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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.

In mathematics, an inverse function or $f^{-1}$ is a function that "reverses" another function. That is, if $f$ is a function mapping $x$ to $y$, then the inverse function of $f$ maps $y$ back to $x$.

2219 questions
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Find the inverse of $f(x) = 3x + \sin(\pi x)$

I understand that to find the inverse you switch the $x$ and $y$ then solve for $y$. However, in this case you end up with $x=3y + \sin(\pi y)$. I know that the inverse exists and is a function since the $f$ is monotonic for all $x$. Is there any…
Prandals
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Find inverse of cos function

Question: Find the inverse of $$f: [\pi,\frac{3\pi}{2}] \rightarrow R, f(x) = \frac{1}{2}\cos(2x)$$ My attempt: $$x=\frac{1}{2}\cos(2y)$$ $$\cos(2y)=2x$$ $$2y=\arccos(2x)$$ $$f^{-1}(x)=\frac{1}{2}\arccos(2x)$$ But the solution is…
user342661
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Did I do the inverse of this quadratic function in the correct way?

$$n(t)=17t^2-20t+700$$ $$y=n(t)$$ $$y=17t^2-20t+700$$ Switching $y$ and $t$, $$t=17y^2-20y+700$$ $$t=(17y^2-20y)+700 -(Eq. 1)$$ Completion of squares: $$\mathbf a=\sqrt{17}y$$ $$\mathbf b=???$$ $$\mathbf 2ab=-20y$$ $$\mathbf…
Siddharth Venu
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What is the inverse of this binary, bijective function?

The following function $\operatorname{encode}$ is a binary, bijective function: $$\operatorname{encode}(x,y) := \binom{x+y+1}{2} + x = z$$ This function comes up in the context of theoretical computer science in order to prove that LOOP-computable…
r0estir0bbe
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Show that one bijection is the inverse of the other bijection if the two (non-identity) bijections commute

Suppose you have two bijections $\eta, \alpha: S \to S$. Both are not the identity maps on $S$, and that $$\eta\alpha = \alpha\eta$$ Can we conclude that $\alpha = \eta^{-1}$? Many thanks in advance!
Shawn
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Difficult simplification Inverse function

This function passes the horizontal line test so there does exist an inverse for it. The problem I am running into is simplifying the equation when I interchange y and x. $$ y=x^3+4x-1 $$ I have got it into the form of the inverse pre-requisite:…
Obaid Jaffery
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Does the function $\sqrt[4]{4−4}$ have an inverse?

The function: $\sqrt[4]{4−4}$, $x≤1$ is supposed to have an inverse function of: $(−^4/4)+1$. But the puzzle is that the function $(−^4/4)+1$ fails the horizontal line test so how on earth could it be the inverse function of $\sqrt[4]{4−4}$. Is…
Markus Maximus
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Create Inverse of an vector function

I have a function which maps a vector to a scalar. $$ f(q, r) = 135 + q + \frac {r( 39 - |r| )}{2} $$ The domain for q is: $$ q \in \mathbb{Z}, -9 <= q <= 9 $$ The domain for r is: $$ r \in \mathbb{Z}, -9 <= r <= 9 $$ The range of this function…
user1239299
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Which rules apply when building the inverse function (multiple parameters)

I have a function $f(x) = (3*a)*x-3*b$ I know here the inverse function is $f^{-1}(y) = (3*a)^{-1} * (y + 3b)$ I don't understand the steps that would lead to this or which math rules I may have forgotten.
wishi
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Value of sum of inverse functions

\begin{align*}&\cot^{-1}(\tan 2x)+\cot^{-1} (-\tan 3x) \\ &=\cot^{-1} (\cot(\tfrac{\pi}{2}-2x))+\cot^{-1} (\cot(\tfrac{\pi}{2}+3x))\\ &=\tfrac{\pi}{2}-2x+\tfrac{\pi}{2}+3x\\ &=\pi -x.\end{align*} Where did I go wrong in the process? The range of the…
madness
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inverse of $f:(0,4/9) \to \mathbb{R}$ such that $f(x) = x - x^{3/2}$

I'm trying to find the inverse of $f(x)=x-x^{3/2}$ in the interval from zero to its maximum (at $x=4/9$). WolframAlpha gives me an expression involving complex terms, but this is no good, as the formula will be part of a moodle calculated question…
eflaschuk
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Inverting the Hue calculation

From Wikipedia, the calculation of a hue from an RGB color input is given as: $$h_{rgb} = \mathrm{atan}\left( \frac{\sqrt{3} \cdot (G - B)}{ 2 \cdot R - G - B} \right).$$ Where R, G and B are in the range [0, 1], and h ranges from [0, 2π] I'm trying…
john16384
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Inverse $n=\frac{e^x}{\log x}$

$$n=\frac{e^x}{\log x}$$ I've been thinking for several hours about how to find the inverse function, but I always get to a nested function. They can help me or recommend some literature that allows me to solve this problem. In theory there is its…
Estudiante Inversa
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How to invert a function that contains a sign function?

I have the following function: $$f(w,y-x) = \operatorname{sgn}[y-x]|y-x|^w$$, where $w \geq 0, (y-x) \in[-1,1]$, which results in $f(w,y-x) \in [-1,1]$. I would like to find its inverse function with respect to $x$. But I got stuck when trying to…
user1769197
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Approximate Inverse of $(1 - e^{-ax})(1 - e^{-bx})(1 - e^{-b(\alpha - x)})$

I must invert the following function, to obtain $x$ as a function of $y$: $y(x) = f(x)g(x)$ where: $f(x) = (1 - e^{-ax})$ $g(x) = (1 - e^{-bx})(1 - e^{-b(\alpha - x)})$ $a
Jack Rolph
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