determine f^1(x) from first principles if: f (x)=2x-x^2

Question

How do I get to the 3rd step?

Answer 1

Let y=f(x)=2x-x²=-1+2x-x²+1=-(x-1)²+1.

Therefore (x-1)²=1-y, x-1=±√(1-y), x=1±√(1-y).

g(y)=1±√(1-y), g(x)=1±√(1-x), g(x)=f⁻¹(x)=1±√(1-x). So f(x) leads to two functions for the inverse:

f⁻¹(x)=1+√(1-x) and f⁻¹(x)=1-√(1-x).