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.
1,632
questions
0
votes
1answer
25 views
If the area of △PQR is A, give the expressions that complete the equation for the measure of ∠R.
Not sure if this is the wrong image or equation, as there is no numbers to go along with the question
1
vote
1answer
25 views
$\mathcal{B}$ an algebra $\implies f^{-1}(\mathcal{B}) $ an algebra
Im trying to prove the following statement:
Let $f: \Omega \to E$.
$\mathcal{B}$ an algebra on $E$ $\implies f^{-1}(\mathcal{B}) $ an algebra on $\Omega$.
To show that $f^{-1}(\mathcal{B})$ is an ...
2
votes
0answers
21 views
Proof of integral of inverse function without differentiability
The standard method for finding the integral of an inverse function involves substituting the function into the integral, and in the process, taking its derivative. However, this assumes that the ...
0
votes
2answers
39 views
The axiom of choice for a category
I am currently studying the counterpart of axiom of choice in ETCS which is the axiom that every surjective function has a right inverse.
In category of sets the surjective functions are epimorphsims ...
0
votes
1answer
30 views
Need help on understand a question on inverse functions
I need help understanding what this question is asking and I am not sure what to do. The question is given below
For each number y find the maximum value of $yx - 2x^4$. This maximum is a function $G(...
2
votes
1answer
41 views
Restricting Domain and Range in Inverse Trigonometric Function
After an explanation of the restricted domains and ranges of inverse trigonometric functions, I.M. Gelfand's Trigonometry gives the following exercise:
Show that $$\sin(\arccos b) = \pm \sqrt{1-b^2}$$ ...
1
vote
1answer
25 views
Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?
Let's consider the rational functions whose numerator and denominator of the function term are coprime.
Which kinds of rational functions of one variable have an inverse relation that contains a ...
1
vote
2answers
44 views
Proving that if $y=\operatorname{arcsec}(x)$ then $\frac{dy}{dx}=\frac{1}{x\sqrt{x^2-1}}$
I'm trying to prove this formula however, I cannot seem to figure out how to single out the $x$ and remove its power. I would very much appreciate your help towards this.
$$y=\operatorname{arcsec}(x) \...
0
votes
1answer
17 views
Finding a general form of the inverse of a function
I have obtained the asymptotic expansion of some function at infinity and the leading orders are of the following form
$$\rho (r)=\frac{r^q}{2 q}+\frac{3 (q-1) r^{2 q-1}}{8 q (2 q-1)}+\frac{3 r^{2 q-1}...
0
votes
1answer
31 views
Series expansion at branch cut
Basically, I have obtained the function $\rho (r)$ below as a result of integrating
$$\rho(r)=\int_{b}^{r}\frac{dx}{\sqrt{1-(b_{0}/x)^{1-q}}}$$
which results to
$$\rho=\frac{2b}{1-q}\sqrt{1-\left(\...
0
votes
1answer
23 views
studying for clep calculus exam inverse functions and derivates
I was taking a practice clep calculus exam I found online and I do not understand how the correct answer was derived.
$f(x) = x^3 + x$. and $h(x)$ is an inverse function of. find $h'(2)$??? I know ...
1
vote
1answer
47 views
How to inverse the function $\frac13e^x+\ln(3x+1)$
What would be the strategy and final result of inverting the function
$$f(x) = \frac{1}{3}e^x+\ln(3x+1)$$
1
vote
0answers
32 views
What is the inverse of $G: (x,y) \rightarrow (x, 1-ye^x+xe^y)$
How can I calculate the inverse of
$$
\begin{align*}
G: \mathbb{R}^2 &\rightarrow \mathbb{R}^2\\
(x,y) &\mapsto (x, 1-ye^x+xe^y)
\end{align*}
$$
I actually just need the inverse in a ...
3
votes
1answer
69 views
Prove that if $f^{(k)}(f^{-1}(x))$ exists, and is nonzero, then $(f^{-1})^{(k)}(x)$ exists.
I think I found an error and just want to confirm.
