Check the local inevitability of the function f defined by f(x,y) =(x²-y²,2xy) at (1,1) Find a domain for the function f in which f is invertible.

To check the local invertibility of the function \( f(x,y) = (x^2 - y^2, 2xy) \) at \( (1,1) \), we use the Inverse Function Theorem.  

1. Compute the Jacobian Matrix  
   The Jacobian matrix \( J_f(x,y) \) is given by:  
   \[
   J_f(x,y) =
   \begin{bmatrix}
   \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\
   \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}
   \end{bmatrix}
   \]
   where  
   \[
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