Can we give the operational properties of function period in detail For example, f (x) = f (x + T) or other In addition, is f (x-a) = f (x + a) a periodic operation?

The properties of periodic function are as follows: (1) if t (≠ 0) is the period of F (x), then - t is also the period of F (x). (2) if t (≠ 0) is the period of F (x), then NT (n is any non-zero integer) is also the period of F (x). (3) if T1 and T2 are the periods of F (x), then T1 ± T2 is also the period of F (x). (4) if f (x) has the minimum positive cycle

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1. How to prove the periodic problem of two functions? If f (x) is an odd function and the equation f (a + x) = f (A-X) holds for all x ∈ R, it is proved that the period of F (x) is 4a If f (x) is symmetric with respect to (a, Y0) and x = B, it is proved that the period of F (x) is 4 (B-A)
2. Two questions about the proof of periodic function 1. It is known that f (x) is an odd function, and the image of F (x) is symmetric with respect to the line x = 2. It is proved that f (x) is a periodic function 2. Let f (x) be an even function defined on R whose image is symmetric with respect to the line x = 1. For any x 1, x 2 belonging to [0,0.5], f (x 1 + x 2) = f (x 1) * f (x 2) is proved to be a periodic function
3. Let f (x) = cos (2 π − x) + 3cos (π 2 − x), then the minimum positive period of the function is () A. π2B. πC. 2πD. 4π
4. How to find the minimum positive period in positive metaphysical function?
5. How to find the minimum value and the minimum positive period of a function
6. On the small problem of function period If f (x-4) = - f (x), f (x) is an odd function, why can we get f (X-8) = f (x) -F (x-4) = f (X-8) why
7. What is the fundamental period of a function
8. Several expressions of periodic function? For example, f (x + 2) = f (x) f (X-2) = f (x) f (x + 2) = - f (x) f (X-2) = - f (x)... And so on! Are they periodic functions? If so, what are the axis of symmetry and the period? What if they are even or odd functions? What are the axis of symmetry and the period? What are the expressions of periodic functions? Detailed answer to chase points!
9. What are the common forms of periodic functions Excuse me?
10. How to prove that Dirichlet function has no limit?
11. What are the properties of periodic function? How to find periodicity? If we know that the period of the function is 3, what conclusions can we get?
12. Properties of periodic function
13. The problem of periodic function in Mathematics If the function y = f (x) satisfies for any real number x in the domain of definition, 1. F (x + a) = - f (x), then y = f (x) with T=_____ It is a period 2. F (x + a) = 1 / F (x), then y = f (x) with T=_____ It is a period 3. F (x + a) = - 1 / F (x), then y = f (x) with T=_____ It is a period 4. F (x + a) = [1 + F (x)] / [1-f (x)], then y = f (x) with T=_____ It is a period 2a 2.2a 3.2a 4.2a But how did it come about?
14. A mathematical problem, periodic function If f (x) is a function defined on the set of real numbers and f (x + 2) = f (x + 1) = f (x), f (1) = Lg3 / 2, f (x) = LG15 Prove by definition that it is a periodic function
15. On the mathematical problems of periodic function Given that the even function f (x) defined on R satisfies f (x) = - f (4-x) and when x ∈ [2.4], f (x) = log2 (x-1) [2 is the base, X-1 is the real number], what is the value of F (2010) + F (2011)? By the way, since he said f (x) is an even function, is there f (4-x) = f (x-4) and why?
16. A problem about periodic function (see supplement) Given the function f (x) = sinwx coswx, if there is a real number X1 such that f (x1) ≤ f (x) ≤ f (x1 + 4) holds for any real number x, what is the minimum value of positive number W? The answer is π / 4. I mainly want to know why it is enough to take half a period between the maximum and the minimum. I always feel that for the interval (K π, K π + π / 2), there is exactly half a period, but not any f (x) has a function value corresponding to this interval Why is half a period between F (x1) and f (x1 + 4) enough? Does it have to be the maximum and minimum?
17. A function, on the period of the problem What is the minimum positive period of the function y = sin2x,
18. Ask a question about periodic function F (x) is a bounded function on R, f (x + 13 / 42) + F (x) = f (x + 1 / 6) + F (x + 1 / 7), find the smaller positive period of F (x) The answer in the reference book is 1 / 42, but the process is wrong. I can only prove that 1 is the period of F (x) How to find out that 1 / 42 is f (x) period,
19. A problem about the period of function If f (x) is an odd function on R and an increasing function on (- 1,0), and f (x + 2) = - f (x), find the period of F (x)? The answer is f (x) = - f (x + 2) = f (x + 2 + 2) = f (x + 4). Why can't f (x) = - f (x + 2) be used directly, and the period is 2
20. A problem of periodic function, Given that the domain of definition of function y = f (x) is r, f (x + 2011) = f (x + 2010) + F (x + 2013) holds for any real number X. if f (1) + F (2) = 1 and f (1) + F (2) + +If f (2013) = 0, then f (2007) =?