题目内容：

1、对任意实数x,y.均满足f(x+y^2)=f(x)+2[f(y)]^2且f(1)不等于0.则f(2009)等于?
2、已知函数f(x)满足:对一切实数x,y恒有f(x+y)-f(y)=(x+2y+1)x,且f(1)=0 求（1）f(0)的值 (2)若一切x属于（0,1/2）不等式f(x)+2

优质解答

1.
(1) 代入x=1,y=0,得f(1)=f(1)+f(0),解得f(0)=0;
(2) 代入x=0,y=1,得f(1)=f(0)+2[f(1)]^2,解得f(1)=0.5;
(3) 所以f(2009) = f(2008+1^2) = f(2008)+2[f(1)]^2 = f(2008)+0.5;
= f(2007)+0.5+0.5 = ...= f(1)+2008*0.5 = 1004.5
2.
(1) 代入x=1,y=0,得f(1)-f(0)=2,因f(1)=0,解得f(0)=-2;
(2) 先求f(x)的表达式,代入y=0,得f(x)-f(0)=(x+1)x,因f(0)=-2,故
f(x)=(x+1)x-2;
(3) (0,1/2)时,f(x)+2