It is proved that the function f (x) = x / x square + 1 is a decreasing function on [1, + ∞)

A mathematical problem related to function,
Let f (x) be an increasing function on [- 1,1], and f (- 1) = - 1
If there is f for all x belonging to [- 1,1] and any a belonging to [- 1,1]
(x) Less than or equal to T ^ 2-2at + 1, then the value range of T is? The answer is f (- 1) = - 1, satisfying that f (x) is an odd function and increases on [- 1,1]
Should meet
max{f(x)}=f(1)=1≤t^2-2at+1
Let g (a) = T ^ 2-2at + 1 = - 2T * a + T ^ 2 + 1
Note that this is a linear equation with respect to a, so g (a) depends on a = 1 or a
Take the minimum value when a = - 1
because
min{g(a)}≥1
therefore
G (1) = T ^ 2-2t + 1 ≥ 1 and G (- 1) = T ^ 2 + 2T + 1 ≥ 1
So the value range of T is {t | t ≤ - 2 or T = 0 or t ≥ 2}. I want to know why it is a linear equation about a,

G (a) = T ^ 2-2at + 1 = - 2T * a + T ^ 2 + 1 there are two variables on the right side of this expression, we can regard it as a binary function about (variable t, variable a); at the same time, we can also regard a as a constant first, at this time, the function can be considered as a unary quadratic function about t; at the same time, we can also regard t as

Mathematical problems about functions. Remember to answer and explain why
1. It is known that the parabola y = ax & # 178; + BX + C has two intersections with the X axis, then the root of the quadratic equation AX & # 178; + BX + C = 0 is______ .
2. The opening direction and opening size are the same as y = 3x & # 178; the parabola expression with vertex (0,3) is_ .
3. Please select a set of values of a, B and C that you like, so that the image of quadratic function y = ax & # 178; + BX + C (a ≠ 0) satisfies the following conditions at the same time: (1) opening downward. (2) when x > 2, y decreases with the increase of X, and when x < 2, y increases with the increase of X. The expression of such quadratic function is .
4. The parabola y = a (x + 2) (X-8) with the opening upward intersects with the X axis and points a and B, intersects with the Y axis and points C,
If ∠ ACB = 90 °, then the coordinate of point C is .

A:
1.x1=(-b-(b^2-4ac)^0.5)/(2a) x2=(-b+(b^2-4ac)^0.5)/(2a)
2: Let the expression y = 3x ^ 2 + BX + C be substituted into the vertex coordinates; then there is 3 = C; when x = - B / (2a) has a vertex, then 0 = - B / 6, so B = 0
Therefore, the expression of this parabola is: y = 3x ^ 2 + 3
3: Since the opening is downward, a

There is a pool with a capacity of 300 cubic meters. Let the flow rate of injected water be q (m3 / min) and the time required to fill the pool be t (min). In this changing process, which variables and constants are involved? Is q a function of T? If so, please write the function analytic expression of Q with respect to t

Q = 300 / T, 300 is a constant, Q and T are variables. This is an inverse proportional function. It's very simple

Determine the relationship of the following function: 1. The image of the function passes through (1, - 1) and is parallel to the line y = - 2x + 5

From the meaning of the question, let's set the linear function relation as follows:
y=-2x+b
The function is brought into the formula through the point (1, - 1)
-1 = - 2 * 1 + B: B = 1
So the function relation is y = - 2x + 1
I hope I can help you

The image of a function and the line y = - x + 6 intersect at point (5, a), and the line y = 2x-3 does not intersect

Substituting (5, a) into y = - x + 6
a=-5+6
a=1
So the intersection coordinates are (5,1)
The line y = 2x-3 does not intersect, indicating that the two lines are parallel, so the slope is equal to 2
According to the point oblique formula, the analytic formula of a function is
y-1=2(x-5)
That is y = 2x-9

If we know that the image of a linear function y = KX + B passes through point a (2,1) and has no intersection with the line y = 2x + 5, then the value of B is______ ．

∵ the graph of a function y = KX + B has no intersection with the straight line y = 2x + 5, ∵ k = 2, ∵ passes through the point a (2,1), ∵ 1 = 2 × 2 + B, the solution is b = - 3, so the answer is: - 3

The image of a certain function passes through point a (5,1) and has no intersection with the line y = 2x-3
1. Find the expression of this function
2. Find the coordinates of the intersection point a and B of this function and X, Y axes
3. If the P coordinate of a point on the line is (x, 9), find the value of X

It has no intersection with y = 2x-3, that is, it is parallel to y = 2x-3
Let the analytic formula be y = 2x + B
Substituting point a (5,1) into
1=10+b b=-9
So y = 2x-9
2 y=2x-9
When x = 0, y = - 9
When y = 0, x = 9 / 2
So the point of intersection with X axis is (9 / 2,0) and the point of intersection with y axis is (0, - 9)
Substituting point P (x, 9) into
9=2x-9
x=9

If the ordinate of the intersection point of the image of the first-order function and the line Y1 = 2x-1 is 3, and there is no intersection point with the line y2 = 8x-5, the analytic expression of the first-order function is obtained

Let me give you a general idea
Suppose that the ordinate of the intersection point with the straight line Y1 = 2x-1 is 3, so we can find out that the intersection point is (2,3), and because there is no intersection point with the straight line y2 = 8x-5, it is parallel to it (do you know the parallel characteristic of 2 straight lines), so the slope k is equal, so let the straight line be y = 8x + B, and bring (2,3) in to get b = - 13
So this line is y = 8x-13

The analytic expression of the function is obtained when the image of a function y equal to KX + B passes through the point (- 3, - 5) and the ordinate of the intersection point with y equal to 2x + 1 is 5
I want to use it immediately,

The ordinate of the intersection point with y equal to 2x + 1 is 5. It can be seen that the function passes through the point (5,11) and then the function passes through (- 3, - 5). By substituting it, the analytic formula is y = 2x + 1

It is proved that the function f (x) = x / x square + 1 is a decreasing function on [1, + ∞)