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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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The integral on the inverse of a function

Let $f$ be a continuous, strictly decreasing, real-valued function such that $\int_{0}^{+\infty}f(x)\,dx$ is finite and $f(0) = 1$. In terms of $f^{-1}$, $\int_{0}^{+\infty}f(x)\,dx$ is? The answer is "equal to $\int_{0}^{1}f^{-1}(y)\,dy$" Okay…
Caprikuarius
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Inverse of a sum of functions

If $h(x) = f(x) + g(x)$, what is $h^{-1}(x)$ in terms of $f^{-1}(x)$ and $g^{-1}(x)$ ? Also, what are other useful inverse identities that you can give me? I know the basics like $(f(g(x)))^{-1} = g^{-1}(f^{-1}(x))$
IUissopretty
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Inverse of a function of a 3rd order

can someone help me how to find the inverse the following function? $$z(\zeta)=\frac{1}{\zeta}+m_1\zeta+m_2\zeta^2+m_3\zeta^3$$ In my case, $z$ is a complex number and cannot be zero. And $m_k$ is a constant. How do I get the inverse of that…
BeeTiau
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closed form inverse of $1/(1-\exp(x)) + 1/x$

In an article I am writing I wrote: 'The expression $$\mu = \frac{1}{1 - \exp(\lambda)} + \frac{1}{\lambda}$$ does not have a pleasant inverse giving us an expression for $\lambda$ in terms of $\mu$. However, [some stuff on the quality of the…
Vincent
  • 10,614
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How to find the range of $ (\arcsin x)^2 + (\arccos x)^2 $ without using derivatives?

I am expected to find the range of the following function: $$ (\arcsin x)^2 + (\arccos x)^2 $$ I individually added up the range of $(\arcsin x)^2$ $(\arccos x)^2$ But that gave wrong answer ? What to do with this problem?
user601454
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Inverse of $x\cdot\sin(x)$

In the function $$f(x) = x\cdot\sin(x)$$ every codomain value $y$ occurrs infinitely many times. So, in principle, there are "inverse" functions $f^{-1}$ such that $$f^{-1}(y)=x \mbox{ and } f(x)=y$$ for every $y \in \mathbb{R}$. (I know that…
Duke
  • 131
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Finding the Inverse of a Nested Square-Root Function

I'm a CS grad student working on a project where one of the issues at the moment comes from trying to find the inverse to the following function: $f(x)=\sqrt{-a-x^2+2\sqrt{a x^2+x^4}}$ I already know that this function only has real values (and…
Bob
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Finding inverse of $g:\mathbb N × \mathbb N \to \mathbb N$

Let function $g:\mathbb N × \mathbb N \to \mathbb N$ $$g(x,y) = \frac{(x+y)(x+y+1)}{2}+y$$ I want to prove that $g$ is bijective. I tried to prove it is injective by contrapositive, but I had some difficulties proving its surjection. Also, I don't…
chengi666
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What can be said about $f^{-1}$ when $f$ has no inverse?

Let $f: A \to B$ be given by $f(x) = |x|$, such that $A = [-1,0] \cup [1,2]$ and $B = [0,2]$. I understand that $f$ does not have an inverse, since $f(-1) = f(1)$. However, albeit my textbook is saying $f$ does not have an inverse $f^{-1}: B \to A$,…
Lenora
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Domain and Range contradiction?

$$y = \log(1 +(k - 1)^2)$$ The domain is all reals and the range is $y \geq 0.$ First, the inverse. Swapping the variables: $k = \log(1 + (y - 1)^2)\\e^k = 1 + (y - 1)^2\\e^k - 1 = (y - 1)^2\\y = (e^k - 1)^{1/2} + 1$ This inverse has the…
John
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Find the inverse of $f(x,y,z) = (\frac{x}{x^2+y^2+z^2}, \frac{y}{x^2+y^2+z^2}, \frac{z}{x^2+y^2+z^2})$

I attempted to switch to cylindrical / spherical coordinates, but I keep getting stuck. Note that $(0,0,0)$ is not in the domain or codomain.
Andrew
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How to inverse $f(x)=x^\frac{1}{x-1}$

I have tried a couple of approaches to inversing it, but I really don't know any strategies or methods. I don't even know if it's possible or how to figure out if it is possible, so I'm asking how to know if you can inverse or not as well.…
IMTIREDSHUTUP
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A symmetric point of the inverses

I have the graphs of $y = F(x) = e^x$, $y = G(x) = ln (x)$, and $L : y = x$ drawn. Of course, $y = F(x)$ and $y = G(x)$ are inverses to each other and therefore they are symmetric about $L$. Let $P(p, q)$ be a point on $y = G(x)$. Through $P$, I…
Mick
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Does the usual procedure for finding the inverse function also prove that the function is invertible?

Students are usually taught to find inverse functions using a procedure such as this from Stewart (Calculus, 2016, p. 58): Question. Does the above procedure also prove that $f$ is invertible? Let's take Stewart's example: In the above example,…
user694131
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Question about an inverse function

I need to find the rule of the inverse of the function $f(x) = 6x + 4$. Should I factorize is to $\frac{x}{6} - \frac{2}{3}$ or should I leave it as $\frac{x-4}{6}$. Thanks
Joann Sammut
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