已知函数f(x)满足f(2x+1)=xx+x,求f(x)

已知函数f(x)满足f(2x+1)=xx+x,求f(x)
已知函数f(x)满足f(2x+1)=xx+x,求f(x)

已知函数f(x)满足f(2x+1)=xx+x,求f(x)
f(2x+1)=xx+x
=(1/4)(2x+1)^2-1/4
所以 f(x)=x^2/4-1/4

设T=2X+1,x=（T-1）/2,f(tT-1)/2整个的平方加上T

令t=2x+1,则x=(t-1)/2，
代入f(2x+1)=xx+x，得
f(t)=(t-1)(t-1)/(2*2)+(t-1)/2
=(t*t-1)/4
t用x替换，即
f(x)=(x*x-1)/4

设2x+1=t,则x=(t-1)/2;那么
f（t）=（t-1）^2 /4+(t-1)/2
=(t^2 -1)/2;
即f（x）=(x^2 -1)/2;