This is from Calculus by Michael Spivak, 3rd edition, Chapter 12 (Inverse Functions), problem 21.
12-21. Prove that if $f^{(k)}(f^{-1}(x))$ exists, ...
0
votes
1answer
20 views
Invertible Conditions in Polynomial Rings
Let ${\bf a}(x)\in (\mathbb{Z}/q\mathbb{Z})[x]$, where $q$ is a prime. Suppose that
${\bf a}(1)\equiv 0\pmod{q}$.
Question: How to show that ${\bf a}(x)$ is not invertible in $\frac{\mathbb{Z}/q\...
5
votes
2answers
318 views
Derivative and inverse
Is there a function whos derivative of its inverse equals the inverse of its derivative? This may also hold on a specific interval only.
$$\frac{d}{dx}f^{\langle-1\rangle}(x)=\left(\frac{d}{dx}f(x)\...
0
votes
0answers
19 views
Monotone function and inverse function
Show that the function $y= f(x)= 1+ x^{2} + \arctan x^{2}$ is strictly monotone in $[0,+\infty ]$. If $f^{-1}$ is the reverse function of $f$, calculate the limit $$\lim_{y\rightarrow 1+}\frac{f^{-1}(...
1
vote
0answers
73 views
Show that the inverse function is continuous
Let be $f:I\to\mathbb{R}$ a continuous, bijective function and $I$ an intervall of $\mathbb{R}$. Further, let be $g:J\to I$ the inverse function of $f$ such that for all $y\in J$ we have $f(g(y))=y$ ...
0
votes
0answers
23 views
Solution by separation of variables, invertibility, monotony
Suppose I have a separable ODE $y'=g(t)\,h(y)$, where I can find a solution $y: I \to \mathbb{R},\;t \mapsto y(t)$ by separation of variables. (h, g are continuous).
My intuitive idea is, that the ...
0
votes
2answers
64 views
Range and inverse of $f(x)=\sqrt{4x^2 + 1}$ on $[0,\infty)$
Consider the function $f(x) = \sqrt{4x^2 + 1}$ for $x \in [0,\infty)$. Denote the range of $f$ by $Y$. There is a function $g$ with domain $Y$ so that $g(f(x)) = x$
for every $x \in [0, \infty)$ and $...
0
votes
1answer
39 views
Question about definition of inverse function
So, I've been reading Precalculus books and I found a question in the definition of inverse function,
some of my books say:
Let $f\colon X\to Y$ be a real function, where $X,Y\subseteq\mathbb{R}$ ...
1
vote
1answer
61 views
Proof for sum and difference of angles in terms of tan inverse
We have the following formulas for sum of angles when angles are in terms of tan inverse:
$\tan^{-1}(x)+\tan^{-1}(y)$ =
$\tan^{-1}((x+y)/(1-xy))$, if $xy<1$
$\pi+\tan^{-1}((x+y)/(1-xy))$, if $x&...
1
vote
0answers
40 views
A problem with diffeomorphisms
For $n\leq k$ we define inclusion $i:\mathbb{R}^n \rightarrow \mathbb{R}^k$ as $i(a_1,...,a_n)=(a_1,...,a_n,0,...,0)$
For $k\leq n$ we define projection $\pi:\mathbb{R}^n \rightarrow \mathbb{R}^k$ as $...
2
votes
1answer
58 views
Suppose $g(u) = \int_{2}^{u} \sqrt{1 + t^3} \ dt$. Find the value of $(g^{-1})'(0)$ if it exists.
I want some verification to see if I am doing this question right. Here is the question, and my attempt:
Suppose $g(u) = \int_{2}^{u} \sqrt{1 + t^3} \ dt$. Find the value of $(g^{-1})'(0)$ if it ...
0
votes
2answers
47 views
How can I find the inverse of $y = \frac{x}{1-x^2}$?
In example $1.5$ of Cracking the GRE Subject Test, the authors make the following calculation in one step with no additional commentary:
we interchange $x$ and $y$ and solve for $y$:
\begin{align}
\...
