Inverse function Definition and 197 Threads

I'm having trouble following the proof of the IFT by Spivak. The statement of the theorem was posted in a similar thread: https://www.physicsforums.com/showthread.php?t=319924 He says, "If the theorem is true for \lambda^{-1} \circ f, it is clearly true for f. Therefore we may assume at the...

Find the inverse of the function ƒ(x) = x2. If not possible, explain why?Relevant information. I drew up the function ƒ(x) = x2 on a graph and did the horizontal test. But, I do not understand why the horizontal test excludes attaining the inverse of that function.

I have a "pop" quiz tomorrow in my calc course and my professor is stating that we have to be able to restate the proof of inverse function derivatives. I am looking it over in my book and it's pretty straight forward except for one part that I can't figure out... This is listed step by step \...

Homework Statement Let g:[b,2] -> R where g(x) = 1-x2. If b is the smallest real value such that g has an inverse function, find b and g inverse The Attempt at a Solution I can find the inverse function easily, but I don't understand how I go about finding b. According to the book...

Homework Statement If F(x) = \frac{x}{x + 1}, then the inverse function, f^{-1}, is given by f^{-1}(x) = Homework Equations The Attempt at a Solution I've replaced F(x) with y, and switched the x and y variables. Where I'm having a problem is solving the resulting equation, here's what I've...

if a continious function is monotoniously increasing in an interval , is it necessary that its inverse will also increase monotoniously in that interval?

Homework Statement Suppose that f' exists and is continuous on a nonempty, open interval (a,b) with f'(x) \neq 0 for all x \in (a,b) . I already proved that f is 1-1 on (a,b) and takes (a,b) onto some open interval (c,d). (i)Show that f^{-1} is continuously differentiable on (c,d). (ii)Using...

Homework Statement f is a function with an inverse and it is differentiable. Use f(f-1(x))=x and come up with the formula for the derivative of f-1 Homework Equations The Attempt at a Solution I tried expanding that equation to f'(f-1(x))*f'-1(x) -x but I tested this and it...

Homework Statement Let f (x) have an inverse function g (x), then f(g(x)) = A) 1 B) x C) 1 / x D) f (x) x g (x) E) None of these 2. The attempt at a solution I know the definition of an inverse function is Let f and g be two functions. If f(g(x)) = x and g(f(x)) = x, then...

I've read that a function f given by f:U\rightarrow V is a diffeomorphism if the inverse function f^{-1} exists and is differentiable. I've also read that that function is a local diffeomorphism in a given point p\inU if it can be found a range A around p such that the function f verifies f:A ->...

Homework Statement Find g' (-1/2), where g(x) is the inverse of f(x) = x3 / (x2 + 1) Homework Equations g'(x) = 1/f' [g(x)] The Attempt at a Solution Everything else I tried got really messy, but I feel like I'm going out on a limb by: 1) y=x3 / (x2+1) 2) y=x3x-2 / 1 3) y=x 4) g...

Homework Statement arcsin(e^x) Homework Equations arcsin = 1/(1-x^2)^(1/2) The Attempt at a Solution f'(x)= 1/(1-x^2)^(1/2) * (e^x) + arcsin(e^x) I did product rule and got to this but not sure where to proceed after this point.

Homework Statement Let f(x)=e^(-x)-x ,, x belongs to R Find the domain of f inverse Homework Equations Domain of f inverse = range of f The Attempt at a Solution we have : -inf < x < inf -inf < -x < inf ... (1) 0 < e^(-x) < inf ... (2) By adding (1) and (2) ...

let be a function y=f(x) with poles f(a_{i} ) = \infty for some real 'a' my question is if we define the inverse function g(x) so g(f(x))=x ,then is this true g(\infty)=a_{i} my question is that it seems that g(x) would have several asymptotes as x-->oo how it can be ??

in one dimension one have that for a function f(x) we can define another function g(x) so f(g(x)=x my question or problem is the following, if i have a function of three variables f(x,y,z) then i can define another function g(x,y,z) so f(g(x,y,z))=Id for example for the...

Hi, I have a limited background in differential geometry. I have a problem involving a surface mapping (from R2 to R3) which does not have a square Jacobian. I understand that for a mapping of preserved dimensionality I can compute a matrix inverse which will allow me to map tangent vector...

In a first countable space any point that is adherent to a set S is also the limit of a sequence in S. In my head, this seems obvious, but I can't seem to get it on paper.. I know that is has to do with inverse functions preserving unions and intersections, but can't seem to write the proof out.

