Ask a few mathematical function questions Y = the range of the root sign (xsquare - x + 2) The minimum value of y = x square + 2x + 2 The maximum value of y = 2 - | x |

Ask a few mathematical function questions Y = the range of the root sign (xsquare - x + 2) The minimum value of y = x square + 2x + 2 The maximum value of y = 2 - | x |

Y = √ (x ^ 2-x + 2) = √ [(x-1 / 2) ^ 2 + 7 / 4] ≥ √ 7 / 2, i.e. y ≥ √ 7 / 2
Y = x ^ 2 + 2x + 2 = (x + 1) ^ 2 + 1 ≥ 1, that is y = 1
|The minimum value of X | is 0, and the maximum value of 2 - | x | is 2, that is, y = 2

Help me solve some math function problems
1. The number of foci between the image of function y = f (x) and the line x = m is? 2. If the function f (x-1) = x-3 is known, then the value of F (2) is? 3. If the function y = f (x) satisfies (f) = 2F (1 / x) + X, find the analytic expression of F (x). 4. X1, X2 are the two real roots of the quadratic equation X-2 (m-1) + m + 1 about X, and y = X1 + X2, find the analytic expression of y = f (m) and the domain of definition of this function. 5, If B is a subset of a, find the value range of real number a

1. The number of foci between the image of function y = f (x) and the line x = m is? Y = FM) = 0.2. If f (x-1) = x-3 is known, then the value of F (2) is? Let x = 3, f (2) = 3x3-3 = 6.3. If y = f (x) satisfies (f) = 2F (1 / x) + X, find the analytic expression of F (x). F (1 / x) = 2F (x) + 1 / x, f (x) = 4f (x) + 2 / x, f (x) = - 2 / (3x

Given two functions f (x) = 8x ^ 2 + 16-k + 2007, G (x) = 2x ^ 3 + 5x ^ 2 + 4x (k is a constant), 1) for all x belonging to [- 3,3], f (x) ＜ = g (x) holds, find the value of real number K. 2) for all X1 belonging to [- 3,3], all x2 belonging to [- 3,3], f (x1) ＜ = g (x2) holds, find the value of K

The first question is equivalent to ask for 8x ^ 2 + 16-k + 2007 = - (2x ^ 3 + 5x ^ 2 + 4x) + 8x ^ 2 + 16 + 2007 (personal question: you have not typed the wrong words here? Why do you add a 16 first and then a 2007? Is it adding a 16x?) because the title is not clear, dare not calculate

Find the function analytic formula of the image of the line y = 3x + 2 with respect to the x-axis symmetry, the function analytic formula of the image with respect to the y-axis symmetry, and the function analytic formula of the image rotated 180 ° around the origin

The original function passes through points (0,2) and (- 2 / 3,0)
On the functions of x-symmetric images passing through points (0, - 2) and (- 2 / 3,0)
The analytic expression of the function is y = - 3x-2
On the function passing through points (0,2) and (2 / 3,0) of images with Y-axis symmetry
The analytic expression of the function is y = - 3x + 2
The function of the image rotated 180 ° around the origin passes through points (0, - 2) and (2 / 3,0)
The analytic expression of the function is y = 3x-2

We know two quadratic functions Y1 and Y2 about X, Y1 = a (x-k) 2 + 2 (k > 0), Y1 + y2 = x2 + 6x + 12; when x = k, y2 = 17; and the symmetry axis of the image of quadratic function Y2 is a straight line x = - 1. (1) find the value of K; (2) find the expression of functions Y1 and Y2; (3) in the same rectangular coordinate system, ask whether the image of function Y1 and the image of function Y2 have intersection point? Please give reasons

(1) From Y1 = a (x-k) 2 + 2, Y1 + y2 = x2 + 6x + 12, ∵ y2 = (Y1 + Y2) - Y1, = x2 + 6x + 12-A (x-k) 2-2, = x2 + 6x + 10-A (x-k) 2, and ∵ when x = k, y2 = 17, that is K2 + 6K + 10 = 17, ∵ K1 = 1, or K2 = - 7 (rounding off), the value of K is 1; (2) from k = 1, y2 = x2 + 6x + 12 -

Postman Xiao Wang starts from the county and goes to a & nbsp; by bike; After finishing the delivery work in village a, Xiao Wang met Li Ming on his way back to the county, so he took his bicycle to get on Li Ming and arrived at the county together. As a result, Xiao Wang arrived one minute later than expected. The relationship between the distance between them and the county (kilometers) and the time Xiao Wang spent after leaving the county (minutes) is shown in the figure (a) when Xiao Wang and Li Ming met for the first time, how many kilometers away from the county? (b) How long does it take Xiao Wang to leave the county and return to the county? (c) How long does it take Li Ming from village a to the county seat?

