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

[mtayab1994]

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]

 

[SammyS]

Homework Statement

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

for x=0: [tex]f(y)=f^{-1}(y)[/tex]
for x>0: [itex]\exists xεℝ[/itex]: [tex]x=k^{2}[/tex]
[tex]f(k^{2}+f(0))=-k^{2}+f(0)[/tex]
for x<0 [itex]\exists xεℝ[/itex]: [tex]x=-k^{2}[/tex]
[tex]f(0)=f(k^{2}+f(-k^{2}))[/tex] = [tex]f(-k^{2})-k^{2}[/tex]...

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.