Question

已知函数...求学霸解答…

Answer 1

已知函数y=log(2,x/4)*log(4,x/2)(2<=x<=4)
(1)求输入x=4^(2/3)时，函数y的值；
(2)令t=log(2,x)，求y关于t的函数关系式，t的范围；
(3)求该函数的值域。
(1)解析：令x=4^(2/3)
y=log(2,4^(2/3)/4)*log(4,4^(2/3)/2)
=log(2,4^(2/3-1))*log(4,4^(2/3-1/2))
=log(2,2^(-2/3))*log(4,4^(1/6))
=-2/3*1/6=-1/9
(2)解析：令t=log(2,x)
y=log(2,x/4)*log(4,x/2)=(lnx-2ln2)/ln2*(lnx-ln2)/(2ln2)
=(lnx/ln2-2)*(lnx/ln2-1)/2
=1/2[log(2,x)-2][log(2,x)-1]
=1/2(t-2)(t-1)=1/2t^2-3/2t+1
∵2<=x<=4==>1<=t<=2
(3)解析：y=1/2t^2-3/2t+1=1/2(x-3/2)^2-1/8
∴函数y的值域为[-1/8,0]