Mathematics Stack Exchange
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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 inverse of a function $t = \frac{1}{\sqrt{1+x^2}}$

I have a formula $t = \frac{1}{\sqrt{1+x^2}}$ How is it possible to convert it into $x = +-\frac{\sqrt{1-t^2}}{t}$ I am assuming that it is an inverse function that is calculated by replacing x with t in the original equation, and then solving for…
user70578
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Inverse of a multi-variable function

I want to calcualte the inverse of $f(x,y)=\frac{1}{2}(x^2+y^2, x^2-y^2)$. My book does it like this: $f: (x,y) \to \begin{pmatrix}u\\v\end{pmatrix}=\frac{1}{2}\begin{pmatrix}x^2+y^2\\x^2-y^2\end{pmatrix}$ Now they calculate $u+v=...=x^2 \Rightarrow…
xotix
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Inverse of Multivariate Function

I'm finding myself a bit confused by an example from some lecture notes. They lay out the simple process of finding an inverse of a standard function, say, $y = 2x + 5$, by simply solving for $x$, giving an inverse of $x = \frac{y-5}{2}$, assuming I…
user465188
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How can we find or why can't we find the inverse function of $f(x)=\ln(x)+x$ for $x>0$?

I want to find the inverse function of $f(x)=\ln(x)+x$, for $x>0$, but I heard it can't be done. Is there a way to find the inverse or to show it can't be found?
Alexandre Tourinho
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Is it possible for a function to not have an inverse relation in terms of standard mathematical functions?

I tried to find the inverse of a function on wolfram's inverse calculator, but when I hit enter it said: "No result found in terms of standard mathematical functions," hence my question above. I am well aware of indefinite integrals that have no…
John Zimmerman
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Why the inverse of this functions not working?

If $f(x) = (x+2)^2+2$ then we get 2 inverse functions for this function: $f^{-1}(x) = -2+\sqrt{x-2}$ $f^{-1}(x) = -2-\sqrt{x-2}$ If we input functions and inverse functions in to each other we must get "x" back. But why we don't get "x" when we…
Siddharth
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About the inverse of a simple function

Consider the function $$f(w)=\frac{e^{2\pi iw}-1}{w+1}$$ I guess that this function is surjective except at integer $w$. My argument is: $\frac{1}{w+1}$ is surjective on the whole complex plane. $e^{2\pi iw}-1$ is $1$ periodic. Therefore the…
Szeto
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Is there an inverse function of $f(x) = x^2 + \pi\cos x$?

Is there an inverse function of $f(x) = x^2 + \pi\cos x$? I don't think there is because of the $x^2$, but I don't know how to prove it.
Plz help me
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Finding inverse function of a function with multiple variable

First, I'll say that english isn't my first language and that I'm not studying math in english, so I'll probably say some stuff wrong. Anyway, what I was wondering is how to find the inverse function of a function that has multiple variables?…
Loic Lavigne
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Inverse function of $x \ln (x^2+2)$

What is the inverse function of $f(x)=x \ln (x^2+2)$ ? Assuming it is invertible, and what is the domain?
Lil Read
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Maple Inverse Function $y=x^2-2$

I have this assignment: Consider the function $f : [0, ∞) → [−2,∞)$ defined by $f(x) = x^2 − 2$. Which statement about the inverse function $f^{−1}$ is true (the inverse)? I am having a hard time finding out how to do this. The function…
user504783
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Inverse function with two $x$'s

In the case of the function $f(x)=x^2+x^4$ is there any trick to inversing it? as the only ways I know do not work on it. As you cannot make y the subject through rearranging it. Thanks in advance.
Mathlete
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sum of functions (where one is non invertable) non invertable?

$A,B$ arbitrary sets, $f_i : A \to B$, $i \in \{1,...,n\}$ functions. Now let $f_j$ be non invertable for a $j \in \{1,...,n\}$. Does this already imply that $(\sum_{i=1}^n f_i)(x)$ is non invertable?
Pazu
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Does a function have to be one-to-one for it to have an inverse?

I am asked to find the inverse for $f(x)=x^3+\arctan(x+1)$ I graphed it on Desmos and found that it is one-to-one, so I assumed it does not have an inverse. Is my approach correct?
Hatem Chalak
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Rewriting $f\circ g$

How to we transform $(f \circ g)(x)$ into a single function, for instance: $\tan(\arccos(\frac{1}{x}))$, where the functions are from different families?
Pratyush Dutta
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