Hii !
Can You Help Me For This Exercice :
Find all The Functions $\displaystyle f$ : $\displaystyle \mathbb{R}$ $\displaystyle \longrightarrow$ $\displaystyle \mathbb{R}$ Such as :
$\displaystyle f((f(x)+y)=f(x^2-y)+4f(x)y$
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.