# 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.

1,036 questions
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### Finding an inverse function (sum of non-integer powers)

I have a function: $$f(x)=x^{2.2} + (1-x)^{2.2}$$ It is defined on the interval $[0,1]$. Minimum: $x=0.5, y=2*0.5^{2.2} = 2^{-1.2}$. I want to find an inverse for it. Since the function has two "...
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### Inverse/Reverse of Number of Permutations and of Number of Combinations with Repetitions?

For an engineering application, I need the inverse functions of the computations of the number of combinations and permutations. In the thread How to reverse the $n$ choose $k$ formula? it shows how ...
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### Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$ So in order to show injectivity $f(a)=f(b) \Rightarrow a=b$ so $\frac{a}{1-|a|}=\frac{b}{1-|b|}$. But how do I prove that? ...
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### Help with concepts with the inverse function theorem

I am solving the following problems: Be the transformation $T \in R ^ 2$: by $T(u, v) = (u ^ 2-v ^ 2, uv) = (x, y)$. Calculate the Jacobian of $T$ and conclude whether the transformation is ...
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### Inverse Fourier transform of $F(k) = 1/(k^2+a^2), a>0$

I need help finding the Inverse Fourier transform of: $$F(k) = \frac1{ k^2 + a^2 },~ a>0$$ Here is what I have so far: Singular points at $k^2 = a^2$, namely, at $k = \pm ia$. The inverse ...
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### If $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$ then what is the value of $g'(2)$?

I know that $f(0)=2$, $f'(0)=3$ and $g=f^{-1}$. But how can I find the value of $g'(2)$?
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### Preimage Structures: lists of all inverse images at a given depth (reference request)

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, the j-th such list would be ...
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### Output response from closed loop transfer function using MATLAB

This transfer function is to control the position of Permanent Magnet DC motor. I was able to get the transfer function and now I need to analyze the output for the tuned closed loop for a given input ...
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### Inverse of $\frac{\sin(x)}{x}$

How would one find the inverse of the function $y=\frac{\sin(x)}{x}$? Here are my steps: $y=\frac{\sin(x)}{x}$, $x=\frac{\sin(y)}{y}$, $xy=\sin(y)$, $\arcsin(xy)=y$, After that step, I can’t find a ...
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### Prove that $f(f^{-1}(V))=f^{-1}(f(V))$

Let $f: X\to Y$ be bijective, and $f^{-1}: Y\to X$ be it's inverse. If $V\subseteq Y$, show that the forward image of $V$ under $f^{-1}$ is the same set as the inverse image of $V$ under $f$. I ...
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### Use the complex definition of $\sin z$ to find an expression for $\sin^{-1} z$

Using $$\sin z = \frac{e^{iz}-e^{-iz}}{2i}$$ Prove $$\sin^{-1} z =\frac{1}{i}\ln(iz+\sqrt{1-z^2})$$ Attempted solution: Let $\sin z = u$ and $e^{iz} = v$. \begin{align*}& 2iu = v - \frac{1}{v}\...
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### An additional assumption to the inverse function theorem.

The theorem is given below: And here is the question: Could anyone give me a hint on how to prove the required in the question please?
Consider a random variable $X\sim N\left(\mu,\sigma^2\right)$, and a monotonically increasing non-linear function of $X$, call it $Y=f\left(X\right)$, defined as: Y=f\left(X\right)=\Phi\Big(a-b\,\...
### inverse of $y=\frac{x}{\log{x}}$?
By Prime Number theorem $\pi(x)=\frac{x}{\log{x}}$ for large x Putting $x=p_n$ where $p_n$ denotes $n^{th}$ prime number, We have, $\pi(p_n)=\frac{p_n}{\log{p_n}}$, $\because \pi(p_n)=n$, \$\...