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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.

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Inversion of Gradient Noise Function

Brief As Possible Given any equation $f(x)$ which adheres to a predefined pattern, there is always a way (sometimes not directly calculable) to get $f^{-1}(x)$ (which may be a set) such that $f^{-1}(...
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1answer
19 views

Inverse of function in terms of $f^{-1}(x)$

The question is how do i find the inverse of function $g(x)$ in terms of $f^{-1}(x)$ if $g(x)=f(x)-2$. The answer is $g^{-1}(x)=f^{-1}(x+2)$ but i dont understand how to get there. Thanks.
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4answers
250 views

Looking for a simple interpretation

$f(x) = 10-x$ If I plugin $x=2$, I get $f(2)=10-2=8$. If I want to know what I must plugin to get $8$, again I simply plugin $8$ into $f(x)$ : $f(8) = 10-8=2$. One can conclude $f(x) = f^{-1}(x)$. ...
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1answer
62 views

Sum of $50$ terms of $\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21+…$

Find the sum of the first $50$ terms of the series $$\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21+.....$$ $$ \sum_1^{50}=\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21+.....\\ =\tan^{-1}\frac{1}{3}+\...
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1answer
23 views

Is this pair of equations impossible to solve for x? $y_1 = x_2 - v^{\pm 1}e^{-x_1}$, or equivalently $(x - y)c^{\exp(-x)} = z$

Original: I'm trying to solve the following for $x_1$ and $x_2$, $$ y_1 = x_2 - v\, e^{-x_1} $$ $$ y_2 = x_1 - \frac{1}{v}\, e^{-x_2} $$ in terms of $y_1$, $y_2$, and $v$, which are known and real, ...
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1answer
51 views

Find the value of $3\tan^{-1}\frac{1}{2}+2\tan^{-1}\frac{1}{5}+\sin^{-1}\frac{142}{65\sqrt{5}}$

Find the value of $3\tan^{-1}\left(\dfrac{1}{2}\right)+2\tan^{-1}\left(\dfrac{1}{5}\right)+\sin^{-1}\left(\dfrac{142}{65\sqrt{5}}\right)$ My reference gives the solution $0$ to this problem. My ...
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1answer
42 views

If $f(x)=\int_{3}^x \sqrt{1+t^3} dt$ Find $(f^{-1})'(0)$

Consider the integral $$f(x)=\int_{3}^x \sqrt{1+t^3} dt$$ Using Theorem 7 from my textbook which is $$\frac{1}{f'(f^{-1}(a))}$$ $f'(t)= \frac{2t^3}{2\sqrt{1+t^3}}$ $f^{-1}(t)=\sqrt[3]{t^2-1}$ I ...
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2answers
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Simplified form of $\cos^{-1}\big[\frac{3}{5}\cdot\cos x+\frac{4}{5}\cdot\sin x\big]$, where $x\in\big[\frac{-3\pi}{4},\frac{3\pi}{4}\big]$

Find the simplified form of $\cos^{-1}\bigg[\dfrac{3}{5}\cdot\cos x+\dfrac{4}{5}\cdot\sin x\bigg]$, where $x\in\Big[\dfrac{-3\pi}{4},\dfrac{3\pi}{4}\Big]$ My reference gives the solution $\tan^{-1}\...
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1answer
53 views

Find Random Number Generator following the density $f (x) = \frac{1 + \alpha x}{2}$, $ −1 ≤ x ≤ 1$, $−1 ≤\alpha ≤ 1$

How could random variables with the following density function be generated from a uniform random number generator? $f (x) = \frac{1 + \alpha x}{2}$, $ −1 ≤ x ≤ 1$, $−1 ≤\alpha ≤ 1$. To find the ...
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4answers
21 views

The function $g(x)=3x+\ln2x$ for $x>0$. Find $g'(x)$ and prove that $g(x)$ has an inverse function.

The function $g(x)=3x+\ln2x$ for $x>0$. Find $g'(x)$ and prove that $g(x)$ has an inverse function. So $g'(x)=3+\frac{1}{x} $ for $x>0$ But I have no idea how I'm supposed to prove that this ...
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1answer
37 views

Find all the bijective functions $f:[0,1]\to[0,1]$ such that $x=\frac{1}{2}\big(f(x)+f^{-1}(x)\big)$ for all $x\in[0,1]$.

