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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 $x$.

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Inverse trigonometry: How to find $x$

When $\sin^{-1}\left(\frac{2x}{1+x²}\right)=2\tan^{-1}(x)$? I thought that it was for all $x\in\mathbb{R}$ but it was incorrect, so please help!
Adas Hoper
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Invertible function - have one big question

If I want to state if a function is invertible I have couple of tools. I can check if the function is odd or even. If it is even, it is certainly not invertible, since $f(x)=f(-x)$, it is strictly agains definition of invertible function. If it is…
naruto25
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Is there any way of finding inverse without "switching"?

I have seen a lot of articles and books teaching how to find an inverse of a function by 'switching' $x$ and $y$ and solve for $y$ variables. My understanding of an inverse function is that it undoes whatever the function being given do, i.e.…
sammyyy
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Equation of an Inverse of a function

I have a question about how to obtain an equation of an inverse function. The equation is $y=x^2+2x+3$ and I changed $x$ and $y$ to get $x=y^2+2y+3$ and it is only a matter of rearranging to have $y$ on the left side but I am finding it difficult to…
MathsGoogle
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Inverse of a function containing $(-1)^n$

Is it possible to find the inverse of a function which contains the term $(-1)^n$? Specifically: I want to find the inverse of the following function, which gives the nth number of the form $6k\pm1$: $$ f(n) = 3n + \frac{3}{2} - \frac{(-1)^n}{2}…
SmallestUncomputableNumber
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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 $g: (x_1,x_2) \mapsto y$. If I impose a linear…
Fred Li
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How do I remove $y$ from $xy$ in $y=xy-2x$?

I got an inverse function $y=xy-2x$. But how do I remove $y$ from $xy$?
Kakeru
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Is the function $f(x)=\begin{cases} x+0.2x^2 \sin (1/x) & x \ne 0\\ 0 & x=0\end{cases}$ invertible in a neighborhood of origin?

Is the function $$f(x)=\begin{cases} x+0.2x^2 \sin (1/x) & x \ne 0\\ 0 & x=0 \end{cases}$$invertible in a neighborhood of origin? I just know that $f$ is continuous on $\mathbb{R}$ and…
Thu Le
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Why does applying $\tan$ to $\arctan x$ make it $x$?

Through doing calculus problems I have found that $\tan(\arctan x)$ seems to be equal to $x$. I can't seem to understand why. Can someone please explain this to me? Also, does this rule apply to anything else? This rule, as in, something of the…
Spica
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Find the inverse of the following map

Let $F(p,d)$ denotes the set of all functions from $\{1,\cdots,p\}$ into $\{1,\cdots,d\}$. Consider the following map $G:F(q,d)\times F(p,d) \to F(p+q,d): (f,h) \mapsto f \star h$, where $f\star h\in F(p+q,d)$ such that $(f\star h)(j) = f(j)$ for…
Student
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Proving the inverse function

this is my first time making a post on this website so all feedback is highly appreciated. I have recently started university and I struggle a bit with wording answers as I am not use to it yet. If anyone could take a quick look at my workings and…
Mathlete
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Need help calculating a simple inverse function

Quick question. $$y=\sqrt{2x-x^2}$$ I need the inverse function for some other problem, but I just can't find it. Could you please point me the steps to solve this? Thanks
Joaquin Benitez
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Existence of Inverse Mapping : $f: \;P(\Bbb R) \rightarrow \Bbb R $ where $f(x) = {1\over x^2}-x\cdot arctanx+{1 \over 2}log(1+x^2) $

I'd like to show that below function holds the inverse mapping: $f: \;P(\Bbb R) \rightarrow \Bbb R $ where $f(x) = {1\over x^2}-x\cdot arctanx+{1 \over 2}log(1+x^2) $ To show the existence of inverse mapping, I want to use the property that every…
Daschin
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Is there a composite function with the following inverse function?

Given the following two functions $$ f: \mathbb R \to \mathbb R, \quad f(x)=5-x$$ $$ g: [3,\infty[ \to [0,\infty[, \quad g(x)=\sqrt{x-3}$$ determine whether $f \circ g^{-1} $ can be formed. If it can be formed, then find its composite function, and…
ronzenith
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Consider the function f : R-->R defined by f(x) =x^7+x+1. Show that f has an inverse

We just started doing inverse functions so I'm not very familiar with this concept...
George S
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