Homework Statement The problem is from Sarason, page 44, Exercise IV.14.1. Let f be a univalent holomorphic function in the open connected set G, and let g be the inverse function. Assume that f(G) is open, that g is continuous, and that f\prime\neq 0\forall z\in G. Prove g is...

Homework Statement Find the inverse function for f(x) = x + 4/x Homework Equations The Attempt at a Solution I tried many times but just can't get the answer. Is there any trick to this question? Thank you

The condition for the inverse function, f^(-1) to happen is function , f is one-one . S0 consider this function , f(x)=x^2-5 , which is NOT a one-one function , and f^(-1)=y x=f(y) x=y^2-5 y^2=x+5 f^{-1}(x)=\pm\sqrt{x+5} Seems that the inverse function of f exists without...

Homework Statement Find the exact value of cos(2arcsin(-1/8)) Homework Equations make use of the double angle formula The Attempt at a Solution let arcsin(-1/8)=theta then sin theta= -1/8 a=sqrt63=3sqrt7 and that is as far as i could go...please help? Thank you.

Homework Statement Graph f(x) = sqrt(x^2 - 2x), and find an interval on which it is one-to-one. Find the inverse of the function restricted to that interval.Homework EquationsThe Attempt at a Solution What I can't do is really finding the inverse function. It seems very simple, but somehow I...

Inverse function (Edited) Homework Statement Find the inverse function of : f(x)=e^x-e^{-x}+2 where x \geq 0 Homework Equations All what I did is : y=e^x-e^{-x}+e The Attempt at a Solution How in Earth can I solve this for x ?

Homework Statement Determine the inverse function for the function, f(x)=((2x+1)^3)-2 I think i know the steps but i want to know if my answer is correct 1.Change f(x) to f(y) 2.Re write so x=f(y) 3.Swap x and y variables 4.replace y with f^-1(x) The Attempt at a Solution...

I want to plot inverse function using Mathematica In[1]:=f = Solve[x == a * Log[y/100], y] Out[2] = {{y -> 100 E^(x/a)}} and then? how to use this reult to plot? ThanksAnother question is about the axeslabel, when I set the Frame->True, the AxesLabel can not be displayed correctly, where...

In my math class we are learning about inverse functions and how you can find if the inverse of a function is a function using the first derivative test. Our teacher omitted some of our homework regarding the derivatives of the inverse and he posed the question of how one would practically use...

Ok, i need to explain why this function has an inverse function. fr^-1 fr(x) = 1.5/sinx and state its image set & domain. Is that (0,1/2Pi] Thanks :)

I got a problem for this question. Given the function y=f(x)=6x^3+6x+2. Find f^-1(x). Can anybody tell me how to solve this function?So far I got this: x=6y^3+6y+2 x-2=6y^3+6yI just couldn't solve for y variable. So, please tell what I do wrong? thanks

f(x) = cosh^2(x)+sinh(2x) = y f'(x) = sinh(2x)+2cosh(2x) = 3e^(2x) + e^(-x) = y' Let g(y) be the inverse of f(x): g'(y) = 1 / f'(x) = 1 / [3e^(2y) + e^(-2y)] = e^(2y) / [4e^(2y) + 1] Integrating gives: [ 3^(1/2)/3 ]*arctan[ 3^(1/2) * e^(2y) ] + C Now when I plotted this function it...

On p. 36 of "Calculus on Manifolds" Spivak writes: "If the theorem is true for (\lambda^{-1})\circf , it is clearly true for f." This far I understand. However, he next says: "Therefore we may assume at the outset that \lambda is the identity." I don't understand how this follows...

Hi How do I take the inverse of log (x)/3? If it is just log (x), it seems quite easy to do but I don't know what to do with the division by 3. I saw this equation in a biostatistics article and I just can't understand how to solve it. It's been so long since I did inverse functions and...

Homework Statement if g(5)=-2, then g^-1(-2)=? Homework Equations n/a The Attempt at a Solution i can't remember how to work inverse functions when there is no variable i think its g^-1(-2)=5 but it could be g^-1(-2)=-2

Homework Statement Give an example of a continuously differentiable mapping F:R^n --> R^n with the property that tehre is no open subset U of R^n for which F(U) is open in R^n Homework Equations let U be an open subset of R^n and supposed that the continuously differentiable mapping...

