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【高一数学】求值域f(x)=(x^2-2x+4)/(x^2-2x-3)

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【高一数学】求值域f(x)=(x^2-2x+4)/(x^2-2x-3)
求值域f(x)=(x^2-2x+4)/(x^2-2x-3)
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,+∞)