题目

已知函数f(x)=x的平方+px+q,集合A=(x|f(x)=x),集合B=(x|f(f(x-1)=x+1),当A=(2)时,求集合B.

提问时间：2020-11-07

答案

f(x)=x的平方+px+q
,集合A=(x|f(x)=x),
A=(2)
A的解即为方程f(x)=x的解
x^2+px+q=x
x^2+(p-1)x+q=0
A的解集为x=2
所以方程x^2+(p-1)x+q=0的唯一解为x=2
方程可以写成（x-2)^2=0
x^2-4x+4=0
对比系数可知
p-1=-4 p=-3
q=4
B=(x|f(f(x-1)=x+1),
f(x-1)=(x-1)^2-3（x-1)+4=x^2-5x+8
f(f(x-1))=(x^2-5x+8)^2-3(x^-5x+8)+4=x+1
这个方程貌似比较复杂
最后的解有四个
分别是
5/2+1/6*3^(1/2)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+1/6*((-6*(44+12*69^(1/2))^(1/3)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)-3*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)*(44+12*69^(1/2))^(2/3)+60*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+18*3^(1/2)*(44+12*69^(1/2))^(1/3))/(44+12*69^(1/2))^(1/3)/((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2))^(1/2)
5/2+1/6*3^(1/2)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)-1/6*((-6*(44+12*69^(1/2))^(1/3)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)-3*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)*(44+12*69^(1/2))^(2/3)+60*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+18*3^(1/2)*(44+12*69^(1/2))^(1/3))/(44+12*69^(1/2))^(1/3)/((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2))^(1/2)
5/2-1/6*3^(1/2)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+1/6*i*((6*(44+12*69^(1/2))^(1/3)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+3*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)*(44+12*69^(1/2))^(2/3)-60*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+18*3^(1/2)*(44+12*69^(1/2))^(1/3))/(44+12*69^(1/2))^(1/3)/((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2))^(1/2)
5/2-1/6*3^(1/2)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)-1/6*i*((6*(44+12*69^(1/2))^(1/3)*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+3*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)*(44+12*69^(1/2))^(2/3)-60*((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2)+18*3^(1/2)*(44+12*69^(1/2))^(1/3))/(44+12*69^(1/2))^(1/3)/((-(44+12*69^(1/2))^(1/3)+(44+12*69^(1/2))^(2/3)-20)/(44+12*69^(1/2))^(1/3))^(1/2))^(1/2)