0
votes
3answers
46 views
How to find inverse function of function with parameter?
I'm trying to find the inverse function of this function:
$$
f_a(x) = \frac{x^a}{x^a + (1-x)^a}
$$
$a$ - parameter
$x$ - variable
Is it possible to find if there is a parameter here?
Hope for your ...
2
votes
1answer
49 views
How can I get a function that is “approximately similar” to the part I care about in another function?
I have a function f(x)=$(a+bx)e^{-cx}, x\ge0$ where $a$, $b$, and $c$ are constants, and $c>0$.
How can I get a function that is "approximately the same" as the part after the hump?
I ...
0
votes
3answers
64 views
Prove that $\{Y:Y \subseteq X\}$ is a set.
I am a beginner and self-learning Real Analysis. Problem 3.4.6 from Tao's Analysis-I poses the following question. I'd like to someone to verify my proof, and help ...
-1
votes
4answers
60 views
If $n$ is even, prove $P(x)$ cannot have an inverse function
If $P(x)$ is a polynomial of degree $n$, how can I prove that if $n$ is even, $P(x)$ cannot have an inverse function.
2
votes
4answers
80 views
Inverse function of $x+n^x$
I need to find the inverse function of $f(x)=x+n^x$, where $n$ is a variable in the range $(0, 1]$. I understand that the result would not be a function because of the vertical-line test, so it would ...
0
votes
0answers
23 views
defining the product for the empty set of indices in order to make an equation hold
Let $S$ be a set and let $p_1, \cdots, p_n: S \rightarrow \{0,1\}$ be functions taking only the values $0$ and $1$.
Define the complements as $p_i':= 1-p_i$ and for a set $A \in M$ with $M$ being a ...
0
votes
1answer
38 views
Scholarly references for definition and discussion of left and right inverses of functions
Can someone share some good scholarly references where the left and right inverse of functions are defined, and their properties analysed?
Even better if they are references that also discuss various ...
0
votes
1answer
123 views
If $n$ is even, then $P(x)$ cannot have an inverse
Here is the problem:
Let $P(x)$ be a polynomial of degree $n$. If $n$ is even, prove that $P(x)$ cannot have an inverse over $\Bbb R$.
I am not exactly sure how to proceed from here. But I did start ...
0
votes
1answer
27 views
Given $f(x) = x^3 − 3x^2 − 1$, $x \geq 2$, find $f^{−1} {'} (−1)$
Given $f(x) = x^3 - 3x^2 -1, x\ge2$, find $f^{-1}{'}(-1)$
I know I should show work I have done on this question when I ask on stackexchange but I literally get stucked at the beginning. I cannot ...
2
votes
1answer
55 views
Ideas on how to integrate $\int \frac{dx}{\sqrt{(a-x)(x-b)(x^2+q+px)}}$ where $a>b$
I am trying to integrate $$\int \frac{dx}{\sqrt{(a-x)(x-b)(x^2+q+px)}},$$ where $a>b$, and $p$ and $q$ are real positive constants.
I would be very grateful if I can get some hints on how to ...
0
votes
2answers
39 views
Two answers of $\frac{\mathrm d}{\mathrm dx}\{\cot ^{-1}\left(\frac{1+x}{1-x}\right)\}$
If $$y=\cot ^{-1}\left(\frac{1+x}{1-x}\right)$$ then $\dfrac{\mathrm d y}{\mathrm d x}=?$
By taking $x=\tan y$ ,then
$y=\cot ^{-1}\left(\dfrac{1+\tan y}{1-\tan y}\right)$
$y=\cot ^{-1}\left(\dfrac{\...
2
votes
2answers
45 views
How to contextualize an inverse function for interpolation purposes?
I have a function that describes the growth of a population in a logistic term:
$$N_t = rN_0(1-\frac{N_0}{\kappa})$$
I would like to interpolate the values so that, given a known $y$ I can get the ...
