已知函数f(x)=x2+ax+b,若集合A={x|x=f(x)},B={x|x=f[f(x)]}.若A={-1,3},用列举法表示B.

Question

已知函数f(x)=x2+ax+b,若集合A={x|x=f(x)},B={x|x=f[f(x)]}.若A={-1,3},用列举法表示B.

Answer 1

A={-1,3},
f(x)-x=x^2+(a-1)x+b=0有两个根-1和3
所以 a-1=-(-1+3),b=-1*3
a=-1,b=-3
f(f(x))=x
(x^2-x-3)^2-(x^2-x-3)-3=x
[(x^2-2x-3)+x]^2-[(x^2-2x-3)+x]-3=x
(x^2-2x-3)^2+2x*(x^2-2x-3)+x^2-(x^2-2x-3)-x-3=x
(x^2-2x-3)^2+(2x-1)(x^2-2x-3)+(x^2-2x-3)=0
(x^2-2x-3)(x^2-2x-3+2x-1+1)=0
(x+1)(x-3)(x^2-3)=0
B={-1,3,-√3,√3}