What is Inverse functions: Definition and 103 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. 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...
  2. AN630078

    Composite and Inverse Functions Problem

    1. a. fg(x)=2(1/2(x-1))+1 fg(x)=2(x/2-1/2)+1 fg(x)=x-1+1 fg(x)=x gf(x)=1/2((2x+1)-1) gf(x)=1/2(2x+1-1) gf(x)=x+1/2-1/2 gf(x)=x The functions functions f(x) and g(x) are inverses of each other. This can be demonstarted by f(x)=2x+1 y=2x+1 x=2y+1 x-1=2y (x-1)/2=y Thus, y=1/2(x-1) = g(x) And...
  3. S

    Correct statement about composite and inverse functions

    f-1(f(A)) = A and f-1(f(B)) = B so options (a) and (c) are wrong. For (b), I get A ⊆ A For (d), I get B ⊆ B For (e), I get A ⊆ A So there are three correct statements? Thanks
  4. karush

    MHB Tikx and inverse functions

    ok I have been trying to cut and paste in packages and code to get a simple inverse function to plot but nutin shows up and get error message. if possible I would like no grid but an xy axis with tick only where the graph goes thru the axis and of course a dashed line of x=y some of the...
  5. karush

    MHB Simplifying $\cot^2(x)-\csc^2(x)$: 1

    Write $\cot^2(x)-\csc^2(x)$ In terms of sine and cosine and simplify So then $\dfrac{\cos ^2(x)}{\sin^2(x)} -\dfrac{1}{\sin^2(x)} =\dfrac{\cos^2(x)-1}{\sin^2(x)} =\dfrac{\sin^2(x)}{\sin^2(x)}=1$ Really this shrank to 1 Ok did these on cell so...
  6. Math Amateur

    MHB Exploring Theorem 4.29: Compact Metric Spaces & Inverse Functions

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...
  7. P

    MHB Inverse Functions: Reflection of f(x) & g(x) Logic

    Can anyone explain the logic behind the answer? Taken from HiSet free practice test
  8. 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...
  9. K

    Find Smallest b to Make Invertible Function: Homework Solution

    Homework Statement Find the smallest value of b so that the function f(x) = x^3 + 9x^2 + bx + 8 is invertible. Homework EquationsThe Attempt at a Solution I know that the function has to be only increasing/decreasing, and I think it is needed to find the derivative of the function. I do...
  10. F

    Hi, I have a quick question about inverse functions.

    One of our homework problem asks: If f is a one-to-one function such that f(-3)=5 , find x given that f^-1 (5)=3x-1. Here's how I attempted to solve the problem: -3=3x-1 3x=-2 x=-2/3 Is this the correct way to solve the problem?
  11. opus

    B Inverse Functions vs Inverse Relations

    If we have a relation, ##R##, and it's inverse, ##R^{-1}## they behave such that a point on ##R##, say (a,b), corresponds to the point (b,a) on ##R^{-1}## This is a reflections across the line y=x. This relation does not mean that ##R^{-1}## is a function. For example, Let ##R## be...
  12. C

    MHB Sets and inverse functions?

    Temperatures can be converted from Fahrenheit to Celsius using the function f(x) = 5 /9 (x − 32). (a) Calculate f(59). (b) Find f −1 (x), and verify that f −1 (f(59)) = 59. (c) Let K be the set {x : f(x) = x}. Find all elements of K and list K
  13. T

    Inverse trigonometry question

    Homework Statement If 1/2<=x<=1 , then cos-1[ x/2 + (√(3-3x2))/2] + cos-1 x is equal to? Homework EquationsThe Attempt at a Solution Substituting x=cosθ = cos-1(cosθ) + cos-1 [ cos(θ)/2 + (√(3(1-cos2(θ)))/2] = θ + cos-1 [cos (π/3) cos θ + sin (π/3) sinθ] =...
  14. 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
  15. Geologist180

    Tricky question regarding Inverse functions

    Homework Statement Suppose that f has an inverse and f(-4)=2, f '(-4)=2/5. If G= (1/f-1) what is g '(2) ? If it helps the answer is (-5/32) Homework Equations [/B] f-1'(b)=1/(f')(a) The Attempt at a Solution Im not really sure how to start this problem. I am familiar with how to use the...
  16. Ryaners

    Finding inverse of a Sin function (problem from Mooculus)

    I'm working through the problems in the Mooculus textbook as revision for Calculus I & there seems to be something wrong with how I'm manipulating the function to find its inverse in the following example. Homework Statement The height in meters of a person off the ground as they ride a Ferris...
  17. 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...
  18. P

    Proving Inverse Functions: Multiplicative Relationships

    Is there a way to formally prove that if ##f## and ##g## are multiplicative inverses of each other, then ##f^{-1} (x) = g^{-1} (\frac{1}{x})##?
  19. Cookiey

    Inverse Function of Greatest Integer Function in a Given Domain

    Homework Statement If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x). Homework Equations (None I believe.) The Attempt at a Solution I know that for a function to be invertible, it must be both one-one and onto...
  20. L

