若函数f(x)＝logm(x-3/x+3),定义域为a,b,值域logm(m(b-1)),logm(m(a-1)),求实数m的取值范围.

Question

若函数f(x)＝logm(x-3/x+3),定义域为【a,b】,值域【logm(m(b-1)),logm(m(a-1))】,求实数m的取值范围.

Answer 1

f(x)＝log[(x-3)/(x+3)],定义域为[a,b],
值域[log[m(b-1)],log[m(a-1)]],即[1+log(b-1),1+log(a-1)],
∴3∴f(a)=log[(a-3)/(a+3)]=1+log(a-1),
∴(a-3)/(a+3)=m(a-1),
∴m=(a-3)/[(a+3)(a-1)],
设t=a-3,则t>0,
m=t/[(t+6)(t+2)]=1/(t+12/t+8)<=1/(4√3+8）=（2-√3）/4,当t=2√3时取等号,
∴m的取值范围是（0,（2-√3）/4].