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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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Show that the function $f(x) = \ln(x^2-2x+2)$ on $[1,\infty)$ has an inverse and find an expression for it.

I know that this function has an inverse since it is one to one. The reason it is one to one is because it's derivative is bigger than zero throughout the functions domain. What I got stuck at is the calculation when I tried to find the inverse…
Nilo
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Is $f(g(x))=x$ a sufficient condition for $g(x)=f^{-1}x$?

I am asking that is $f(g(x))=x$ a sufficient condition for $g(x)=f^{-1}x$ (but I only want $f(x)$ to be an algebraic function)? At school I am learning about inverse functions and my teacher said that to check if a function $g(x)$ is the inverse…
BadAtAlgebra
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Inverse of a function not defined vs function not invertible

So let's say that we have a function $f(x) = x^2$ (just to give an example). Is there a difference between saying that f(x) is not invertible and saying that $f^{-1}(x)$ is not defined? To me, it sounds like these are two different ways of saying…
Setin
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On inverse trigonometric functions

Can we always write tan^-1 (x) as cot^-1(1/x),sin^-1(x) as cosec^-1(x) and sec^-1(x) as cos^-1 (x) or are there any restrictions on this fact because of domain ?
Schwarz Kugelblitz
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Inverse of this function with two powers

I'm trying to find the inverse of the function: $$y=\frac{x^f + 1 - (1-x)^\frac{1}{f}}{2}$$ where $f=10c^3+1$ and $x,y$ and $z$ lie in the interval $[0,1]$ I'm using this for mapping of MIDI note velocities and need to be able to convert…
Brad Robinson
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What's the fastest way to prove if a function has an inverse?

I have been searching online for a method that shows whether or not a function has an inverse and the fastest method I can find is that you can prove that $f(x)=f(y)\Longrightarrow x=y$ which proves that the function is one-to-one. For the function…
user532874
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Function inverses

From the definition of a inverse standpoint ($f^{-1}(f(x))=f(f^{-1}(x))=x$), why does interchanging variables ($x$ and $y$) work to find the inverse? It seems logical to me but I cannot come up with a coherent argument as to why.
John Arg
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How to find the inverse function

For this function: $f(x)=x\exp(ax)$ what is the inverse function $f^{-1}(\cdot)$?
skyindeer
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Generalized inverse function for $Y=\min(X, b)$

$X$ is a continuous and non-negative random valuable with CDF $F$, $b>0$, what is the generalized inverse function for $Y=\min(X, b)$
abc196998
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Inverting the maximum of a function? Is this solvable analytically [= by hand?]

Define $$f_a(x) = ax - \log \left[ \frac{x}{5(1-x)} + 1\right], \ x < 1,$$ for some constant $a > 0$. Let $x'$ be the maximum-point of this function, and $f_a(x')$ the max-value. Clearly, both these values will depend on the choice of $a$. I…
Jaood
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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 $h(z)$; $y=h(z)=h(g(y))$. I'm sorry if this is a stupid…
kolobokish
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Inverse of a function with summation

Suppose that I have the following function: $$z(\zeta)=\sum_{k=0}^{n}m_k\zeta^{1-k}$$ How do I get the inverse of that function? i.e. I want to express $\zeta(z)$? In my case, $10\leq n \leq 20$. In my case, $z$ cannot be zero and the constants…
BeeTiau
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Solve for x: $\sin^{-1}(\sin (\frac{2x^2+4}{x^2+1}))<\pi -3$

Solve for x: $\sin^{-1}(\sin(\frac{2x^2+4}{x^2+1}))<\pi -3$ My approach : $\pi -3$ is approximately equal to $\frac{\pi}{22.42}$ and $\frac{2x^2+4}{x^2+1}$ is positive as no condition is imposed on x we can take any value but i am not able to…
Samar Imam Zaidi
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Find the sum of inverse functions

$y=\sin^{-1}(\sin 8)-\tan^{-1}(\tan 10)+\cos^{-1}(\cos 12)- \sec^{-1}(\sec 9)+\cot^{-1}(\cot 6)-\csc^{-1}(\csc 7)$. If $y$ simplifies to $y=aπ+b$, then find $a-b$. My answer is zero. But the answer is 53. Where have I gone wrong?
Samar Imam Zaidi
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How can I find the right inverse of a function and show that a left one doesn't exist

Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a mapping with $f(x)=x^2$ Show that $f$ has a right inverse, $h$, but not a left inverse and find h(0) and h(1).. So from looking at this function, I know it's not injective because suppose $f(a) =…
user8358234
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