How can I explicitly compute the inverse of a vector valued function?

[Carl140]

Homework Statement

Let f= (f_1, f_2, f_3) be a vector valued function defined (for every
point (x_1,x_2,x_3) in R^3 for which x_1 + x_2 + x_3 is not equal to -1) as follows:


f_k (x_1,x_2,x_3) = x_k /( 1+x_1+x_2+x_3) where k =1,2,3.


After some computations I found that the determinant of the Jacobian
matrix is (1+x_1+x_2+x_3)^(-4) (which coincides with the answer of the book).
Then, by the inverse function theorem, it follows that f is one to one
since the determinant is nonzero.

The problem is the following:


Compute f^(-1) explicitly.


How can I do this?


http://en.wikipedia.org/wiki/Inverse_function_theorem


Gives a formula to find the inverse of the jacobian matrix, but I'm trying to find the inverse of the function.

How to do this?

The Attempt at a Solution

I don't see how to find the inverse explicitly, I know it exists because the determinant
of the Jacobian is nonzero everywhere.

 

[Carl140]

Nevermind, its just a system of equations =)

 

[Dick]

In general it can be hard or impossible to find a formula for the inverse. In this case it's easy because it's a simple function. Just use algebra. If f(x,y,z)=(a,b,c) can you find a formula for x, y and z in terms of a, b and c? That's three simultaneous equations in the three variables if you equate the components. Hint: add them.