What is Inverse function: Definition and 197 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. brotherbobby

    State the domain and range for a given function

    Attempt : Let me copy and paste the problem as it appeared in the text. Please note that the given problem appears in part (b), which I have underlined in red ink in this way ##\color{red}{\rule{50pt}{1pt}}## Clearly the domain is ##\boxed{\mathscr{D}\{f(x)\}...
  2. brotherbobby

    B A function and its inverse

    I could only verify this for a few elementary functions. Does a proof exist? Does it go beyond the realms of high school mathematics? Many thanks.
  3. S123456

    I Local inverse of non bijective functions

    Hi, I am having a hard time trying to solve this question. How do I find the local inverse at x0? f (x) = x^4 − 4x^2 Find an expression for f^−1 for f at the point x = −2. Thanks a lot! I would really appreciate any help!!
  4. brotherbobby

    To find the range of a given ##\sin## function

    Attempt : The domain of the function ##\sin(3x^2+1)## is clearly ##x\in (-\infty, +\infty)##. The values of ##x## go into all quadrants where the ##\sin## curve is positive and negative. Hence the range of the function ...
  5. 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!
  6. 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?
  7. PhysicsRock

    I Is the solution to this problem as trivial as I think?

    The problem goes as follows: Let ##M, N## be sets and ##f : M \rightarrow N##. Further let ##L \subseteq M## and ##P \subseteq N##. Then show that ##L \subseteq f^{-1}(f(L))## and ##f^{-1}(f(P)) \subseteq P##. Obviously, I would simply use the definition of a functions inverse to obtain...
  8. 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
  9. karush

    MHB -apc.1.1.5 inverse function

    Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$ ok not real sure what the answer is but I did this (could be easier I am sure} rewrite as $y=(2x+1)^3$ exchange x and rename y to g $x=(2g+1)^3$ Cube root each side...
  10. barryj

    Having trouble finding the derivative of an inverse function

    Summary: Please see the attached problem and solution The answer is 1/5. I have tried various solutions and cannot get 1/5. What is my error? [Moderator's note: Moved from a technical forum and thus no template.]
  11. barryj

    I want to see the plot of an inverse function -- can I use a TI84?

    Summary: I am studying inverse functions and want to see a plot of an inverse function. I hope this is an OK post here. Lets say I have a function y = x^3 + x. This function has n inverse sine the derivitave is always positive and is a one on one function. I can easily graph this function...
  12. karush

    MHB 219 AP Calculus Exam Inverse function

    Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$$(A)\, \dfrac{2}{27} \quad (B)\, \dfrac{1}{54} \quad (C)\, \dfrac{1}{27} \quad (D)\, \dfrac{1}{6} \quad (E)\, 6$ok not sure what the best steps on this would be but assume we first find...
  13. K

    I How to expand inverse function

    If I'm given a function ##f(x)##, say it has continuos first derivative, then I expand it as ##f(x + \Delta x) = f(x) + (df / dx) \Delta x##. If instead, I'm given ##f^{-1}(x)## how do I go about expanding it? Will this be just ##f^{-1}(x + \Delta x) = f^{-1}(x) + (df^{-1} / dx) \Delta x##?
  14. YoungPhysicist

    Find the inverse function of ##f(x) =x^4+2x^2##

    Homework Statement Find the inverse function of ##f(x) =x^4+2x^2, x>0## Homework Equations ##f(f^{-1}(x)) = x## The Attempt at a Solution My only progress so far is ##x^4+2x^2 = x^2(x^2+2)## Then I am stuck. Since my progress is close to nothing so I don’t expect a complete...
  15. barryj

    Purpose of the derivative of the inverse function

    Homework Statement In calculus, I learn that the derivative of the inverse function is g'(x) = 1/ f'(g(x)) Homework Equations So.. The Attempt at a Solution Can someone give me an example of where I need to know this, or is this just a math exercise. Is there a relatively simple physics...
  16. R

