It is known that the function f (x) satisfies f (2x-3) = 4x ^ 2 + 2x + 1 1 to find the analytic expression of F (x) 2 g (x) = f (x + a) - 7x, a is a real number, try to find the minimum value of G (x) in 1,3 closed interval

It is known that the function f (x) satisfies f (2x-3) = 4x ^ 2 + 2x + 1 1 to find the analytic expression of F (x) 2 g (x) = f (x + a) - 7x, a is a real number, try to find the minimum value of G (x) in 1,3 closed interval

1、f(2x-3)=(2x-3)^2+7(2x-3)+13
f(x)=x^2+7x+13
2、g(x)=(x+a)^2+7(x+a)+13-7x
=(x+a)^2+7a+13
Dang-3

Function has f (a + b) = f (a) + F (b) - 1 for any real number a and B, and when x > 0, f (x)
(1) prove that f (x) is an increasing function on R. (2) if f (4) = 5, solve the inequality f (3m ^ 2-7) 0, f (x) > 1

Because f (0) = f (0 + 0) = f (0) + F (0) - 1
So f (0) = 1
Because f (0) = f (x-x) = f (x) + F (- x) - 1
So f (- x) = 2-F (x)
Let a > b, then A-B > 0
There is f (a-b) = f (a) + F (- b) - 1 = f (a) + 2-F (b) - 1 = f (a) - f (b) + 1
Because when x > 0, f (x) > 1
And A-B > 0, so f (a-b) > 1
Therefore, f (a-b) = f (a) - (b) + 1 > 1
That is, f (a) - f (b) > 0 is r for any a > B
So f (x) is a monotone increasing function
Because f (4) = 5, f (2 + 2) = f (2) + F (2) - 1 = 5
That is, f (2) = 3
So the inequality f (3m ^ 2-7)

Given the function f (x) = x + 4x + 3a, f (BX) = 16x - 16x + 9, where x belongs to R, a and B are constants, then the equation f (AX + b)

[analysis]
f(bx)=(bx)^2+4(bx)+3a=16x^2-16x+9
So a = 3, B = - 4
f(ax+b)=f(3x-4)=(3x-4)^2+4(3x-4)+9
=9x^2-12x+9
You can also ask me questions here:

Given the function f (x) = 4x + A / x + 1, X ＞ - 1, a is a constant
(1) If a = 1, try to prove that f (x) ≥ 0
(2) For any x ∈ (1, + ∞), f (x) > 1, the value range of a is obtained

The first question is factorization, when a = 1, f (x) = 4x + 1 / (x + 1), suppose f (x) 0, then find the minimum value of function 4 (x + 3 / 8) ^ 2 when x > - 1, it is easy to know that when x = - 3 / 8 has the minimum value of 0, then the original inequality is changed to A-25 / 16 > 0, that is, the value range of a is (25 / 16, + ∞). Because of oral calculation, the calculation may be wrong, you can calculate it yourself