Finding invertible functions and local inverses

Question

Assignment:

Let f : R²→R² be given by,
f(( x, y)) = (x² - y² , xy)  for(x , y) ∈ R².

Answer the following questions:

(a) Is f invertible?
(b) In which points(a , b) ∈ R² does f have a local inverse i.e., for which points (a, b) does there exist a neighbourhood U of (a, b) such that f is injective on U?
(c) Determine the derivative of the local inverse of f_|uat f((-1,1)) and f((1,-1)).

Provide complete and step by step solution for the question and show calculations and use formulas.