The problem of increasing and decreasing functions and the problem of odd and even functions The definition field of y = f (x) is R. for any a, B belonging to R, f (a + b) = f (a) + F (b). When x > 0, f (x) < 0 holds, f (3) = - 3 1) It is proved that the function is a decreasing function over R 2) Prove that a function is an odd function 3) Functions in [M, n] m, n all belong to the range of n *

The problem of increasing and decreasing functions and the problem of odd and even functions The definition field of y = f (x) is R. for any a, B belonging to R, f (a + b) = f (a) + F (b). When x > 0, f (x) < 0 holds, f (3) = - 3 1) It is proved that the function is a decreasing function over R 2) Prove that a function is an odd function 3) Functions in [M, n] m, n all belong to the range of n *

1. Let x 10, then f (x 2-x 1) 0, so f (x) is a decreasing function;
2. Substituting a = b = 0, f (0) = f (0) + F (0), then f (0) = 0
If f (- x) + F (x) = f [(- x) + x] = f (0) = 0, then f (- x) = - f (x), that is, f (x) is an odd function;
3. If f (x) is a decreasing function on R, then the range on the interval [M, n] is [f (n), f (m)]
In addition, when x ∈ Z, f (3) = f (1 + 1 + 1) = f (1) + F (1) + F (1) = - 3,
That is, f (1) = - 1
F (x + 1) = f (x) + F (1) = f (x) - 1, that is, f (x + 1) - f (x) = - 1 = constant, then the sequence {f (x)} is an arithmetic sequence with F (1) = - 1 as the first term and d = - 1 as the tolerance, then when x ∈ Z, f (x) = f (1) + (x-1) d = - x, so f (n) = - N, f (m) = - M
Then: the range is [- N, - M]