    Shifting and inverse functions

    Homework Statement If we shift a curve to the left, what happens to its reflection in the line y = x? In view of this geometric principle, find an expression for the inverse of g(x) = f(x + c) where f is a one-to-one function. Homework EquationsThe Attempt at a Solution Initially I did this...
  21. 462chevelle

    Solve Inverse Functions: Find (f^-1)'(a) for f(x)=Cos(2x), a=1

    Homework Statement Verify that f has an inverse <- prof told us not to worry about this. Then use the function f and the given real number a to find (f^-1)'(a). f(x) = Cos(2x), 0<=x<=pi/2 where a=1 Homework Equations 1/(f'(g(x))) where g(x)=f^-1(a) d/dx(cos(2x)) = -2sin(2x) The Attempt at a...
  22. Dethrone

    MHB Domain and range of inverse functions (circular and hyperbolic)

    I've always been having trouble with the domain and range of inverse trigonometric functions. For example, let's start with an easy one: $\sin^{-1}\left({x}\right)$ Process: First, I draw out the function of $\sin\left({x}\right)$. Then I look at its range and attempt to restrict it so that it...
  23. A

    MHB Inverse Functions and "Verifying"

    Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: Alright, I'm having quite a bit of trouble with these. So here it goes: For the first one, I did the 3-step procedure to finding the...
  24. chisigma

    MHB How to find the series of inverse functions

    In the Math Challenge Forum it has been requested fo compute the series... $\displaystyle S = \sum_{n=1}^{\infty} \tan^{-1}\ \frac{\sqrt{3}}{n^{2} + n + 3}\ (1)$ ... and that has been performed using the general identity... $\displaystyle \sum_{n=1}^{\infty} \tan^{-1}\ \frac{c}{n^{2} + n +...
  25. C

    MHB Even Functions, Symmetry, Inverse Functions

    Can someone explain why the answer is D a < 0 because it finishes downwards e < O because the y-intercept is in the negatives. b, & d = zero (but i don't get this) c is supposedly > 0 (nor do i get this) According to the solutions the graph is an even function, and symmetrical about the...
  26. applestrudle

    Finding inverse functions when there is X^2+x

    Homework Statement f(x)= 2x + x^2 Homework Equations The Attempt at a Solution I don't know how to make x the subject
  27. A

    Understanding inverse functions

    I need some help understanding inverse functions, we've had a 4-page chapter covering the basics of inverse functions and I understand that. But now we have suddenly gotten these task that I don't understand how to solve, I've read the part on inverses several times, but I still don't...
  28. M

    Question regarding inverse functions

    f(x) where x belongs to all real numbers inverse: f-1(x), where x belongs to all real numbers True or False: The inverse of f(x+3) is f-1(x+3) My ideas: I think that it is false given that when you usually find the inverse of a function, you switch the x and y variables and solve for y again...
  29. chisigma

    MHB A general way to find the inverse functions....

    In www.mathhelpforum.com and interesting question has been proposed by the user misiazeska the 05 20 2013...How to find the inverse of this function?...$$y= 5\ x^{3} - x^{5}$$ ... and the unanimous answer has been '... it doesn't exist any closed formula to find x as function of y...'. In my...
  30. S

    One-to-one functions and inverse functions.

    I was just wondering if inverse functions only apply to one-to-one functions?(Or a function who's domain has been restricted to act as a one-to-one function). Thanks.
  31. B

    MHB Inverse Functions Problem

    f(x)=a+4/(x-a) f-1(x)=(a^2-ax-4)/(a-x) Which of the following is true? The function is the opposite of its own inverse for any value of a. The function is its own inverse for positive values of a only. The function is the reciprocal of its own inverse for positive values of a only. The...
  32. K

    Inverse Functions Homework: Find f^-1, g^-1, Show f^-1 f=x

    Homework Statement 5. (a) The functions f and g are defined by f : x|→ 2x + ln3 (x is a real number) g : x|→ e^3x (x is a real number) i) Find f^-1(x) and g^-1(x) and state their domain of definition ii) Show that f^-1 f = f f^-1 = x (x is a real number) iii) Find the composite function...
  33. A

    Calculus Inverse Functions

    Homework Statement 1. Show that the function f(x) = x5 -x3 +2x is invertible. Compute the derivative of f-1 at 2. The Attempt at a Solution To find f-1 I switched x and y which gave me x = y5 - y3 + 2y this is where i got stuck because I am not sure how to solve for y after that...
  34. MarkFL

    MHB Inverse Functions & Newton's Method: Debmnzl's Question | Yahoo! Answers

    Here is the question: Here is a link to the question: A math function question please help? - Yahoo! Answers I have posted a link there to this topic so that the OP may find my response.
  35. C

    A question regarding inverse functions

    Hi I'm reading a book called Calculus lifesaver, and in the book they state that the inverse of a function f(x)= x3 is the same as f-1(x)=3√x and is the same as f-1(y)=3√y So I did a test, with a simpler function and I can't see how this is true If I have a function f(x)= 2*x Then...
  36. mnb96

    Question on Jacobian with function composition and inverse functions

    Hello, let's suppose I have two functions \phi:U\rightarrow V, and T:V\rightarrow V that are both diffeomorphisms having inverse. Furthermore T is linear. I consider the function f(u) = (\phi^{-1}\circ T \circ \phi)(u), where \circ is the composition of functions. Since T is linear, we...
  37. phosgene

