What is Inverse: Definition and 1000 Discussions

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, i.e., g(y) = x if and only if f(x) = y. The inverse function of f is also denoted as



{\displaystyle f^{-1}}
.As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we would undo each step in reverse order. In this case, it means to add 7 to y, and then divide the result by 5. In functional notation, this inverse function would be given by,





{\displaystyle g(y)={\frac {y+7}{5}}.}
With y = 5x − 7 we have that f(x) = y and g(y) = x.
Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse, it must have the property that for every y in Y, there is exactly one x in X such that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f.

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  1. A

    How to Determine Correct Inverse Trig Angle?

    I understand why certain inverse trig functions have two answers. Like for arcsin(0.5), it could be pi/6 or 5pi/6. I know both angles have the same sin value, that they're both on the same horizontal line on a graph of sin, I get all of that, but two questions about it: 1) In cases where...
  2. Pilou115

    Finding the inverse of 4th rank elasticity tensor

    I'm desperately trying to understand how to get from 2.7.11 to 2.7.16 and cannot find any reference online on how to find the inverse of an elastic tangent modulus (fourth_order tensor). Can someone help me or give me a reference I can check where they do a similar thing? I would really...
  3. ergospherical

    I Calculating the inverse of a function involving the error function

    I have a probability distribution over the interval ##[0, \infty)## given by $$f(x) = \frac{x^2}{2\sqrt{\pi} a^3} \exp\left(- \frac{x^2}{4a^2} \right)$$From this I want to derive a formula for the inverse cumulative density function, ##F^{-1}##. The cumulative density function is a slightly...
  4. C

    Using inverse to find eigenvalues

    For this, I don't understand how if ##(A - 2I_2)^{-1}## has an inverse then the next line is true. Many thanks!
  5. C

    Finding ##A^{-1}## of a matrix given three submatrices

    For this problem, Find ##A^{-1}## given, The solution is, However, in the first image, why are we allowed to put together the submatrices in random order? In general does someone please know why we are allowed to decompose matrices like this? Many thanks!
  6. B

    B Are there two kinds of inverse with respect to closure?

    For every instance of addition or multiplication there is an inverse, closed on the naturals. Not every instance of subtraction and division is defined, so not closed on the naturals. This looks like two kinds of inverse. Instance inverse - the inverse of instances of addition and...
  7. S

    B Does the inverse square law equalize gravity at different spots?

    Inverse square law would reduce the gravity from the parts of Earth that are farthest from our feet. It'll also reduce the gravity from Earth's center by a lesser amount, but would that be lesser enough so the gravity 20 kilometers under our feet is stronger than the core's gravity or even the...
  8. raminee

    A Editing in Freq domain and applying inverse FFT

    Hello All, I am somewhat familiar with FFT and iFFT and its uses. However I have an issue when I edit in Freq domain and try to get back to time domain . I have an audio signal in time domain that I transform to frequency domain using an FFT routine in block sizes of N points. (in my case 256...
  9. C

    Proving inverse of a 2 x 2 matrix is really an inverse

    For this, Dose someone please know how ##ad - bc## and ##-cb + da## are equal to 1? Many thanks!
  10. C

    Help with inverse matrix's

    For this, Dose someone pleas know where they get ##C = CI## from? Also, What dose it mean when A and B commute? Many thanks!
  11. P

    Equation involving inverse trigonometric function

    I came across the mentioned equation aftet doing a integral for an area related problem.Doing the maclaurin series expansion for the inverse sine function,I considered the first two terms(as the latter terms involved higher power of the argument divided by factorial of higher numbers),doing so...
  12. C

    Interchanging x and y for inverse function

    For this, Why are we allowed to interchange x and y? Is it because the equation will still be true? Many thanks!
  13. Bishal Banjara

    I How to obtain other inverse metrics than that of Schwarzschild?