NOTE: I AM NOT DOING THIS FOR HOMEWORK BUT JUST WANT TO KNOW BECAUSE I COULDN'T HELP SOMEONE ELSE WITH THIS PROBLEM. f(x)= x3+x can someone please explain how to do this step by step. thank you.

Dear all, Does anybody knows any the proof for Inverse Function Theorem for single variable function or link where I can find that proof? Thank you in advance

Homework Statement Given f(x) = -x2 - 2x + 3, x < -1 Find f-1. Find f-1f. Homework Equations Nil. The Attempt at a Solution Actually I worked out much of the question already, and I already know that f-1f: x -> x. The problem is, I can't seem to get f-1f(x) = x on...

Greetings all. I was solicited by a friend to find the inverse of a particular function, and I can't for the life of me determine/remember how. The original equation is y = 3+x^2+tan((1/2)*Pi*x) with x on (-1,1). The function is invertible - f' is always > 0 on that interval - but I have...

Homework Statement Find the inverse function f^{-1} of the binary entropy f (given below) on the domain of definition [0;1/2[ (i.e. where f is continuous strictly increasing). The function f is given by: f(x)=-x\log(x)-(1-x)\log(1-x) (where \log is the logarithm base 2) Homework...

In his proof of the IFT, on p. 36 of "Calculus on Manifolds," Spivak states: "If the theorem is true for \lambda^{-1} \circf, it is clearly true for f. Therefore we may assume at the outset that \lambda is the identity. I don't understand why we may assume that. thanks for your help...

The following graph shows the number of daylight hours as a function of the day of the year. Draw the inverse of the graph, and describe what the inverse represents. The graph given is very close to this (which I believe is close enough for this question)...

Homework Statement Find the inverse function. [Find (f^-1)(x)] f(x) = (-2x^5) + 1/3 Homework Equations Find the inverse function. [Find (f^-1)(x)] f(x)=2-2x^2 The Attempt at a Solution (-2x^5) + 1/3 (-1/2x^1/5) + 3 Answer is different, how would I get to (-1/2x + 1/6)^1/5?

Homework Statement If f:x-->y has an inverse function g, then g:y--->x is one to one and onto Homework Equations The Attempt at a Solution Let g be the inverse of f:x--->y I think this must have something to do with an isomorphism because of one to one and onto.

Hi, the statement of Inverse Function Theorem for Complex functions I learned is this: "let f:A->C, g:B->C be continuous s.t. f(A) \subset B. A,B are both open subsets of C, the complex plane. If g(f(z)) = z for all z in A, and g is analytic on f(z) where g'(f(z)) != 0, then f is analytic on...

Homework Statement Suppose that f has an inverse and f (6) = 18, f'(6) = 4/5. If g = 1/(f-1), what is g'(18)? have no idea how to set up problem

What is a good book that gives a clear idea of what these theorems are? I am taking differential geometry, and I would like to try and get a better understanding of them. To be honest, my calculus sequence NEVER went over them, so I need to get these ideas under my belt.

Homework Statement Hello, this is what the question states: Consider the function f(x) = 2x + cos(x). Find all points at which the inverse function has a slpe of 1/2. The Attempt at a Solution What I did was find where the original function has a slope of 2. Those x values would...

Homework Statement Prove or Disprove: A 2x2 matrix \epsilon A+I has a cube root near \left( \begin{array}{cc} 2 & 0 \\ 0 & -5 \end{array} \right).Homework Equations Inverse Function TheoremThe Attempt at a Solution I'm just confused about the "near \left( \begin{array}{cc} 2 & 0 \\ 0 & -5...

Homework Statement The problem is to prove that: if f(g(x))=x ... (1) and g(f(x))=x ...(2) then f(x)=g-1(x) The Attempt at a Solution Differentiating (1) wrt x f'(g(x))*g'(x)=1 f'(g(x))=1/g'(x) As the slopes are reciprocals of each other, hence f(x)=g-1(x) Is this as...

Homework Statement If f(x) = x + cos(x), find f-inverse of (1) Homework Equations **first time asking a forum question, please inform me of any errors in posting this question The Attempt at a Solution (1) y = x + cos(x) => if y = f(x) (2) x = y + cos(y) => if x = f(y) (3)...

Homework Statement \int\frac{4}{x(x+3)} Homework Equations The Attempt at a Solution I can get to s certain point and I know I need to do substitution but, everytime I try a substitution it just creates a more difficult problem. 4\int(x^{-1}(x+3)^{-1}) I've tried...