Question

【高一数学】求值域f(x)=（x^2-2x+4）/（x^2-2x-3）

Answer 1

f(x)=(x²-2x+4)/(x²-2x-3)

=(x²-2x-3+7)/(x²-2x-3)

=1+7/(x²-2x-3)

=1+7/(x²-2x+1-4)

=1+7/[(x-1)²-4]

(x-1)²-4≥-4

所以1/[(x-1)²-4]>0或1/[(x-1)²-4]≤-1/4

所以7/[(x-1)²-4]>0或7/[(x-1)²-4]≤-7/4

f(x)=1+7/[(x-1)²-4]

f(x)>1或者f(x)≤-3/4

所以值域是

(-∞,-3/4]∪(1,+∞)