4
votes
1answer
36 views
Composition of inverse trigonometric functions with trigonometric functions
I found in this Wikipedia article a useful table showing the algebraic expressions for the composition of trigonometric functions with inverse trigonometric functions, along with a picture explaining ...
1
vote
1answer
26 views
Confusion regarding the domain of an inverse function?
In my textbook, the following example is given. I am confused as to why the domain for $f^-1(x) = x^2+5$ is "all real $x$ greater than $0$". Doesn't this function, take all real x-values? Or ...
0
votes
2answers
83 views
Find the inverse of $f(x)=\sqrt{3x^2 +1}$
Find the inverse of $f(x)=\sqrt{3x^2 +1}$.
We don't know if $f$ is invertible so we have to prove it, but how?
$f$ bijective $\Leftrightarrow $ $f$ invertible
Is $f$ injective? $ x \neq y \Rightarrow ...
0
votes
1answer
41 views
Finding the Inverse Cumulative Distribution function for the Rice distribution
I'm interested in implementing an algorithm for producing random samples according to some Rician distribution (sometimes known as the Ricean distribution, see).
I intend to produce these psuedo-...
-1
votes
2answers
126 views
How to simplify $cosh(arccoth(x))$? [closed]
I was given the problem to simplify $\cosh(\text {arccoth} (x))$ for $|x| > 1$, and I was just wondering how I would do that.
0
votes
1answer
35 views
Given a function and compute the definite integral of its inverse
So I have a function
$$ f(x)=x^{3}-12x^{2}+69x+6 $$
The question askes me to find the relationship between $$\int_{1}^{2} f(x)dx$$ $$and$$ $$\int_{64}^{104} f^{-1}(x)dx$$
And compute the value of $\...
0
votes
0answers
21 views
How to find the inverse of $U(T) = \frac{R_2(T)}{R_1 + R_2(T)} \cdot U_1$
I'm trying to find an inverse function of:
$$U(T) = \frac{R_2(T)}{R_1 + R_2(T)} \cdot U_1$$
where $R_2(T) = 1330 \cdot e^{3531.55 \cdot \left(\frac{1}{T + 273.15} - \frac{1}{279.75}\right)}$ and $R_1 =...
0
votes
1answer
76 views
counterexample for : $f$. [closed]
It is given that $f$ is a open set in $R^3$ and consider $C^2$ function $f: A \rightarrow R^n $ with $(a) \neq 3$ for every $a \in A$. I want to give counterexample for : $f$ maps every closed set in ...
2
votes
2answers
62 views
Let $f(x)=e^x+x^3+x-3$. This function is invertible. What is the value of $f^{-1}(-2)$?
I am having trouble with this question. I don't have any calculus background.
I think $f^{-1}$ means the inverse of $f(x)$
Does that mean I have to find the inverse of $f(x)$ first? But again, I am ...
-2
votes
2answers
43 views
Determine if $f(x) = \frac{100}{1+2^{-x}}$ is one-to-one [closed]
I currently have $$f(x) = \frac{100}{1+2^{-x}}$$ and I'm trying to determine if it is one-to-one. I have looked at the other posts however I'm not sure as to how to deal with the $-x$.
1
vote
4answers
65 views
How to find the inverse function of $f(x)=\sin^2\left(\frac{2x+1}{3}\right)$ and find its domain.
$f(x)=\sin^2\left(\frac{2x+1}{3}\right)$ which is restricted on $-\frac{3\pi+1}{2}\le x< -\frac{3\pi+2}{4}$
I know I have to switch the $f(x)$ and the $y$:
$x=\sin^2\left(\frac{2f^{-1}(x)+1}{3}\...
1
vote
0answers
33 views
Limit of an involution at the extremities
Let $f:]a,b[\to ]a,b[$ an involution not necessarily continuous . It's true that
$$\lim_{x\to b^-} f(x)=a\iff \lim_{x\to a^+} f(x)=b\quad ?$$
I can't find any counterexample
Addition: Question ...
0
votes
1answer
45 views
