# Thread: Find all The Functions !

1. ## Functional equation !

Hii !

Can You Help Me For This Exercice :

Find all The Functions $\displaystyle f$ : $\displaystyle \mathbb{R}$ $\displaystyle \longrightarrow$ $\displaystyle \mathbb{R}$ Such as :

$\displaystyle f((f(x)+y)=f(x^2-y)+4f(x)y$

2. Originally Posted by Perelman
Hii !

Can You Help Me For This Exercice :

Find all The Functions $\displaystyle f$ : $\displaystyle \mathbb{R}$ $\displaystyle \longrightarrow$ $\displaystyle \mathbb{R}$ Such as :

$\displaystyle f((f(x)+y)=f(x^2-y)+4f(x)y$
Put $\displaystyle y=-f(x)$:

$\displaystyle (1)\qquad f(0) = f(x^2 + f(x)) - 4(f(x))^2.$

Put $\displaystyle y=x^2$:

$\displaystyle (2)\qquad f(f(x)+x^2) = f(0) + 4f(x)x^2.$

Combine (1) and (2) to get $\displaystyle f(x)\bigl(x^2-f(x)\bigr) = 0.$

That gives you two solutions: $\displaystyle f(x) = 0$ and $\displaystyle f(x)=x^2$. I'll leave you to think about whether or not there are also hybrid solutions, where f(x) is 0 for some values of x, and x^2 for other values of x.

3. A particular solution is $\displaystyle f(x)=x^2$.