Find all bijective functions $ f : [0,1] \to [0,1]$ that satisfy the equation $$x=\frac{1}{2} \big(f(x) +f^{-1} (x)\big)\,\forall x \in[0,1]\,.$$ I honestly don't know how to approach this. ...
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Finding the inverse of a CDF [on hold]

I have a Beta(3, 1) distribution with CDF: Fx(x) = {0 if x <0 {x^3 if 0 < x < 1 {1 if x > 1 I need to find the inverse of that ...
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2answers
46 views

If the image of $f$ is equal the domain then the function has an inverse function [on hold]

Is the above statement right? If it is, then is it a sufficient condition for inverse function to exists? Also does the statement domain $=$ image imply that the function is a bijection?
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3answers
44 views

Solving inverse function

$$f : \Bbb R \rightarrow \Bbb R $$ $$f^{-1}(2x-7) = x-1, f(a-1) = 5$$ Determine $a$. The inverse of the function $f^{-1}(2x-7)$ is written as $$f\biggr (\dfrac{x+7}{2}\biggr ) = x-1$$ ...
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0answers
36 views

Complete expression for $3\cos^{-1}x$

Complete expression for $3\cos^{-1}x$ My Attempt Let $a=\cos^{-1}x\implies x=\cos a$ $$ \cos3a=4\cos^3a-3\cos a=4x^3-3x=\sin\bigg[\sin^{-1}\Big(4x^3-3x\Big)\bigg]\\ 3a=3\cos^{-1}x=2n\pi\pm\cos^{-1}\...
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3answers
70 views

Find the inverse of $f(x)=x^3+x+1$.

$$f(x)=x^3+x+1$$ I didn't learn this at school and I want to know how I can get the inverse of this function. Do you use differentiation? I have this solution but I don't understand what it ...
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1answer
33 views

$\cos^{-1}x+\cos^{-1}y=\cos^{-1}\Big(xy-\sqrt{1-x^2}\sqrt{1-y^2}\Big)$ true for all $x$?

$\cos^{-1}x+\cos^{-1}y=\cos^{-1}\Big(xy-\sqrt{1-x^2}\sqrt{1-y^2}\Big)$ true for all $x$ ? My Attempt Let $a=\cos^{-1}x$, $b=\cos^{-1}y\implies$$\cos a=x$, $\cos b=y$ and $a,b\in[0,\pi]\implies a+b\...
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1answer
57 views

Expression for $\cos^{-1}x\pm\cos^{-1}y$

As mentioned in Proof for the formula of sum of arcsine functions $\arcsin x+\arcsin y$ for $\sin^{-1}x+\sin^{-1}y$ $$ \sin^{-1}x+\sin^{-1}y= \begin{cases} \sin^{-1}( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;...
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4answers
67 views

$\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\frac{x+y+z-xyz}{1-xy-yz-zx}$ true for all $x$?

$\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\dfrac{x+y+z-xyz}{1-xy-yz-zx}$ true for all $x$ ? This expression is found without mentioning the domain of $x,y,z$, but I don't think its true for all $x,y,...
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2answers
37 views

Is $f(f^{-1}(x))=x \quad \land \quad f^{-1}(f(x))=x$ true for all inverse trigonometric functions?

Based on: $f(f^{-1}(x))=x \quad \quad\land\quad \quad f^{-1}(f(x))=x$ Are all the following definitions true? $\arcsin(\sin x)=x$ $\sin(\arcsin x) = x$ $\arccos (\cos x) = x$ $\cos (\arccos x) = x$ $...
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1answer
29 views

How do I check if a function has an inverse function?

From what I've learnt, a function $f$ has an inverse function $f^{-1}$ if the function $f$ is injective (one-to-one, horizontal rule applies). How can I check if a function has an inverse in the ...
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3answers
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Learning $\arcsin, \arccos, \arctan$ - how to?

Sorry for asking such question. I have a very basic understanding of $\arcsin, \arccos, \arctan$ functions. I do know how their graph looks like and not much more beyond that. Calculate: Which ...
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2answers
47 views

Difference between arcsin and inverse sine.

