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.

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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
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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 ...
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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 ...
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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 ...
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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(...
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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 $$\sin(\arccos b) = \pm \sqrt{1-b^2}$$ ...
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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 ...
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2answers
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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) \...
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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}...
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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(\...
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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 ...
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How to inverse the function $\frac13e^x+\ln(3x+1)$

What would be the strategy and final result of inverting the function $$f(x) = \frac{1}{3}e^x+\ln(3x+1)$$
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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 ...
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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, ...
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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\...
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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)\...
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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}(...
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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 $f(g(y))=y$ ...
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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 ...
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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 $...
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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 $X,Y\subseteq\mathbb{R}$ ...
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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&...
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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 $...
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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 ...
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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} \...
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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 ...
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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 ...
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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 ...
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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.
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4answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{\...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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-...
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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.
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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 $\...
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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 =...
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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 ...
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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 ...
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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 $-x$.
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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}\...
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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 ...
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