Finding the Inverse Function and Its Derivative - Analyzing f(x) at Point (1,1)

In summary, The conversation is about finding the inverse of a given function f(x1,x2) and its derivative at a specific point (1,1). The first question involves showing the existence of an inverse by proving that the determinant of the derivative at (1,1) is not equal to 0. The second question involves using the chain rule and finding the derivative of the inverse function at (1,1). The conversation also includes a clarification on how to obtain the inverse function and its derivative.
  • #1
56
0
Hello:

My 2 questions are underlined below. I have another problem but will first wait for an answer on this before posting the other question.


Problem Statement

Consider the function f:R2->R2 defined by
f(x1,x2) = ( exp (x1-x2) + x[tex]^{2}_{1}[/tex]x2 + x1(x2-1)[tex]^{4}[/tex],
1 + x[tex]^{2}_{1}[/tex] + x[tex]^{4}_{1}[/tex] + (x1x2)[tex]^{5}[/tex] )

Q1- Find Df(x1,x2) at point (1,1) and show that there are open sets (1,1)[tex]\in[/tex] U and f(1,1)[tex]\in[/tex] V such that f:U->V has an inverse g

Q2- Compute Dg(f(1,1))

Solution

First, I can compute the matrix Df(x) by taking the partial derivatives wrt x1, x2 and then take its determinant evaluated at x= (1,1).
If I evaluate it at (1,1) it comes out to A = [3 0 ; 11 5]

Second, the determinant of A is 15 which is != 0. So an inverse exists at this point.

My question for Q1 above: is it sufficient to prove that if det !=0 then an inverse exists. Is this all the question is askign. What does the question mean when asking to show there are open sets (1,1) [tex]\in[/tex] U ...

My question for Q2 above: Dg(f(1,1)) = inv(A) evaluated at (1,1). Is this correct

Thanks

Asif
 
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  • #2
Q1 is just the inverse function theorem.
Q2 is a special case of the chain rule.
 
  • #3
For Q2:

For a chain rule, don't I need a matrix for g. How do I get this?

Is this g matrix, the inverse of f evaluated at (2,4) <-- I get this from f(1,1) = (2,4)
And multiply this by Df(1,1)

Thanks

Asif
 
  • #4
Well, you can formulate the chain rule in terms of the jacobian evaluated at p.

J(f○g)(p) = J(g)(f(p))*J(f)(p)
 
  • #5
Thanks... yes I think got the Jacobian part.
However, what I am not getting is how do I get 'g'.

BTW, I assume you are saying p=f(1,1)Thanks

Asif
 
  • #6
No, my p in your case is (1,1).

What happens on the left side when g is f's inverse?
What can you then do to solve for J(g)(f(p))?
 
  • #7
When g is f's inverse, all I have to do is take inverse of f matrix evaluated at (1,1). This will give me the Jg(f(p)). Is this right?
What I had missed was the Jf(p)

Thanks

Asif
 
  • #8
If g is f's inverse, then J(f○g)(p) = Id
So
Id = J(g)(f(p))*J(f)(p)
Then
J^(-1)(f)(p) = J(g)(f(p))
 

What is an inverse function?

An inverse function is a function that reverses the action of another function. It essentially "undoes" the original function, taking the output of the original function and using it as the input for the inverse function. This allows us to retrieve the original input from the output.

How do you find the inverse of a function?

To find the inverse of a function, we follow the steps of first replacing the function's output with a variable (usually denoted as "y"), then solving for "x" (the original input). The resulting equation is the inverse function.

What is the notation for an inverse function?

The notation for an inverse function is "f^-1(x)", where "f" represents the original function and "x" represents the input. This notation is read as "f inverse of x".

What is the domain and range of an inverse function?

The domain of an inverse function is the range of the original function, and the range of an inverse function is the domain of the original function. This is because the input and output of the original function become the output and input, respectively, of the inverse function.

What is the relationship between a function and its inverse?

A function and its inverse are symmetric about the line y=x. This means that if we graph both the original function and its inverse, they will be reflections of each other across the line y=x. Additionally, the composition of a function and its inverse (f(f^-1(x)) or f^-1(f(x))) will always result in the input value.

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