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.
0
votes
0answers
3 views
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}(...
0
votes
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.
3
votes
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 $f(x) = f^{-1}(x)$. ...
1
vote
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 $$\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21+.....$$
$$
\sum_1^{50}=\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21+.....\\
=\tan^{-1}\frac{1}{3}+\...
0
votes
1answer
23 views
Is this pair of equations impossible to solve for x? $y_1 = x_2 - v^{\pm 1}e^{-x_1}$, or equivalently $(x - y)c^{\exp(-x)} = z$
Original:
I'm trying to solve the following for $x_1$ and $x_2$,
$$ y_1 = x_2 - v\, e^{-x_1} $$
$$ y_2 = x_1 - \frac{1}{v}\, e^{-x_2} $$
in terms of $y_1$, $y_2$, and $v$, which are known and real, ...
0
votes
1answer
51 views
Find the value of $3\tan^{-1}\frac{1}{2}+2\tan^{-1}\frac{1}{5}+\sin^{-1}\frac{142}{65\sqrt{5}}$
Find the value of $3\tan^{-1}\left(\dfrac{1}{2}\right)+2\tan^{-1}\left(\dfrac{1}{5}\right)+\sin^{-1}\left(\dfrac{142}{65\sqrt{5}}\right)$
My reference gives the solution $0$ to this problem.
My ...
2
votes
1answer
42 views
If $f(x)=\int_{3}^x \sqrt{1+t^3} dt$ Find $(f^{-1})'(0)$
Consider the integral $$f(x)=\int_{3}^x \sqrt{1+t^3} dt$$
Using Theorem 7 from my textbook which is $$\frac{1}{f'(f^{-1}(a))}$$
$f'(t)= \frac{2t^3}{2\sqrt{1+t^3}}$
$f^{-1}(t)=\sqrt[3]{t^2-1}$
I ...
1
vote
2answers
42 views
Simplified form of $\cos^{-1}\big[\frac{3}{5}\cdot\cos x+\frac{4}{5}\cdot\sin x\big]$, where $x\in\big[\frac{-3\pi}{4},\frac{3\pi}{4}\big]$
Find the simplified form of $\cos^{-1}\bigg[\dfrac{3}{5}\cdot\cos x+\dfrac{4}{5}\cdot\sin x\bigg]$, where $x\in\Big[\dfrac{-3\pi}{4},\dfrac{3\pi}{4}\Big]$
My reference gives the solution $\tan^{-1}\...
2
votes
1answer
53 views
Find Random Number Generator following the density $f (x) = \frac{1 + \alpha x}{2}$, $ −1 ≤ x ≤ 1$, $−1 ≤\alpha ≤ 1$
How could random variables with the following density function be generated
from a uniform random number generator? $f (x) = \frac{1 + \alpha x}{2}$, $ −1 ≤ x ≤ 1$, $−1 ≤\alpha ≤ 1$.
To find the ...
1
vote
4answers
21 views
The function $g(x)=3x+\ln2x$ for $x>0$. Find $g'(x)$ and prove that $g(x)$ has an inverse function.
The function $g(x)=3x+\ln2x$ for $x>0$. Find $g'(x)$ and prove that $g(x)$ has an inverse function.
So $g'(x)=3+\frac{1}{x} $ for $x>0$ But I have no idea how I'm supposed to prove that this ...
0
votes
1answer
37 views
Find all the bijective functions $f:[0,1]\to[0,1]$ such that $x=\frac{1}{2}\big(f(x)+f^{-1}(x)\big)$ for all $x\in[0,1]$.
Find all bijective functions $ f : [0,1] \to [0,1]$ that satisfy the equation $$x=\frac{1}{2} \big(f(x) +f^{-1} (x)\big)\,\forall x \in[0,1]\,.$$
I honestly don't know how to approach this. ...
-3
votes
0answers
21 views
Finding the inverse of a CDF [on hold]
I have a Beta(3, 1) distribution with CDF:
Fx(x) = {0 if x <0
{x^3 if 0 < x < 1
{1 if x > 1
I need to find the inverse of that ...
0
votes
2answers
46 views
If the image of $f$ is equal the domain then the function has an inverse function [on hold]
Is the above statement right? If it is, then is it a sufficient condition for inverse function to exists?
Also does the statement domain $=$ image imply that the function is a bijection?
0
votes
3answers
44 views
Solving inverse function
$$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$$
...
1
vote
0answers
36 views
Complete expression for $3\cos^{-1}x$
Complete expression for $3\cos^{-1}x$
My Attempt
Let $a=\cos^{-1}x\implies x=\cos a$
$$
\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}\...
2
votes
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 ...
0
votes
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$$\cos a=x$, $\cos b=y$ and $a,b\in[0,\pi]\implies a+b\...
1
vote
1answer
57 views
Expression for $\cos^{-1}x\pm\cos^{-1}y$
As mentioned in Proof for the formula of sum of arcsine functions $\arcsin x+\arcsin y$ for $\sin^{-1}x+\sin^{-1}y$
$$
\sin^{-1}x+\sin^{-1}y=
\begin{cases}
\sin^{-1}( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;...
