Question about Proof of Inverse Function

[transphenomen]

I am reading the proof of the Inverse Function Theorem in baby Rudin and I have a question about it. How does associating a function phi(x) (equation 48) with each point y tell us anything about if f(x) is one-to-one? I'll show the proof below. Also, if f'(a) = A, and f(x)=x², what would A^-1 be?

 

[sponsoredwalk]

The exact same equation had me freaking out for a while.

You can think of φ(x) = x + A⁻¹(y - f(x)) as coming from f(x) = f(x₀) + f'(x)(x - x₀),
& it's a way of introducing the concept of a fixed point so we can use the contraction
principle. You basically want to show that |φ'(x)| < ½ & then use theorem 9.19
|φ(b) - φ(a)| ≤ M(b - a) & with M = ½ you have a contraction mapping, thus by the
contraction principle you have that for every y it's associated function φ must be 1-1.