    The Schwarzschild solution could simply be expressed as $$ds^2=-(1-2GM/r)dt^2+(1-2GM/r)^-1dr^2+r^2d\omega^2$$. Is it possible that we could obtained a new metric into the form as $$ds^2=-(1-2GM/r)^-1dt^2+(1-2GM/r)dr^2+r^2d\omega^2$$? If possible, what are the steps and procedures that should be...
  14. N

    B Why does ##F## often appear as inverse square laws such as Newtonian gravity?

    ...y and Coulomb's law diverge as ##r\rightarrow##0? I mean, if a point light source emits light omnidirectionally, the intensity converges at the source, right? THIS is how I should've worded my previous post!
  15. P

    I Can an inverse function of a special cubic function be found?

    Like this one: ##f(x)=3x^3 -18x^2 +36x##.Is there any way to find the inverse of this function without using the general solution to cubic equation?
  16. J

    I Existence and Uniqueness of Inverses

    Existence: Ax = b has at least 1 solution x for every b if and only if the columns span Rm. I don't understand why then A has a right inverse C such that AC = I, and why this is only possible if m≤n. Uniqueness: Ax = b has at most 1 solution x for every b if and only if the columns are...
  17. brotherbobby

    Proving that the inverse of a rational number exists

    Problem statement : I cope and paste the problem as it appears in the text below. Attempt : Not being a math student, I try and prove the above statement using an "intuitive" way. Let us have a rational number ##b = \frac{n}{m}##. Multiplying with ##a## from the right, we see ##ab =...
  18. M

    B Proof of inverse square law for gravitation?

    Newton arrived at "there is a force that drives a planet around the star by examining kepler's laws but how did he arrive to inverse square law by kepler's third law (##T^2=\frac {4\pi r^3}{GM}##)? Thank you.
  19. S

    B Direct and inverse problems

    It seems to me that all mathematics, to one degree or another, is devoted to solving inverse problems. Let's take five equations as an example: 1) ax+b=0 2) ax^2+bx+c=0 3) ax^3+bx^2+cx+d=0 4) ax^4+bx^3+cx^2+dx+e=0 5) ax^5+bx^4+cx^3+dx^2+ex+f=0 To solve the first problem, you need to spend one...
  20. D

    If f(x)=(e^x+e^-x)/2, what is the inverse function?

    Hi everyone This is the solution for the problem. I don't understand how they got from To This was my attempt at a solution I can't seem to get rid of one of the y terms and am left with one on each side. Could someone explain the solution to me please? Thanks
  21. H

    I How can I convince myself that I can find the inverse of this matrix?

    If I have a ##n\times n## matrix $$ U= \begin{bmatrix} u_{11} & u_{12} &u_{13} & \cdots u_{1n} \\ 0 & u_{22} & u_{23} & \cdots u_{2n} \\ 0&0 &u_{33} &\cdots u_{3n}\\ \vdots & \vdots &\vdots & \cdots \vdots \\ 0 & 0 & 0 &\cdots u_{nn} \end{bmatrix} $$ Now, I don't want to use the fact that it's...
  22. ergospherical

    I Hermitian Inverse: Exploring Eigenvalues of ##H^{-1}##

    ##H## is an ##n\times n## Hermitian matrix with eigenvectors ##\mathbf{e}_i## and all eigenvalues negative. It's claimed that ##G = \int_{0}^{\infty} e^{tH} dt## is such that ##G = H^{-1}##. I was looking at\begin{align*} G\mathbf{e}_i &= \int_0^{\infty} \sum_{n=1}^{\infty} \frac{t^n}{n!} H^n...
  23. H

    Sifting property of a Dirac delta inverse Mellin transformation

    Hi, I have to verify the sifting property of ##\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} e^{-sa}e^{st} ds## which is the inverse Mellin transformation of the Dirac delta function ##f(t) = \delta(t-a) ##. let ##s = iw## and ##ds = idw## ##\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iwa}e^{iwt}...
  24. F

    A Need help about a demo with inverse weighted variance average

    I have a problem of understanding in the following demo : In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions...
  25. C

