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.