    Derivatives of integrals and inverse functions

    Homework Statement Find the derivative of: 1. f(x)=arccos(5x^3) 2. f(x)=∫cos(5x)sin(5t)dt when the integral is from 0 to x Homework Equations Chain rule, dy/dx=dy/du*du/dx The Attempt at a Solution For the first one, I can just take 5x^3 as u and then apply the chain rule...
  38. H

    Analysis - Inverse Functions

    Homework Statement : Define f: ℝ→ℝ by f(x)=x^2. Find f^-1(T) for each of the following: (a) T = {9} (b) T = [4,9) (c) T = [-4,9] The attempt at a solution: So, the inverse of f should be f^-1(T)=+/-√(x). Therefor: (a) f^-1(9)= +/- 3 (b) f^-1(4)= +/- 2, f^-1(5)= +/- √(5), f^-1(6)= +/- √(6)...
  39. S

    Confused about derivatives of inverse functions

    Hi. I am reviewing for an exam, and this is a topic that I did not go through very thoroughly. I understand how to calculate the derivative of an inverse function when I am given a point, as I simply use the equation (1/f((f^-1))'. So if I am given the equation y=x^3, for example, and am asked...
  40. C

    MHB Understanding inverse functions

    I have a graph f(x) = 3(x + 1)^2 - 12 , I have sketched this graph (Not shown) hand it is a parabola with a y- intercept at - 9. the vertex being - 12. The image set is a closed interval {- 12, infinity} Sorry no square brackets and no sign for infinity. I am asked to explain why the function...
  41. B

    Inverse functions and null set.

    Ok, I understand an inverse function sends a variable in the range to the corresponding value in the domain, but am not sure if what I'm thinking is correct... : For example: Let A be the set A = \{1,2,3,7,8\} ; B = \{4,5,6\} and the function f map A to B s.t f(1) = 4 f(2) =...
  42. M

    A question about restrictions of inverse functions

    Homework Statement Hi. I found in the answears that the inverse of function f(x)=3-\sqrt{x-2} is f^{-1}(x)=(3-x)^{2}+2 only if we restrict it to {x:x\leq3}. I understand that the restriction is needed because the found inverse is a parabola (and thus not one-to-one function). My general...
  43. P

    MHB Application for Derivative of Inverse Functions?

    Why are derivatives of inverse functions important? My students are giving me questions like: When would using the theorem be useful? Can't you just find the inverse function and take its derivative? I'm sure many of you know the type of question: "Who cares?" My answers are that the theorem...
  44. T

    Proving an equivalence relation using inverse functions

    Homework Statement Let f : A → B be a function and let Γ ⊂ B × B be an equivalence relation on B. Prove that the set (f × f)^-1 (Γ) ⊂ A × A (this can be described as {(a, a′) ∈ A × A|(f(a), f(a′)) ∈ Γ}) is an equivalence relation on A.Homework Equations The Attempt at a Solution Let...
  45. T

    Inverse functions and One to One

    Homework Statement For exercises 49 and 50 let f(x) = (ax + b)/(cx + d) 50. Determine the constants a, b, c, d for which f = f-1 Homework Equations I found in question 49 when they asked to find f-1 that: f-1 = (dx - b)/(a - cx) This was also the answer at the back of the book but...
  46. W

    Inverse functions and tangent lines

    Homework Statement 2 problems, i solved both of them but I am not 100 % I am right Find all points on the curve y=x-2cosx where the tangent line to the curve is parallel to the line y=x and write an equation of the tangent line at such point Let f^-1 be the inverse of the one -to- one...
  47. M

    Inverse Function of g(x) at 4 - Solve Algebraically

    Problem: If g(x) = 3 + x + e^x find Inverse of g at 4 My work: 4 = 3 + x + e^x 1 = x + e^x This is where I stop... I can look at it and see that x = 0 But I don't know how to find the solution algebraically...
  48. S

    Integration with inverse functions

    Homework Statement let f(x)=(4t^3+4t)dt(between 2 and x) if g(x) = f^(-1)(x), then g'(0)=? Homework Equations The Attempt at a Solution f'(x) = 4x^3+4x annd i already don't know where to go from here.. help?
  49. F

    Simplifying Inverse Functions

    Homework Statement f(x) = 3x^2-6 We are asked to solve for the inverse function of the above function. Homework Equations The Attempt at a Solution y=3x^2-6 x=3y^2-6 \frac{3y^2=x+6}{3} y^2 = \frac{x+6}{3} \sqrt{y^2}= \frac{\sqrt{x+6}}{3} y=...
  50. P

    Composite + Inverse Functions

    Hello guys :smile: Given that: f(x) = e2x and g(x) = (2x-1) Find: (g ° f)-1(x) So, what I did first was to put f into g: 2 x e2x - 1 = y 2 x e2x = y + 1 e2x = y + 1 / 2 (ex)2 = y + 1 / 2 ln (y+1/2) = x Is that ok? Thanks, Peter G.