(a) It can be seen from the figure that when they meet, they are 4km away from the county town. (b) Xiaowang takes 30 minutes to take a rest in village a and return. When they meet Li Ming 1km away from the county town, it takes 20 minutes. When they return, the speed of Xiaowang is (6-1) △ 20 = 0.25 (km). According to the original speed of Xiaowang, 1 △ 0.25 = 4 (min), then the time of Xiaowang is = 30 + 30 + 20 + 4 + 1 = 85 (min). (c) Li Ming can see from the figure that when they leave the city It takes 80 minutes from 5km to 1km. Li Ming's speed = (5-1) △ 80 = 0.05 (km), Li Ming's second meeting from village a with Xiao Wang takes 5 △ 0.05 = 100 (min), and Li Ming's total meeting time from village a is 100 + 5 = 105 (min). A: it can be seen from the figure that the meeting time is 4km away from the county, Xiao Wang takes 85 minutes from the county to return to the county, and Li Ming takes 105 minutes from village a to the county Minutes

Urgent a mathematical function problem!
If f (x) belongs to R for any x, f (x-1) = f (x + 1), f (- x) = f (x), and if x is not equal to y, x, y belongs to (- 1,0), there is always [f (x) - F (y)] / (X-Y)

f(e)

2Sin (quarter school + a) sin (quarter school - a) = cos2a
Add: Pai is a primary school student, which is equal to 3.141592653. Who can prove it for me,

Expand with the sum / difference formula: I use 45 degrees to represent the quarter
simple form
=2(sin45cosa+cos45sina)(sin45cosa-cos45sina)
=2 (radical 2 / 2 cosa + radical 2 / 2 Sina) (radical 2 / 2 cosa - radical 2 / 2 Sina)
=2(1/2 cosa^2 - 1/2 sina^2)
=cosa^2-sina^2
=cos2a

The proof of mathematical function in grade one of senior high school
1. The function f (x) = - X & sup3; + 1 is a monotone decreasing function in the interval (- ∞, 0)
2. The function f (x) = x + 1 / X is a monotone decreasing function in the interval (0,1) and a monotone increasing function in the interval [1, + ∞)
3. Prove that the function f (x0 = - X & sup2; + X is a monotone increasing function on (- ∞, 1, 2)
4. Prove that the function y = x + 1 / X-1 is a monotone decreasing function on (- ∞, 1)
Not only the answer, the process is more important!
Online, etc!!!!
There is no mistake in the first question. There is a minus sign in the front

It is easy to prove by definition
Set X10
So monotonic decreasing
2 / 3, 4 use the same method, do not need factorization
It's easier to use the derivative method

It is known that cosx = cos α, cos β, Tan (x + a) / 2tan (x-a /) 2 = Tan ^ 2 (β / 2)

Given cos x = cos α cos β, prove Tan [(x + α) / 2] Tan [(x - α) / 2] = Tan & # 178; (β / 2)
It is proved that the left side = {sin [(x + α) / 2] sin [(x - α) / 2]} / {cos [(x + α) / 2] cos [(x - α) / 2]} [with the integral sum difference formula]
={cos [(x + α) / 2 - (x - α) / 2] - cos [(x + α) / 2 + (x - α) / 2]} / {cos [(x + α) / 2 - (x - α) / 2] + cos [(x + α) / 2 + (x - α) / 2]} [simplification]
=(COS α - cos x) / (COS α + cos x) [substituting the known condition cos x = cos α, cos β]
=(COS α - cos α cos β) / (COS α + cos α cos β) [approximate]
=(1-cos β) / (1 + cos β) [with angle doubling formula]
=[2sin²(β/2)]/[2cos²(β/2)]
=Tan & # 178; (β / 2) = right side