If the function f (x) is a monotone function over the domain D, and there exists an interval [a, b] &; D (where a < b), such that when x ∈ [a, b], the value range of F (x) is exactly [a, b], then the function f (x) is said to be a positive function over D, and the interval [a, b] is called an equal domain interval (1) Given that f (x) = x 12 is a positive function on [0, + ∞), find the equal domain interval of F (x); (2) Try to explore whether there is a real number m, so that the function g (x) = x2 + m is a positive function on (- ∞, 0). If there is, ask for the value range of the real number m; if not, please explain the reason

If the function f (x) is a monotone function over the domain D, and there exists an interval [a, b] &; D (where a < b), such that when x ∈ [a, b], the value range of F (x) is exactly [a, b], then the function f (x) is said to be a positive function over D, and the interval [a, b] is called an equal domain interval (1) Given that f (x) = x 12 is a positive function on [0, + ∞), find the equal domain interval of F (x); (2) Try to explore whether there is a real number m, so that the function g (x) = x2 + m is a positive function on (- ∞, 0). If there is, ask for the value range of the real number m; if not, please explain the reason

(1) The expression of F (x) is not clear. Take F (x) = x ^ 2 as an example
Let x ^ 2 = x (x ≥ 0)
Then x = 0 or x = 1
And f (0) = 0, f (1) = 1
Obviously, when x ∈ [0,1], f (x) ∈ [0,1]
So the interval of F (x) is [0,1]
(2) Note that G (x) is a decreasing function when x ∈ (- ∞, 0]
If M ≥ 0, then G (x) ≥ 0
Obviously, when x ∈ (- ∞, 0], G (x) cannot be a positive function
If M