this is an injective function, then rotating it around the ...
0
votes
1answer
42 views

Find the poles and residues in an awkward Laplace inversion

Assume that part c) has been proved and ignore parts c) & d). To invert the Laplace transform we would do $\displaystyle u(x,t)=\frac{1}{2\pi ...
1
vote
1answer
27 views

local invertibility does not imply global invertibility

What is an example of a smooth function with continuous derivatives, that is locally invertible but not globally, and the reason for that is not injectivity. My first idea was $f:\mathbb{R}^{2}\to ...
2
votes
1answer
35 views

I need to evaluate the Inverse Trig Integral

The integral is $$\int \frac{x\,dx}{(3+2x+2x^2)}.$$ I'm stuck with breaking the denominator into $u^2+a^2$.
2
votes
5answers
96 views

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

What approach would be ideal in finding the inverse of $f(x) = \frac{x+2}x$?
0
votes
1answer
19 views

Class of the inverse function

The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class. But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ ...
1
vote
2answers
78 views

Derivatives of component inverse functions

I might have missed the point of the following questions. Anyone kindly give a suggestion? Let $f:\mathbb{R}_\mathbf{x}^3\to\mathbb{R}_\mathbf{y}^3$ and ...
0
votes
1answer
47 views

Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
1
vote
2answers
85 views

Tangent line of the inverse function of $y = e^x + x$

I've been sitting on this problem for a while, hopefully you guys could give me a lead on what the hell is going on :) Let $f(x) = e^x + x$ Find the tangent line to $f^{-1}(y)$ (the inverse ...
0
votes
2answers
39 views

Find the function $h(x) = g(2g^{-1}(x))$

Show that the function $g(x) = x^4 + x^3 + 1$ is one-to-one on [0, 2]. In addition, for the function $h(x) = g(2g^{-1}(x))$, find h′(3). For the first part, I manage to prove that g(x) is increasing ...
0
votes
2answers
68 views

How do I find $(f^{-1})'(a)$? [closed]

if $$f(x) = 3x^3 + 3x^2 + 6x + 9 $$ $$a = 9$$ and also $$f(x) = 2x^3 + 3\sin x + 3\cos x$$ $$a = 3$$ I know I have to find the inverse but I think I’m getting overly complicated answers and my ...
1
vote
1answer
134 views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
0
votes
1answer
49 views

Could someone please help me solve this calculus problem?

For f(x)= integral 1/sqrt(4+e^t) dt from 0 to lnx, with x>0, find (f^-1)'(0) (that is, the derivative of the inverse of f at 0)
2
votes
5answers
93 views

Derivative of $ h(t)= \sin (\cos^{-1}t$)?

Can someone please explain the steps/rules I need to preform to find the derivative of $h(t)= \sin (\cos^{-1}t)$? I tried to used the product rule, and realized it was obviously a failure. Using ...
0
votes
3answers
116 views

Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...
0
votes
0answers
123 views

Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
2
votes
1answer
116 views

Implicit Function Theorem to show no function can be one to one

Apply Implicit Function Theorem to show that no $C^1$ function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ can be one to one near any point of its domain. Repeat the proof by using Inverse Mapping Theorem ...
2
votes
1answer
101 views

Apply Implicit Function Theorem

Apply Implicit Function Theorem to show that no $C^1$ function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ can be one to one near any point of its domain. Repeat the proof by using Inverse Mapping Theorem ...
0
votes
1answer
26 views

Chain rule with inverse function

In a proof, my professor shows: $ s = g^{-1}(u) $ $ ds = \frac{dg^{-1}(u)}{du} du $ , by the chain rule If I were to apply the chain rule to calculate ds, I would not get the du in the denominator. ...
2
votes
3answers
125 views

Invertible function $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $

How can I prove that $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $ is invertible. First I choose variable $x$ for $y$ and tried to switch and simplified the function but I am stuck. Need some help ...
2
votes
0answers
43 views

Integrating inverse functions

I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...
0
votes
1answer
60 views

Maximal value of domain for a function by looking at inverse function.

