It is known that a set M is a set of functions f (x) satisfying the following two properties simultaneously ① F (x) is a monotone increasing function or a monotone decreasing function in its domain of definition; ② There is an interval [a, b] in the domain of F (x), such that the range of F (x) on [a, b] is [A / 2, B / 2] (1) Judge whether the function f (x) = √ x belongs to m? And explain the reason. If so, request the interval [a, b]; (2) If the function f (x) = √ (x-1) + T ∈ m, find the value range of real number t Be as detailed as possible

It is known that a set M is a set of functions f (x) satisfying the following two properties simultaneously ① F (x) is a monotone increasing function or a monotone decreasing function in its domain of definition; ② There is an interval [a, b] in the domain of F (x), such that the range of F (x) on [a, b] is [A / 2, B / 2] (1) Judge whether the function f (x) = √ x belongs to m? And explain the reason. If so, request the interval [a, b]; (2) If the function f (x) = √ (x-1) + T ∈ m, find the value range of real number t Be as detailed as possible

1) F (x) increases monotonically in the domain of definition
From √ a = A / 2, √ B = B / 2
A = 0, B = 4
So f (x) belongs to m, [a, b] is [0,4]
2) F (x) is monotone increasing, and the domain of definition is x > = 1
The equation √ (x-1) + T = x / 2 needs two unequal positive roots not less than 1
x-1=(x/2-t)^2
That is g (x) = x ^ 2 / 4 - (T + 1) x + T ^ 2 + 1 = 0
Delta = T ^ 2 + 2T + 1-T ^ 2-1 = 2T > 0, t > 0
If the axis of symmetry x = 2 (T + 1) > 1, t > - 1 / 2 is obtained
g(1)=1/4-t-1+t^2+1=t^2-t+1/4=(t-1/2)^2>=0
Therefore, it only needs t > 0