    MHB Finding an Inverse Function

    Hello, I am trying to find the inverse of f(x) = x \div (x+4) IIRC i need to replace f(x) with y and solve for x. I've tried to do y = x \div (x+4) becomes y(x+4) = x then xy + 4y = x but I can't reduce the amount of x to one. What am I doing wrong in this problem?
  17. D

    A Inverse function of the Nyquist-Shannon sampling theorem

    I'm currently carrying out an analysis on waveforms produced by a particular particle detector. The Nyquist-Shannon sampling theorem has been very useful for making an interpolation over the original sample points obtained from the oscilloscope. The theorem (for a finite set of samples) is given...
  18. facenian

    I Question about inverse function

    Let ##f:U\subset R^n\rightarrow V\subset R^n## be a biyective function of class ##C^m(m\geq 1)##, ##U## and ##V## are open sets in ##R^n##. I know from the inverse funtion theorem that when ##J(f)\neq 0## in a point of the the domain a local inverse exists, however, given the above conditions...
  19. A

    Inverse function of tanh()

    Homework Statement ##f:= tanh = \frac{e^x-e^{-x}}{e^x+e^{-x}}## Prove that ##f^{-1}(x)= \sum\limits_{k=0}^{\infty} \frac{x^{2k+1}}{2k+1}## for all x in (-1,1) The Attempt at a Solution I also found the inverse function to be: ##f^{-1}(x)= \frac{1}{2}ln(\frac{1+x}{1-x})## I tried working...
  20. K

    A Integral with an inverse function limit

    Hello, I have tried the integral below with Mathematica and it gives me the following solution: ##\frac{d}{dc}\int_{z^{-1}(c)}^{1} z(x)dx = -\frac{c}{z'(z^{-1}(c))}## I am not quite sure where it gets it from...I think it can be separated and with differentiation the first part will be zero...
  21. J

    Derivative of Inverse Function

    Homework Statement Suppose ##f(x) = x^5 + 2x + 1## and ##f^{-1}## is the inverse of function f. Evaluate ##f^{-1}(4)## solution: 1/7 Homework Equations ##(f^{-1}(x))=\frac{1}{f'(f^{-1}(x))}## The Attempt at a Solution I attempted to use my calculator's solve function to get the solution of...
  22. K

    Derivative of an inverse function

    Homework Statement $$\int\frac{dx}{x\sqrt{x^2-1}}=\left\{ \begin{array} {lc} \sec^{-1}(x)+C_1 & {\rm if}~x>1 \\ \sec^{-1}(-x)+C_2 & {\rm if}~x<-1 \end{array} \right.$$ Why the second condition ##\sec^{-1}(-x)+C_2~~{\rm if}~x<-1## ? Homework Equations Derivative of inverse secant...
  23. K

    Snell's law and inverse function

    Homework Statement Snell's law is: $$\frac{\sin\theta_1}{c_1}=\frac{\sin\theta_2}{c_2}$$ $$\frac{c_1}{c_2}=n_{12}$$ Express ##\theta_2## as a function of ##\theta_1## Find the largest value of ##\theta_1## for which the expression for ##\theta_2## that you just found is...
  24. K

    Inverse Function Homework: Simplifying Sin^-1(2Sin^-1(0.8))

    Homework Statement Simplify: $$\sin^{-1}(2\sin^{-1}0.8)$$ Homework Equations Inverse sine: ##y=\sin^{-1}(x)~\rightarrow~\sin(y)=x## $$\sin^2(x)+\cos^2(x)=1$$ The Attempt at a Solution The inner parenthesis: ##\sin y=0.8## . In the drawing it's alpha's sine. Now i double the α and the...
  25. I

    Discover Inverse Functions for Equations with Exponents and Logarithms

    Homework Statement a.)y=sqrt(2^x -1) . I tried: b.)y=log(sqrt(2^x -2)) and c.)y=log^3 (2-sqrt(x)). Homework EquationsThe Attempt at a Solution x=sqrt(2^y -1) x^2 = 2^y -1 2^y = x^2 +1 y=log2(x^2 +1) y=2 log2(x+1) is that correct result? regarding b and c I am just lost :/. Will appreciate...
  26. Theia

    MHB How can I find the inverse function for a given approximation?

