Finding Functions that Satisfy a Specific Relationship: A Math Olympiad Problem

[mtayab1994]

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)

 

[SammyS]

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.

http:
//www.physicsforums.com/showthread.php?t=556487

Is there anything new in what you're posting this time?

 

[mtayab1994]

This question has been previously discussed.

http:
//www.physicsforums.com/showthread.php?t=556487

Is there anything new in what you're posting this time?

yes for x>0 and for x<0 i want to know if what i did is correct.

 

[micromass]

Please post in the old thead.