## Homework Statement

$$f\left(x^{2}+f(y)\right)=y-x^{2}$$

## 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: $$f(y)=f^{-1}(y)$$

for x>0: $\existsxεℝ$: $$x=k^{2}$$

$$f(k^{2}+f(0))=-k^{2}+f(0)$$

for x<0 $\existsxεℝ$: $$x=-k^{2}$$

$$f(0)=f(k^{2}+f(-k^{2}))$$ = $$f(-k^{2})-k^{2}$$ which entails:

$$f(-k^{2})=f(0)+k^{2}$$ =$$-(-k^{2})+f(0)$$

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## Homework Statement

$$f\left(x^{2}+f(y)\right)=y-x^{2}$$...

for x=0: $$f(y)=f^{-1}(y)$$
for x>0: $\exists xεℝ$: $$x=k^{2}$$
$$f(k^{2}+f(0))=-k^{2}+f(0)$$
for x<0 $\exists xεℝ$: $$x=-k^{2}$$
$$f(0)=f(k^{2}+f(-k^{2}))$$ = $$f(-k^{2})-k^{2}$$...
This question has been previously discussed.

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This question has been previously discussed.

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