    Hello all I was doing some approximation to solve another problem, but got stuck when trying to figure out a suitable inverse functions for this: a = \frac{\cos x}{3x^2 - \pi^2}, where 0 \le x \le \pi. What I need is the two functions x(a) at least near a \approx -0.086 \pm 0.01 but I'm not...
  27. M

    F bijective <=> f has an inverse

    Homework Statement Proof that: f has an inverse ##\iff## f is a bijection Homework Equations /definitions[/B] A) ##f: X \rightarrow Y## If there is a function ##g: Y \rightarrow X## for which ##f \circ g = f(g(x)) = i_Y## and ##g \circ f = g(f(x)) = i_X##, then ##g## is the inverse...
  28. M

    Finding the Inverse Function of f(x) = 1−3x−2x^2 on Domain [-2, -1]

    Homework Statement Let f(x) = 1−3x−2x^2 , x ∈ [−2, −1]. Use the Horizontal Line Test to show that f is 1–1 (on its given domain), and find the range R of f. Then find an expression for the inverse function f −1 : R → [−2, −1]. The Attempt at a Solution I have already done the horizontal line...
  29. G

    B What is the process for finding 2y = x + 2 in an inverse function?

    Please take a look in below image. How do they get 2y = x + 2?
  30. T

    Solution Set for cot-1(x)2 -(5 cot-1(x)) +6 >0?

    Homework Statement Solution set of the inequality (cot-1(x))2 -(5 cot-1(x)) +6 >0 is? Homework EquationsThe Attempt at a Solution Subs cot-1(x)=y We get a quadratic inequality in y. y2-5y+6>0 (y-2)(y-3)>0 Using the wavy curve method, the solution set is...
  31. Kupkake303

    How to find the inverse of a function

    T(t) = Ts+(98.6 – Ts)e-kt rewrite in the form t=g-1(T) In trying to understand how to find the inverse of this but am having a hard time, please advise. Thanks, Kupkake303
  32. G

    MHB Linearization of this equation / Inverse function

    Hello, I need to find the inverse function of the following equation y = a * ((exp(-b * x)) + (c * (1 - (exp(-b * x))))) Where a, b and c are constants. I have experimental points that fit to this equation and I want to use these values in the inverse funtion to linearize it. I have tried...
  33. G

    MHB Define $f: \mathbb{Z} \to \mathbb{Z}: f^{-1}(\left\{0,1,2\right\})$

    Define $f: \mathbb{Z} \to \mathbb{Z}$ via $f(n) = n^2$ for all $n \in \mathbb{Z}$. Why does $f^{-1}(\left\{0,1,2\right\}) = \left\{0,-1,1\right\}$? The definition I'm using is $f^{-1}{(T)} = \left\{a \in A: f(a) \in T \right\}$ so we have $f^{-1}({ \left\{0,1,2\right\} }) = \left\{n \in...
  34. N

    Proving an inverse function

    if then to prove an inverse of this exists the following has been done to show that it is one to one what is the basis of equating the 2 square roots ?
  35. S

    MHB Find Inverse of f(x) = ln(x-1)

    find the inverse function of f(x)=ln(x-1), x>1
  36. Unichoran

    Finding an inverse function

    Homework Statement Hello,I have some problems with my Pre-Calculus homework. The task is: You get paid 8$ per hour plus 0.85$ per unit you produced. 1.Set up an equation for it. 2.Find the inverse function. 3.What does each variable in the inverse function mean? Homework Equations See below...
  37. J

    Checking for Accuracy in homework

    I noticed the scan was cut off on the second image at the bottom right, but I came up with x= 31/5 My first test in Calc I begins tomorrow and I want to know that I'm headed in the right direction. I think I understand to some extent how logarithms can be expanded and condensed though I'm...
  38. kandelabr