    B Inverse of a Vector: Find the Correct Form

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  26. B

    Python Sklearn LabelEncoder inverse transform

    Hi everyone! I need to inverse an label transform with sklearn. I found this example on web: from sklearn.preprocessing import LabelEncoder np.random.seed(1) y = np.random.randint(0, 2, (10, 7)) y = y[np.where(y.sum(axis=1) != 0)[0]] array([[1, 1, 0, 0, 1, 1, 1], [1, 1, 0, 0, 1, 0...
  27. S

    Inverse square law of gravitation and force between two spheres

    I recently encountered this problem on a test where the solution for the above problem was given as follows: $$F= \frac{Gm_1m_2} {r^2} $$ (1) but $$ m=\frac{4}{3}\pi R^3 $$ substituting in equation (1) $$F= \frac{{G(\frac{4}{3}\pi R^3\rho})^2 }{2R^2} $$ where r=radii of the two spheres m=mass...
  28. D

    I Inverse Functions: x=f(y) and X=f^-1(y)

    Consider the case of a real function f for which f inverse exists. 1) We we are not used to having the y-axis (vertical axis) to denote the independent variable which it does in x=f-1(y). We rotate the system through positive 90 degree and reflect about the vertical to change the sense of the...
  29. L

    Inverse Laplace transform

    \mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]=Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=2]+Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=-2] From that I am getting f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})...
  30. D

    B Compton Scattering and Inverse Compton Scattering

    Hi, Is there a more general equation than the Compton equation that allows one to determine whether an electron will Compton scatter or inverse Compton scatter? If so, where can I find it (or what is it?) Thanks.
  31. brotherbobby

    Solving an equation for ##x## involving inverse circular functions

    Problem Statement : Solve for ##x## : Attempt : If I take ##x=\tan\theta##, the L.H.S. reads $$\tan^{-1}\frac{1-\tan\theta}{1+\tan\theta}= \tan^{-1}\left[\tan\left(\frac{\pi}{4}-\theta \right) \right ]=\frac{\pi}{4}-\theta.$$ On going back to ##x## from ##\theta##, the given equation now...
  32. B

    B Is the highest frequency possible the inverse of Planck time?

    Is the highest frequency possible the inverse of Planck time? Separate or connected question, what's the highest frequency achievable practically today?
  33. A

    Proving this equation -- Limit of a sum of inverse square root terms

    Hi I was working on a physics problem and it was almost solved. Only the part that is mostly mathematical remains, and no matter how hard I tried, I could not solve it. I hope you can help me. This is the equation I came up with and I wanted to prove it: $$\lim_{n \rightarrow+ \infty} {...
  34. Vividly

    I Question about Inverse Derivative Hyperbola function

    Im confused about a certain part of solving an equation. So I used the hyerbola formula to find the answer but I think I did the math wrong. X^2-y^2=c^2 X=1 Y= (2x^5-1)^2 I did the calculations as you can see in the picture but I know I messed up on the square root part. When you square one...
  35. chwala

    Find the domain of the inverse of a function

    This is a textbook problem: now for part a) no issue here, the range of the function is ##-1≤f(x)≤299## now for part b) i got ##x≥-1## but the textbook indicates the solution as ##x≥0## hmmmmm i think, that's not correct...
  36. S

    When do "inverse" ETFs cover their shorts?

    Exchange traded funds like TZA and RWM attempt to get results that are the "inverse" of certain stock indexes. I understand they do this by shorting stocks. When do they cover their shorts? -before the end of each trading day?
  37. M

    MHB Approximation of eigenvalue with inverse iteration method

    Hey! :giggle: We have the matrix $\begin{pmatrix}2 & 1/2 & 1 \\ 1/2 & 3/2 & 1/2 \\ 1 & 1/2 & 2\end{pmatrix}$. We take as initial approximation of $\lambda_2$ the $1.2$. We want to calculate this value approximately using the inverse iteration (2 steps) using as starting vector...
  38. Eclair_de_XII

    If the inverse image of the image of a set via some function is equal....

