Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [inverse]

Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

An inverse is an operation that reverses the effect of another operation. This is a broad concept that arises in many areas of mathematics.

  • Multiplicative inverse: $2^{-1} = 1/2$
  • Inverse function: $\sin^{-1}x$ is the inverse of sine
  • Inverse matrix $A^{-1}$
  • Left and right inverses of group elements, of operators between linear spaces, etc.
4444 questions
6
votes
1 answer

Is there a name for this type of inverse?

I have a function $f : A \to B$ and an inverse $f^{-1} : B \to A$, and the only property of the inverse is that $(f \circ f^{-1} \circ f)(x) = f(x)$. In particular, it is not necessarily true that $(f^{-1} \circ f)(x) = x$. I normally associate…
Mike Izbicki
  • 875
4
votes
3 answers

If a function maps an input to its inverse, is it bijective?

I read in my textbook that a function is a bijection if and only if it has an inverse. Is it the same thing to say a function $f: X → X$ is a bijection if $f(x) = x^{-1}$? If $a = x$ and $b = x^{-1}$, then I'd have $f(a) = b$, and in this situation…
Chris
  • 1,129
  • 1
  • 9
  • 18
3
votes
1 answer

Linear algebra proof that AB = On with A invertible only if B = On

$A,B \in Mn(R)$ so that $AB=0n$ and $A$ is an invertible matrix. Proof that $B=0n$ by definition $A$ is invertible so: $\exists C \in Mn : AC=CA=In$ so $A \ne 0n$ Then $AB=0n$ if $B=0n$ Here I can only say if and not if and only if because the…
user144037
  • 71
3
votes
3 answers

Yet another inverse function to calculate

Is it possible to evaluate the inverse of this function, in order to obtain for each $y\in\mathbb R^+$ an explicit value of $f^{-1}(y)$? Thanks in advance! $f(\delta)=(\frac{1}{Z})\delta^{-\alpha}e^{-\beta d\delta}$ [Function from this article,…
jackb
  • 231
3
votes
0 answers

Finding the number of the real roots of $a^x=g(x)$ where $g(x)$ is the inverse function of $f(x)=a^x$

Question : Let $a$ be a constant which satisfies $0\lt a\lt 1$. Letting $g(x)$ be the inverse function of $f(x)=a^x$, then find the number $N$ of the real roots of $f(x)=g(x)$. Motivation : This is the question which I created. First, I was thinking…
mathlove
  • 139,939
3
votes
1 answer

Writing inverse without using piecewise function

I am trying to find the inverse function for the given function f(x) = 3x - |x| + |x - 2|. I have already found that the inverse function can be expressed as the piecewise function f^{-1}(x) = \begin{cases} \frac{x - 2}{3} & \text{if } x < 2 \\ x -…
Luke
  • 86
  • 4
3
votes
1 answer

Reciprocal of the reciprocal of zero

By straightforward evaluation, $$(0^{-1})^{-1}=(NaN)^{-1}=NaN$$ where $\frac{1}{0}$ is taken to equal $NaN$ (not a number), or undefined or indeterminate. However, the laws of exponents state that $$(x^m)^n=x^{mn}$$ so the original equation should…
Dodo
  • 175
2
votes
2 answers

Inverse of $x(x+2)$ given $x\ge -1$

Consider the function: $y=x(x+2)$ . Consider its domain to be $x \geq -1$ . Graphically it makes sense that the inverse of this function is $-1 + \sqrt{x+1}$. But how to compute it analytically? Thanks
user170607
  • 21
2
votes
3 answers

Inverting a quartic equation of state

I have the following equation (which is an adaptation of the Beattie-Bridgeman Equation of State): $$ P = \frac{RT}{V} + \frac{B}{V^2} + \frac{C}{V^3} + \frac{D}{V^4} $$ This is a function of the form $P = f(V)$ as R, T, B, C and D are all constant…
user102033
2
votes
2 answers

Proof linear algebra - Inverse

Verify that A=$\begin{bmatrix}1&-1\\0&2\end{bmatrix}$ satisfies $A^2-3A+2I=0$. Use this fact to show that $A^{-1}= \frac{1}{2}(3I-A)$ I know it does satisfies the first equation/already proved it.
user840664
2
votes
1 answer

iteration method for inverse matrix

a) Consider the iterative scheme \begin{equation} x_{n+1} = x_n + c (Ax_n - I) \tag{1} \end{equation} When the process converges, show that this scheme (where $c$ is an appropriately chosen real number) can be used to calculate $A^{-1}$ . b) for…
crazy
  • 89
2
votes
3 answers

Finding the Inverse of a 5th-Order Polynomial Function

Find the inverse of: $F(x)=7x^5-5x^3-3x^2+2x$.
Lance
  • 21
2
votes
4 answers

is this function invertible ??

given the function $$ f(x)= x+\cos(x)+\sin(\cos(x)) $$ (1) is this invertible ?? i mean it exists another function $ g(x) $ so $$ f(g(x))=x $$ my guess is that for big $ x \gg 1 $ the function 'x' is asymptotic to $ g(x) \sim x $ since for big 'x'…
Jose Garcia
  • 8,506
2
votes
2 answers

Is the preimage of a bounded set also bounded?

I need to prove the following statement: Let $f:\mathbb{C}\rightarrow\mathbb{C}$ a continuous function and $B \subseteq \mathbb{C}$ bounded, implies, that the set $A=f^{-1}(B)$ to be bounded. I do know that the statement is true for a continuous…
nando
  • 53
1
vote
1 answer

How to get the inverse function of this one?

Let's have function $$ \psi (x) = -\frac{1}{ax} - \frac{b}{a^2}\ln(x) + \text{const} + O(x). $$ I have read that the inverse function is written in a form $$ \psi^{-1}(t) = -\frac{1}{at} - \frac{b}{a^3}\frac{\ln(t)}{t^2} + \frac{c}{t^2} +…
John Taylor
  • 1,755
  • 11
  • 18
1
2 3 4 5 6 7 8