It is known that the function f (x) defined on R satisfies f (- x) = - f (x), f (x-4) = - f (x), and is a decreasing function in the interval [0,2]. If the equation f (x) = k has four different roots in the interval [- 8,8], then the sum of the four roots is () A. ±4B. ±8C. ±6D. ±2

It is known that the function f (x) defined on R satisfies f (- x) = - f (x), f (x-4) = - f (x), and is a decreasing function in the interval [0,2]. If the equation f (x) = k has four different roots in the interval [- 8,8], then the sum of the four roots is () A. ±4B. ±8C. ±6D. ±2

∵ f (- x) = - f (x), ∵ f (x) is an odd function, ∵ f (x-4) = - f (x), that is, f (x + 8) = f (x), ∵ f (x) is a periodic function with period 8. According to f (- x) = - f (x), f (x-4) = - f (x), we can get the symmetry of F (x-4) = f (- x), ∵ f (x) with respect to the straight line x = - 2

A mathematical problem: the definition field of F (x) is r, f (x + y) = f (x) + F (y) - 1, and X

1. Let x = y = 0, f (0) = f (0) + F (0) - 1, f (0) = 1
2. Let x1

The definition field of mathematical problem f (x) is d. f (x) satisfies the following conditions,
If f (x) is defined as D and f (x) satisfies the following conditions, then f (x) is said to be a closed function: (1) f (x) is a monotone function in D; (2) there exists [a, b] ∈ D, such that the range of f (x) on [a, b] is [a, b], f (x) = (2x + 1) open radical + k is a closed function, and the value range of K is————
The answer is - 1 ＜ K ≤ - &

Obviously, the domain of function definition is d = [- 1 / 2, + ∞), and on D, the function is an increasing function. Therefore, if there is [a, b] ∈ (should be contained in, but not belong to) d, so that the range of F (x) on [a, b] is [a, b], then f (a) = a, and f (b) = B, therefore, √ (2x + 1) + k = x has two different real roots, x = A and x = B (both greater than or