Given the function f (x) = 2mx2-2 (4-m) x + 1, G (x) = MX, if at least one of F (x) and G (x) is positive for any real number x, then the value range of real number m is () A. （0，2）B. （0，8）C. （2，8）D. （-∞，0）

Given the function f (x) = 2mx2-2 (4-m) x + 1, G (x) = MX, if at least one of F (x) and G (x) is positive for any real number x, then the value range of real number m is () A. （0，2）B. （0，8）C. （2，8）D. （-∞，0）

When m ≤ 0, when x is close to + ∞, the function f (x) = 2mx2-2 (4-m) x + 1 and G (x) = MX are both negative, which obviously does not hold. When x = 0, because f (0) = 1 > 0, when m > 0, if - B2A = 4-m2m ≥ 0, that is, 0 < m ≤ 4, the conclusion is obviously true; if - B2A = 4-m2m < 0, as long as △ = 4 (4-m) 2-8m = 4 (M -

On the problem of mathematical abstract function in senior one
Given the function f (x + y) = f (x) + F (y) + 2Y (x + 1), and f (1) = 1
If x is a positive integer, try to find the expression of F (x)

Consider f (x) as the x-th term of a sequence
Because f (x + 1) = f (x) + F (1) + 2 (x + 1)
So f (x + 1) - f (x) = 2x + 3
And then we add them up
We can get f (x) = f (x) - f (x-1) + F (x-1) - f (X-2) +... + F (2) - f (1) + F (1)
=2[(x-1)+(x-2)+...+1]+3(x-1)+1
=x(x-1)+3(x-1)+1
=x^2+2x-2

Great Xia, please
It is known that when x.y is not equal to 0, f (x times y) = f (x) + F (y), and f (2) = 1?
Isn't f (1 / 2) equal to - 1?

F (1) = f (x * 1 / x) = 2F (1) f (x) = - f (1 / x), so this function should be a log function
Answer log2 (6)

I've just learned abstract function. Although it's abstract, I'd like to understand it very much. The teacher talks about f (x) = 5x + 1 and f (2x + 2) = 5x + 1 in class. Please tell me the connection and difference between F (x) and f (2x + 2), and the relationship between definition field and value field. The more detailed, the better. It's best to give some examples
I want to ask how to do this problem, f (X-Y) - f (y) = (x + 2Y + 1) x, f (1) = 0, find f (x)

Generally speaking, f can be regarded as a "program" for the operation of the independent variable after its value is taken. For example, f in F (x) = 5x + 1 can be regarded as 5 * () + 1. In this way, when the independent variable x = m, f (M) = 5 * (m) + 1 = 5m + 1, and f (2x + 2) = 5x + 1, it can be understood as the internal "program" reflected by F when the independent variable 2x + 2 is taken