Question

已知函数f(x)=log^2(x+1)/(x-1)+log^2(x-1)

已知函数f(x)=log^2(x+1)/(x-1)+log^2(x-1)+log^2(p-x),p＞1.
（1）求f(x)的定义域 (2)求f(x)的值域 写出详细解答。谢谢

Answer 1

⑴定义域等价于求不等式组：
(x+1)/(x-1)＞0
x-1＞0
p-x＞0
解得：1＜x＜p

⑵f(x)=log^2(x+1)/(x-1)+log^2(x-1)+log^2(p-x)
=log^2[(x+1)/(x-1)·(x-1)·(p-x)]
=log^2[-x^2+(p-1)x+p]
函数g(x)=-x^2+(p-1)x+p在(1，p)的值域为0＜g(x)＜(p+1)^2/4，从而f(x)的值域为f(x)＜2log2^(p+1)-2