I first learned that arcsin and inverse sine are two ways of saying the same thing. But then I was thinking about the inverse sine function being a function, so it must be limited in it's range from -...
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1answer
19 views

$f: \mathbb{R}^2 \to \mathbb{R}^2$ with $f(x,y) := (e^x\cos y, e^x\sin y)$ locally/globally reversible and Jacobian matrix

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ with $f(x,y) := (e^x\cos y, e^x\sin y)$ I have to do the following things: Prove that $f$ is locally reversible everywhere. Is $f$ globally ...
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24 views

Inverse derivative

Note: $b(\cdot)$ is a function, $G(\cdot)$ is a cdf. Given that $b(v) = \beta$ and $b(v) = \cfrac{1}{G(v)} \int_{0}^{v}y_1 g(y_1)dy_1$ where $G(y_1) = Prob(Y_1 \leq y_1) = F^{n-1}(y_1)$ where F is a ...
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1answer
13 views

Why the value of the CDF within the corresponding range keeps the same even after applying a Nonlinear Transformation?

For transformations of random variables, why the value of the CDF within the corresponding range keeps the same even after applying a Nonlinear Transformation? For example. X ~ U(0, 2), the PDF of X ...
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1answer
26 views

Is there inverse to right for this function?

We have function $g: \mathbb R\to[11/4, \infty)$, with $g(x)=x^2-3x+5$ and we have to check if there exists $h:\mathbb R \to(-1, \infty) $ with property $g \circ h=1_{[11/4, \infty)}$, where $\circ$ ...
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1answer
49 views

How do I find the inverse of a derivative at $x=-1$?

Let $f(x)=x^3-3x^2-1$, $x\geq2$. Find the value of $\left(df/dx\right)^{-1}$ at $x=-1$. This is the work that I've done so far: Found $f'(x)=3x^2-3x$ Found the inverse of $f'(x)$ ►$3y^2-6y-x=0$ ...
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1answer
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Prove the inverse relation $f^{-1}$ of a function $f : A\rightarrow B$ is a function from B to A if and only if $f$ is bijective.

This proof is from "Mathematical Proofs: A Transition to Advanced Mathematics"(4th Ed.) on page 268. I understand how $f^{-1} : B\to A$ being a well-defined function implies that $f$ must be injective,...
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1answer
36 views

Integrating using inverse functions

Someone the other day told me about the idea of evaluating integrals using horizontal instead of vertical bars (apparently something to do with Lebesque integration but thats way too complicated for ...
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1answer
6 views

Inverse function of a bivariate function with linear constraint

Suppose we have a bijective function $f: x \mapsto y$ with an inverse $f^{-1}$, then from $y$ we can solve for $x$ with: $$f^{-1}(y)=x.$$ Now suppose that we have a bivariate surjective function $g: (...
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inverse function of a squared

i'm stuck on trying to find the inverse of the following function: $f($$x) = {\sqrt{x^2-4x}}$ What i did so far is: $y = {\sqrt{x^2-4x}}$ Swap $x = {\sqrt{y^2-4y}}$ Remove the root by squaring ...
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0answers
39 views

Why does $Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$ exhibit a bijection with the pairs (x,y)?

Consider $$Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$$ I discovers it essentially zigzags along the grid with $\mathbf N$ vs $\mathbf N$ (natural numbers). So intuitively, given any P(x,y) we follow ...
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1answer
38 views

Inverse of the asymptotic expansion of Gauss Hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below). Basically I want to series expand $\rho$ for large $r$ (i.e. as $r\to \infty$) and then ...
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2answers
47 views

Is $f(x)=x+\sin x$ a homeomorphic function?

I want to determine $f(x) = x+\sin x$ is homeomorphic or not on $\mathbb{R}$? A bijective continuous function is homeomorphic if its inverse is also continuous. I know that $f$ is bijective. Also $...
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1answer
16 views

Partial derivative of function of inverse function

I have got a probelm with the following task: $\frac{\partial }{\partial x}f(f^{-1}(x,t),\tau)$, where $f\in\mathscr{C}^{\infty}(\mathbb{R^2})$. My attemp is $\frac{\partial }{\partial x}f(f^{-1}(x,t),...
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0answers
10 views

Inverse function in integral form

Here is my question: Suppose my function is defined in terms of Riemann integral in the following form $$z=g(y)= \int_{}^{}f(x,y)dx$$. Is there any explicit formula of inverse function $h(z)$; $y=h(...
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2answers
73 views

Why are $\ln x$ and $e^x$ considered to be each others' inverses?

