Mathematics Stack Exchange
Mathematics Stack Exchange

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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Finding Inverse function

$$x^3+8x+3=y$$ How am I suppose to make $y$ the subject? I'm not sure how to find its inverse, for example how to I find $f^{-1}(12)$
Mon
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Can a scalar-valued multivariate function be invertible?

Consider the scalar-valued multivariate function: $$ z = f(x,y) $$ Where $x,y,z \in \mathbb{R}$. If $f$ maps $\mathbb{R}^2$ to $\mathbb{R}$, can an inverse function $f^{-1}$ that maps $\mathbb{R}$ to $\mathbb{R}^2$ exist? If so, what are some…
mhdadk
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1st and 2nd derivative of inverse function: implicitly

I know this question has been answered but my case may be different: I have given the inverse function of $f(x)$ evaluation on one point: $f^{-1}(x_0)$. In order to calculate this I'd set $f^{-1}(x_0) = f(x)$ and solve for $x$. Let's call the…
Leon
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Inverse of Piecewise-Defined function

I'm stuck on this problem: Let $f$ be defined by f(x){ \begin{array}{cl} 2-x & \text{ if } x \leq 1, \\ 2x-x^2 & \text{ if } x>1. \end{array} Calculate $f^{-1}(-3)+f^{-1}(0)+f^{-1}(3)$. It's difficult because I've never dealt with inverses in…
random guy 2000
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Possible to reverse this operation?

Say I have a vector f of length n and I obtain all pairwise products of its elements: vector[(n*(n-1))/2] f_prods ; int i_prod = 0 ; for(i_n in 1:(n-1)){ for(j_n in (i_n+1):n){ i_prod += 1 ; f_prods[i_prod] = f[i_n]*f[j_n] ; …
Mike Lawrence
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Find the inverse of $f(x)=\sqrt{3x^2 +1}$

Find the inverse of $f(x)=\sqrt{3x^2 +1}$. We don't know if $f$ is invertible so we have to prove it, but how? $f$ bijective $\Leftrightarrow $ $f$ invertible Is $f$ injective? $ x \neq y \Rightarrow f(x) \neq f(y) $ $f(x) = f(y) $ $\sqrt{3x^2 +1} =…
Spectree
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Solve equation with $\arccos$ and sqrt

I have an equation to calculate $A$ from $r$ and $h$: $$ A= \arccos\left(\frac{r-h}{r}\right)r^2 - (r-h)\sqrt{r^2-(r-h)^2} $$ But now $A$ and $r$ are known, and I want to solve for $h$ instead. So far, I solve this equation numerically, but I was…
matth
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Domain of composite of function and its inverse

I'm looking for the domain of definition of the composite function of one function and its inverse. Let $$H(x)= \frac{e^x -1}{e^x +1} $$ and its inverse $$G(x)=\ln{ \frac{x+1}{1-x}}$$ Is $-1
ZchGarinch
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Interpretation of inverse of a composite function

I was thinking about how to find the inverse of a composite function. I referred to several great answers on this site on others' questions and they all pointed out that to take the inverse of a composite function is the same as inversing all the…
Vamsi Krishna
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why is $f^{-1} (x)$ , the reflection of $f(x)$ over the line $y=x$?

I can understand the algebra but I just can not understand the intuition. For example consider $y=x^2$, I just don't understand how $x^2 =y$ is a reflection over the line $y=x$.
tryst with freedom
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inverse function to find the image

How can I prove that the image of $f(x) = 2\tan(x) - 1/\cos(x)$ which is defined in $(-\pi/2, \pi/2)$ is $\mathbb{R}?$ I thought to find the inverse function of $f$ but I have no idea how to do it.
cbdes
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Range of sum of two functions.

How to find range of $\tan^{-1}x + 10$ My attempt : I thinks that range of sum of two functions $f$ and $g$ is either range of $f$ + range of $g$ or Union of range of $f$ and range of $g$. Please help me.
Baljeet
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Problem with inverting function to form $y=f(x)$

As in the title. Here are my attempts and the given function: Given function: $y=\frac {\log_2{x}-3}{2x} $ Inverse function: $$ x= \frac {\log_2{y}-3}{2y} $$ $$2xy=\log_2{y}-3 $$ $$ 2^{2xy}=\frac y8 $$ At this step i'm…
1qwertyyyy
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Two functions that satisfy inverse function theorem

I have this problem: Let $f,g\in C^1(\mathbb{R}^2,\mathbb{R}^2)$ function that satisfy inverse function theorem in $(2,5)$ such that $f(2,5)=g(2,5)=(2,5)$.. $f-g$ satisfy inversion function theorem in $(2,5)$? What I can't understand is what does …
Archimedess
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Inverse Image function

Let $A= [{5}, \infty[$ , $B=[{1}, \infty[$ and $f: A \to B, f(x)=x^2-{10}x+{26}$ Determine the inverse image $f^{-1}:\ B \to A, f^{-1}(x)$ How should I proceed with this? Thanks in advance
Emilycodes
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