已知集合A={x丨-2≤x≤a},B={y丨y=2x+3,x属于A},C={z丨z=x的平方,x属于A}且C是B的子集,求a的取值

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2025-12-17 04:20:28
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回答1:

A={x| -2≤x≤a},
B={y| y=2x+3 , x∈A}
C={z| z=x^2, x∈A}

C= {x| 0≤x≤4 or 0≤x≤a^2 }
B= { x| -1 ≤x ≤ 2a+3 }
C is subset of B, then

4≤2a+3 and 2a+3≥a^2
a≥1/2 and a^2-2a-3≤ 0
a ≤1/2 and -1≤a≤3

=> 1/2≤a≤3

另解:
①若-2<=a<=0,C={x|a^2<=x<=4}
C是B的子集,有a^2>=-1,4<=2a+3, a无解
②若0C是B的子集,有4<=2a+3, 解得1/2<=a<=2
③若a>2,C={0<=x<=a^2}
C是B的子集,有a^2<=2a+3, 解得2结合①②③得 1/2<=a<=3