## The relationship between function and curve and equation The curve must be an equation? The equation must be a curve? A function must be an equation? A function must be an equation?

(1) Curve and equation are two different concepts. Curve is a geometric concept, equation is a mathematical concept, and curve can be expressed by equation. Therefore, it can not be said that curve must be equation or equation must be curve. (2) the image of function is not necessarily curve, or straight line, and curve does not necessarily correspond to function

### Given that the abscissa of the intersection of line L and line y = 2x + 1 is 2, and the ordinate of the intersection of line y = x + 1 is 2, find the expression of line L. (note the format)

Let l be y = KX + B

The abscissa is 2, where y = 2x + 1

y=2*2+1=5

So l passed (2,5)

The ordinate is 2, where y = x + 1

2=x+1

x=1

So l passed (1,2)

So 5 = 2K + B

2=k+b

So k = 3, B = - 1

So l is y = 3x-1

### Periodic function If the image of the function f (x) defined on R is symmetrical about the point (- 3 / 4,0), and f (x) = - f (x + 3 / 2), f (1) = 1, f (0) = 2, then the value of F (1) + f (2) + F (3) +... + F (2011) is __

F (x) = - f (x + 3 / 2), i.e. f (x + 3 / 2) = - f (x + 3 / 2) + 3 / 2] = - f (x + 3 / 2) = f (x), so the function period is 3. Because the image of function f (x) is symmetrical about point (- 3 / 4,0), then f (x) + F (- 3 / 2-x) = 0, and f (x + 3 / 2) = f (x), so f (- 3 / 2-x) = f (x

### Given that the function f (x) satisfies f (x + 2) = f (x), f (2 + x) = f (2-x) and X ∈ [2,3], f (x) = (X-2) 2, find the analytical formula of F (x) on [4,6]

F (2 + x) = f (2-x), it is obtained that the f (x) image is symmetrical about x = 2;

F (x) = (X-2) 2 on X ∈ [2,3] is symmetric about x = 2, and the analytical formula of function f (x) on [1,2] is also f (x) = (X-2) 2;

That is, the function f (x) = (X-2) 2 on [1,3];

From F (x + 2) = f (x), we know that the period of F (x) is 2, and [1,3] is exactly a period of F (x). The image of translating one period to the right and then another period is as follows:

It can be seen from the figure that the analytical formula F (x) = (x-2-2) 2 = (x-4) 2 on [4,5], and the analytical formula on [5,6] is f (x) = (x-2-2) 2 = (X-6) 2;

That is, the analytical formula of F (x) on [4,6] is: F (x) =

(x−4)2 x∈[4，5]

(x−6)2 x∈(5，6] ．

### Comprehensive application of function Let the function f (x) = (x-1) ^ 2 (x + b) e ^ x, if x = 1, be a maximum point of F (x), then the value range of real number B is______

Firstly, we can see that when x = 1, no matter how much B is taken, f (x) = 0, which means that near x = 1, f (x) is always negative, so that x = 1 is a maximum point

1： When B is not equal to - 1, (x-1) ^ 2 is always greater than or equal to 0, and e ^ x is always greater than 0, so in the neighborhood of x = 1 (x + b)

### Mathematical problems of function equation Log3 (x ^ 2-10) = 1 + log3 (x) the 3 after solving the log is the subscript. I can't type it!

Move log3 (x) on the right to the left to get log3 (x ^ 2-10) - log3 (x) = 1 + log3 (x)

So log3 [(x ^ 2-10) / x] = 1, so [(x ^ 2-10) / x = 0, so x = √ 10 or - √ 10. Also, because X in log3 (x) cannot be negative, x = + √ 10

Note: √ is the root sign