0
votes
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,...
0
votes
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?
$\arcsin(\sin x)=x$
$\sin(\arcsin x) = x$
$\arccos (\cos x) = x$
$\cos (\arccos x) = x$
$...
0
votes
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 ...
2
votes
3answers
45 views
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 ...
0
votes
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 -...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 $\circ$ ...
1
vote
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 $f'(x)$ ►$3y^2-6y-x=0$
...
0
votes
1answer
13 views
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,...
2
votes
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 ...
0
votes
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: (...
-1
votes
2answers
27 views
inverse function of a squared
i'm stuck on trying to find the inverse of the following function:
$f($$x) = {\sqrt{x^2-4x}}$
What i did so far is:
$y = {\sqrt{x^2-4x}}$
Swap
$x = {\sqrt{y^2-4y}}$
Remove the root by squaring ...
0
votes
0answers
39 views
Why does $Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$ exhibit a bijection with the pairs (x,y)?
Consider
$$Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$$
I discovers it essentially zigzags along the grid with $\mathbf N$ vs $\mathbf N$ (natural numbers). So intuitively, given any P(x,y) we follow ...
0
votes
1answer
38 views
Inverse of the asymptotic expansion of Gauss Hypergeometric function
I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below). Basically I want to series expand $\rho$ for large $r$ (i.e. as $r\to \infty$) and then ...
0
votes
2answers
47 views
Is $f(x)=x+\sin x$ a homeomorphic function?
I want to determine $f(x) = x+\sin x$ is homeomorphic or not on $\mathbb{R}$?
A bijective continuous function is homeomorphic if its inverse is also continuous.
I know that $f$ is bijective. Also $...
0
votes
1answer
16 views
Partial derivative of function of inverse function
I have got a probelm with the following task:
$\frac{\partial }{\partial x}f(f^{-1}(x,t),\tau)$, where $f\in\mathscr{C}^{\infty}(\mathbb{R^2})$. My attemp is
$\frac{\partial }{\partial x}f(f^{-1}(x,t),...
0
votes
0answers
10 views
Inverse function in integral form
Here is my question:
Suppose my function is defined in terms of Riemann integral in the following form
$$z=g(y)= \int_{}^{}f(x,y)dx$$.
Is there any explicit formula of inverse function $h(z)$; $y=h(...
0
votes
2answers
73 views
Why are $\ln x$ and $e^x$ considered to be each others' inverses?
From my understanding the definition of a function's inverse is as follows.
Take a function $f$ which has the inverse $f^{-1}$. This would mean that $f(f^{-1}(x)) = x$ and that $f^{-1}(f(x)) = x$ for ...
1
vote
1answer
16 views
Simplifying a proof that $f$ and $g$ are mutually inverse functions knowing that $f$ is an injection
To prove that $f$ and $g$ are mutually inverse bijections $A\to B$ and $B\to A$, it is necessary to prove:
that really $f:A\to B$ and $g:B\to A$;
for every $x_0\in A$ and $y=f(x_0)$ we have $x_1=x_0$ ...
0
votes
1answer
23 views
Set theory, functions and inverses
I'm doing an intro course on set theory and have the question if the inverses of the surjective functions in the sets A=｛a, b｝and B= {c, d, e} are also functions.
So far, I thought that the inverse ...
0
votes
1answer
15 views
Describe the preimage of any point $(b,c) \in \mathbb{R}^2$ for $F(x, y) = (x+y,xy)$
I plotted the line and the hyperbola implied by the first and second coordinates respectively. I realize that the preimage can be
Empty for $c > 0 \land b < c$
Size one for $c = 0 \lor (b = c \...
4
votes
3answers
61 views
Showing $\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_0^1 \mid F^{-1}(u)-G^{-1}(u)\mid du$ with $F$, $G$ CDF functions
Let's $X$ and $Y$ have CDF functions admitting moment of order $1$.
Let's be $F$ cdf of $X$ and $G$ cdf of $Y$.
I want to show that $$\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_{0}^{1} \mid F^{-1}(...
0
votes
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,...
0
votes
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
$ \...
1
vote
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:
$$f(x) = 3x^4 \\
f(-x) = 3(-x)^4 ...
1
vote
2answers
62 views
Inverse factorial function
I am wondering what is the inverse/opposite factorial function?
e.g inverse-factorial(6)=3
Furthermore, I am intrigued to know the answer to:
a!=π
find a
I would really appreciate if anyone could ...
1
vote
3answers
23 views
Having trouble getting started with a proof involving functions
The proof looks like this:
Suppose that a function $f(x) = ax + b$ is its own inverse, meaning that $f^{-1}(x) = ax + b$ also. Prove that we must have either have $a = -1$ or $a = 1, b = 0$.
From ...
-1
votes
2answers
58 views
Inverse of $\ln(e^x-3)$
so the whole concept about inverses is a little foggy.
Say you have function $f(x)=\ln(e^x-3)$ and you want to know the inverse function, then:
$$\ln(e^x-3) = y$$
$$e^x-3 = e^y$$
$$e^x = e^...
1
vote
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 ...
0
votes
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 ...