    A Least-squares - fitting an inverse function

    I have a set of data points that I must fit to an inverse function like this: y(x) = a/(x+b) + c My problem is that least-squares fitting using this equation is extremely unstable and heavily dependent on initial guess. No matter how accurate parameters I start with, the algorithm often goes...
  39. J

    Inverse function of a quarter circle gives me same function

    Is this normal? it doesn't seem correct. The equation for the portion of circle with radius 1 unit in the 1st Quadrant is: ## y = f(x) = \sqrt{1-x^2} ## Domain is 0<x But when I calculate f'(x) I also get ## f'(x) = \sqrt{1-x^2} ## I thought inverse functions always reflect about y=x. Please...
  40. arpon

    What is the inverse function of x + sin x ?

    What is the inverse function of ##x+sin x## ?
  41. T

    Derivative of an inverse function

    Homework Statement I will post a picture of the problem and then the second picture will be my work. The problems are the first two. Homework EquationsThe Attempt at a Solution I didn't know how to do this at first so I don't know if I am doing it correctly now. Also I don't know the correct...
  42. D

    MHB Continuity of inverse function at endpoints

    Hello! *Let $f$ be a strictly increasing continuous function on a closed interval $[a, b]$, let $c = f(a), d = f(b)$, and let $g:[c, d] → [a, b]$ be its inverse. Then $g$ is a strictly increasing continuous function on $[c, d]$.* How can it be shown that $g$ is continuous at its endpoints $c$...
  43. Math Amateur

    MHB Inverse Function Theorem for One Real Variable

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of the Inverse Function Theorem (IFT) for real-valued functions of one real variable. Stoll's statement of the IFT for Derivatives and its proof read as follows: In the above proof we read: "...
  44. S

    MHB Inverse Function Formula

    One semester I was asked to find the inverse of $\,f(x) \:=\:\dfrac{3x - 5}{2x+1}$ Later, I had to find the inverse of $\,f(x) \:=\:\dfrac{2x+7}{4x-3}$ It occurred to me that a general formula would a handy tool. Especially since I planned to teach Mathematics and I might be teaching this very...
  45. W

    Formula of an inverse function

    Homework Statement Find the formula of the inverse function of f(x)=300/(3+15e^.05x). Homework Equations f(x)=300/(3+15e^.05x) The Attempt at a Solution I'm definitely way off but I got .05y(5x)+ln100=lnx. What I did was multiple the denominator by the y(cross mltiplication)...
  46. C

    MHB Generating an inverse function from the given one

    Hi, I have a relationship $$P \cong \Bigg[\Big(K_1\rho^{\frac{5}{3}}\Big)^{-2}+ \Big(K_2\rho^{\frac{4}{3}}\Big)^{-2}\Bigg]^{-\frac{1}{2}}$$I need to find the inverse as $$\rho= \rho(P)$$. I made a detailed calculation and came up to this $$y^5+\Big(\frac{P}{K_2}\Big)^2 y+...
  47. A

    How to find inverse function

    Consider the function g(x) represented by the table below: x -6 -4 -2 0 2 4 6 g(x) -4 -2 4 0 6 -6 2 Complete the table of values for the INVERSE, g^{-1}(x), in the table below: x -6 -4 -2 0 2 4 6 g^{-1}(x)
  48. bsmithysmith

    MHB Continuity of the Inverse Function

    I just started Calculus 1, a summer quarter that's compressed and I'm having trouble understanding a theorem that state continuity of the inverse function. Within my textbook, it mentions "If f(x) is continuous on an interval I with range R, and if inverse f(x) exists, then the inverse f(x) is...
  49. B

    Inverse function theorem over matrices

    Homework Statement I have a function f:M_{n×n} \to M_{n×n} / f(X) = X^2. The questions Is valid the inverse function theorem for the identity matrix? It talks about the Jacobian at the identity, but I have no idea how get a Jacobian of that function. Can I see the matrices as vectors and...