It is known that f (x) is an even function defined on R, and the image of y = f (x) is symmetric with respect to the line x = 2. It is proved that f (x) is a periodic function

It is known that f (x) is an even function defined on R, and the image of y = f (x) is symmetric with respect to the line x = 2. It is proved that f (x) is a periodic function

The image of y = f (x) is symmetric about the line x = 2, so f (2-x) = f (2 + x), so f (x) = f (4-x), f (x) is the even function defined on R, f (4-x) = f (x-4), so f (x) = f (x-4), so f (x) period is 4

If FX is an odd function and F1 = 1 / 2, f (x + 2) = FX + F2, find F5

F (- 1) = - 1 / 2, and f (1) = f (- 1) + F (2)
So f (2) = 1,
f（5）=f（3）+f（2）=f（1）+2*f（2）=5/2

Let a = (cosx, SiNx), B = (cosy, siny), if | √ 2A + B | = √ 3 | - A - √ 2B |, then cos (X-Y)=----------

It is suggested that cos (X-Y) = --- is a point B, and the modules of a and B are all 1 | √ 2 A + B | = √ 3 | - 2 B |, just square them

A piecewise function problem of judging parity needs detailed explanation,
F (x) = x ^ 2 + 10 + 21, X belongs to [- 5, - 2] x ^ 2-10x + 21, X belongs to the parity of [2,5]

∵x∈[-5,-2],∴-x∈[2,5]
∴f(-x)=(-x)^2-10(-x)+21=x^2+10+21=f(x)
So it's even

If f (x) = x2 + 2x & nbsp; & nbsp; (x ≥ 0) g (x) (x < 0) is an odd function, then f (g (- 1))=______ ．

According to the meaning of the question, when x < 0, f (x) = g (x), f (x) is an odd function, G (- 1) = f (- 1) = - f (1) = - (12 + 2 × 1) = - 3, then f (g (- 1)) = f (- 3) = - f (3) = - (32 + 2 × 3) = - 15; so the answer is - 15

How to judge the parity of function?

Let's see if the domain is symmetric about the origin
If not, there is no parity
If it's symmetrical
Then f (- x) = - f (x) is an odd function
F (- x) = f (x) is an even function

On the parity of piecewise function, why do we need to - x when x > 0, when x0

If a piecewise function is discussed in different intervals, its expression is different. Therefore, if x > 0, x0, f (x) = x ^ 2-2x + 3
When x0, - x

How do you judge the parity of piecewise function,
(except by drawing)

Let me give you a simple example: F (x) = x ^ 2, (x is greater than or equal to 0) f (x) = - x ^ 2 (x is less than 0). First of all, when x is greater than 0, f (x) = x ^ 2, then - x is less than 0, f (- x) = - (- x) ^ 2 = - x ^ 2 = - f (x) when x is less than 0, f (x) = - x ^ 2, then - x is greater than 0, f (- x) = - (- x) ^ 2 = x ^ 2 = - f (x) when x is equal to 0

Speed:
If f (x) is an odd function and f (x + 4) = f (x), and f (x) = x when 0 ≤ x ≤ 1, then the value of F (7.5) is_____

f(7.5)=f(4+3.5)=f(3.5)
f(3.5)=f(4+(-0.5))=f(-0.5)
F (x) is an odd function, f (x) = - f (- x)
f(-0.5)=-f(0.5)
X = 0.5 on 0 ≤ x ≤ 1
f(0.5)=0.5
-f(0.5)=-0.5

Odd and even functions in senior one
The function f (x) = (m-1) x ^ 2 + 2mx + 4 is an even function. Write out the monotone interval of the function and prove it
Please bring some brief process, thank you

The function f (x) = (m-1) x ^ 2 + 2mx + 4 is even, M = 0
f(x)=-x^2+4.
When x ≤ 0, f (x) increases monotonically; when x ≥ 0, f (x) decreases monotonically