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## Homework Statement

[tex]f\left(x^{2}+f(y)\right)=y-x^{2}[/tex]

## Homework Equations

Find all functions f that satisfy the relationship for every real x and y.

## The Attempt at a Solution

is this correct reasoning?

for x=0: [tex]f(y)=f^{-1}(y)[/tex]

for x>0: [itex]\existsxεℝ[/itex]: [tex]x=k^{2}[/tex]

[tex]f(k^{2}+f(0))=-k^{2}+f(0)[/tex]

for x<0 [itex]\existsxεℝ[/itex]: [tex]x=-k^{2}[/tex]

[tex]f(0)=f(k^{2}+f(-k^{2}))[/tex] = [tex]f(-k^{2})-k^{2}[/tex] which entails:

[tex]f(-k^{2})=f(0)+k^{2}[/tex] =[tex]-(-k^{2})+f(0)[/tex]