The function g:[–a,a]→ R, g(x)=sin(2(x-π/6))has an inverse function.The maximum possible value of a is: From what I understand the domain of g(x) is the range of g'(x). So I would try to find the ...
2
votes
0answers
24 views

probability subspaces that make entropy function equal to a constant value

Given the entropy fucntion: $$ H = -\sum_i^n p(i) \ln(p(i))\,.$$ where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a ...
1
vote
2answers
36 views

Show that this is one to one continuous and find its inverse which is continuous as well.

Let's define $\phi: \Bbb R^2 \to S$ for $S$ is subset of $\Bbb R^3$ For constant $a,b,c,d$ and $c\not =0$ $$\phi(x,y)=(x,y, \frac{d-ax-by}{c})$$ I want to show that the function $\phi$ is 1-1 ...
0
votes
1answer
26 views

Parametric Curves Existence of Tangent

If $\frac{dy}{dt}$ and $\frac{dx}{dt}$ exist, then does $\frac{dy}{dx}$ always exist when $\frac{dx}{dt} \not=0$? Indeed, this is a very simple question. Sorry but I'm just a beginner for ...
0
votes
1answer
62 views

Laplace Transformation spring question

Here is the question: http://i.imgur.com/XAH2UnX.jpg I can't seem to get the answer. Are those values in the writing like 1N/m even relevant? Can someone give me some direction? Thanks!
0
votes
4answers
127 views

If $g(x)=3+x+e^x$, then how do I find $g^{-1}(4)$?

If $g(x)=3+x+e^x$, then how do I find $g^{-1}(4)$? I took $g(x)=y$ and tried to solve the problem, but i could not get the solution.So, please help me by providing me the solution to my question.
1
vote
0answers
222 views

Calculus inverse tangent line

Question is: Let $ f(x)=(1/2)x^3+6x+4 $ and let $y=f^{-1}(x)$ be the inverse function of f. Determine the x-coordinates of the two points on the graph of the inverse function where the tangent line ...
0
votes
3answers
31 views

Inversing a function

I'm having some problems calculating the inverse of this function: $f(u,v)=(u+v,v-u^2)$, its domain is $D=\{(u,v)$ in $\Bbb R^2 : u>0\}$ Thanks in advance.
0
votes
1answer
59 views

Find x in polynomial given value of inverse

I'm studying for a test and this question has me really stumped: $f(x) = 2x^3+5x+3$. Find x if $f^{-1}(x) = 1$ I don't know how I am supposed to figure out the inverse of this polynomial. I used ...
1
vote
1answer
59 views

Computing the inverse Laplace transform of this?

What's the correct way to go about computing the Inverse Laplace transform of this? $$\frac{-2s + 1}{(s^2+2s+5)}$$ I Completed the square on the bottom but what do you do now? $$\frac{-2s + ...
3
votes
3answers
991 views

How do I find the inverse function of a polynomial with $x^5$?

I've been stumped on this problem for hours and cannot figure out how to do it from tons of tutorials. Please note: This is an intro to calculus, so we haven't learned derivatives or anything too ...
3
votes
2answers
110 views

Formula for Nth Derivative of Matrix Inverse

I was looking for an equation for the nth derivative of a matrix inverse, ie $\frac{d^n \bf{A}^{-1}}{dx^n}$ I know that the first derivative $\frac{\text{d} \bf{A}^{-1}}{\text{d}x} = -\bf{A}^{-1} ...
3
votes
1answer
111 views

A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
2
votes
2answers
40 views

How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$

Question: Let $E=\{(x,y): 0<y<x \}$ set $f(x,y)=(x+y, xy)$ for $(x,y)\in E$ a) How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$ And how to find formula for ...
3
votes
2answers
154 views

Proof with functions and inverse - Spivak

How does he know that $f^{-1}$ is one-one? Doesn't he have to prove that? Or is he applying his first theorem in the chapter to $f$? That is $f$ is a function if and only if $f^{-1}$ is ...
2
votes
2answers
103 views

Intermediate Value Theorem, least upper bounds, Spivak

I have only taken an excerpt from the book from Spivak 3rd edition page 220 in his "Inverse Function" chapter. At the end of the 3rd paragraph, he says that Then $f$ takes on some value ...