From my understanding the definition of a function's inverse is as follows. Take a function $f$ which has the inverse $f^{-1}$. This would mean that $f(f^{-1}(x)) = x$ and that $f^{-1}(f(x)) = x$ for ...
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1answer
16 views

Simplifying a proof that $f$ and $g$ are mutually inverse functions knowing that $f$ is an injection

To prove that $f$ and $g$ are mutually inverse bijections $A\to B$ and $B\to A$, it is necessary to prove: that really $f:A\to B$ and $g:B\to A$; for every $x_0\in A$ and $y=f(x_0)$ we have $x_1=x_0$ ...
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1answer
23 views

Set theory, functions and inverses

I'm doing an intro course on set theory and have the question if the inverses of the surjective functions in the sets A={a, b}and B= {c, d, e} are also functions. So far, I thought that the inverse ...
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1answer
15 views

Describe the preimage of any point $(b,c) \in \mathbb{R}^2$ for $F(x, y) = (x+y,xy)$

I plotted the line and the hyperbola implied by the first and second coordinates respectively. I realize that the preimage can be Empty for $c > 0 \land b < c$ Size one for $c = 0 \lor (b = c \...
4
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3answers
61 views

Showing $\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_0^1 \mid F^{-1}(u)-G^{-1}(u)\mid du$ with $F$, $G$ CDF functions

Let's $X$ and $Y$ have CDF functions admitting moment of order $1$. Let's be $F$ cdf of $X$ and $G$ cdf of $Y$. I want to show that $$\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_{0}^{1} \mid F^{-1}(...
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1answer
49 views

One to one functions and inverse

It lists one to one functions: $g=\{(-5,-3),(2,5),(3,-9),(8,3)\}$ $h(x)= 3x-2$ And it asks to find the following: $g^{-1} (3) = h^{-1}(x)= (h * h^{-1})(-5)=$ I really need help with this problem,...
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0answers
11 views

Integral of inverse fuctions for noncontinuous distributions

Let $ F:[a,b]\rightarrow[0,1] $ be an arbitrary (noncontinuous) distribution function. Denote with $Q(p)=inf\{x:p \leq F(x)\}$ the associated quantile function. I would like to use that $ \...
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2answers
46 views

Is $\sqrt{x}$ an even function

I'm going through some pre-calculus and I am presented with this rule. If $f(x)$ is an even function, then $f(x) = f(-x)$ So with the following example I have: $$f(x) = 3x^4 \\ f(-x) = 3(-x)^4 ...
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2answers
62 views

Inverse factorial function

I am wondering what is the inverse/opposite factorial function? e.g inverse-factorial(6)=3 Furthermore, I am intrigued to know the answer to: a!=π find a I would really appreciate if anyone could ...
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3answers
23 views

Having trouble getting started with a proof involving functions

The proof looks like this: Suppose that a function $f(x) = ax + b$ is its own inverse, meaning that $f^{-1}(x) = ax + b$ also. Prove that we must have either have $a = -1$ or $a = 1, b = 0$. From ...
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2answers
58 views

Inverse of $\ln(e^x-3)$

so the whole concept about inverses is a little foggy. Say you have function $f(x)=\ln(e^x-3)$ and you want to know the inverse function, then: $$\ln(e^x-3) = y$$ $$e^x-3 = e^y$$ $$e^x = e^...
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1answer
24 views

Fulfilling conditions of Inverse function(?)

Let's assume that $$h(g(t))=t$$ What conditions ae needed to say that $$g(h(t))=t$$ Is satisfied too? (Provided that $h$ and $g$ are continuous and derivatable, but not knowing whether they have ...
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1answer
39 views

using sinh(x) to find series representation of arcsinh(x)

From "Complex Variables Demystified", 2008, page 102: Given: $$sinh(z)=\frac{e^z-e^{-z}}{2}$$ find the series representation for arcsinh(x). Solution: (1) The Maclaurin theorem can be used to ...