We can know that a≥- 1.
Because y=2x- 1 is a monotone function.
therefore
B={yl y=2x- 1，x∈A} = {y| -3 ≤y≤2a- 1} = [-3，2a- 1]
z= x？ Is a parabola with an axis of symmetry x=0 and an upward opening.
The discussion about a is as follows:
(1) When a≤0,
For z, the maximum value is obtained when x =-1, and the minimum value is obtained when x=a, where c = {ZL z = the square of x, x ∈ a} = [a? , 1]
At this time, if c is contained in b,
Then 2a- 1≤ 1 and-1≤a≤0 must be satisfied.
Comprehensive-1≤a≤0
(2) When 0≤a≤ 1
For z, x=- 1 takes the maximum value, and x=0 takes the minimum value, and c = {ZL z = the square of x, x ∈ a} = [0, 1].
At this time, if c is contained in b,
Then 2a- 1≤ 1 and 0≤a≤ 1 must be satisfied.
There are 0≤a≤ 1.
(3) When a≥ 1
For z, x=a takes the maximum value, x=0 takes the minimum value, and c = {ZL z = the square of x, x ∈ a} = [0, a? ]
At this time, if c is contained in b,
2a- 1≤ a？ And a≥ 1.
2a- 1≤ a？ => Answer? - 2a + 1 ≥0 = >(a- 1)？ ≥0 holds.
Synthetically get a≥ 1.
Synthesize (1), (2) and (3) to obtain
When a ≥- 1, b always contains C.