    ##A=(-1,1)## ##B=[0,1)## Define ##f:A\longrightarrow B## by ##f(x)=x^2## Set ##X=A##. ##f(X)=\{f(x):x\in X\}=\{x^2:x\in(-1,1)\}=B## ##f^{-1}(f(X))=\{x\in A:f(x)\in f(X)\}=\{x\in (-1,1):x^2\in B\}=X## Now choose a non-zero point ##y\in f(X)##. There are two pre-images of this point: ##x,-x\in...
  39. brotherbobby

    Prove an equation involving inverse circular functions

    (I must confess that, in spite of working through the chapter on inverse circular functions, I could barely proceed with this problem. Note what it asks to prove : ##x\sqrt{1-x^2}+\ldots## and how much is that at odds with the formula (1 above) of adding two ##sin^{-1}##'s, where you have...
  40. Mirod

    Zee QFT problem I.4.1: inverse square laws in (D+1)-dimensions

    I tried to do it for 2+1 D (3+1 is done in the text, by writing the integral in spherical coordinates and computing it directly). In 2+1 D I wrote it as: E = - \int \frac{d^2 k}{ (2\pi)^2 } \frac{e^{kr cos\theta}}{k^2 + m^2} = - \int_0^{\infty} \int_0^{2\pi} \frac{d k d\theta}{ (2\pi)^2 }...
  41. H

    What is the Inverse Laplace Transform of e^(-sx^2/2)?

    My attempt at finding this was via convolution theorem, where we take F(s) = 1/s^2 and G(s) = e^(-sx^2/2). Then to use convolution we need to find the inverses of those transforms. From a table of Laplace transforms we know that f(t) = t. But I am sort of struggling with e^(-sx^2/2). My 'guess'...
  42. N

    A Trace of the inverse of matrix products

    Hello, I am puzzled about the following condition. Assume a matrix A with complex-valued zero-mean Gaussian entries and a matrix B with complex-valued zero-mean Gaussian entries too (which are mutually independent of the entries of matrix A). Then, how can we prove that...
  43. M

    MHB Inverse laplace transform pf infinite product

    I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?
  44. S

    Solution of inequality of composite function involving inverse

    I can solve (i), I got x = -1.6 For (ii), I did like this: $$(f^{-1} o ~g)(x)<1$$ $$g(x)<f(1)$$ But it is wrong, the correct one should be ##g(x) > f(1)##. Why? Thanks
  45. Eclair_de_XII

    Proving continuity of inverse cube function

    The proof is given in two steps 1. Prove the lemma. 2. Use lemma to prove result. %%1-Lemma%% Assume ##a\neq0##. Define ##g:(-(|a|+1),|a|+1)\longrightarrow \mathbb{R}## by ##g(x)=\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}##. Then ##g## is bounded from below by some positive number ##m##...
  46. N

    I Get the time axis right in an inverse Fast Fourier Transform

    Hi I would like to transform the S-parameter responce, collected from a Vector Network Analyzer (VNA), in time domain by using the Inverse Fast Fourier Transform (IFFT) . I use MATLAB IFFT function to do this and the response looks correct, the problem is that I do not manage to the time scaling...
  47. G

    Prove a multiplicative inverse exists (complex number)

    I have as a solution for part one: c=(a)/(a^2 + b^2) d=(-b)/(a^2 + b^2) Which matches with the solution manual. The manual goes on to give the solution for part b: (a+bi) * ( (a)/(a^2 + b^2) - ((b)/(a^2 + b^2))i ) = 1 I'd simply like to know where the 'i' at the end of the second...
  48. M

    Engineering Robust Stability: criterion for inverse multiplicative uncertainty

    Hi, I have a question that I am quite confused about. Please note this is at the undergraduate level. Question: Given the transfer function with inverse multiplicative uncertainty \bar G (s) = \frac{G(s)}{1+\Delta \cdot W(s) \cdot G(s)} and the fact that the system is connected in feedback...