How to find the minimum value and the minimum positive period of a function

Simplify first Analyze again to find the model

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1. 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
2. What is the fundamental period of a function
3. 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!
4. What are the common forms of periodic functions Excuse me?
5. How to prove that Dirichlet function has no limit?
6. Besides Dirichlet functions, which functions are Riemann non integrable and Lebesgue integrable,
7. The famous Dirichlet function is defined like this What are the independent variables and the dependent variables of this function What are the definition and value ranges of this function Please write the function values when x = - 1, root 2, 6.4 and 3.1415 respectively
8. Dirichlet function On Baidu Encyclopedia The Dirichlet function over the real number field is expressed as: D(x)=lim(n→∞){lim(m→∞)[cosπm!x]^n} 　(1)　 It can also simply express the form of piecewise function d (x) = 0 (x is irrational) or 1 (x is rational) (2) The process of finding (1) and pushing (2)
9. Some problems on Dirichlet function For: 1. Is a periodic function, 3 is a period of it; 2. The equation f (x) = cosx has rational roots; 3. The equation f [f (x)] = f (x) has the same solution set as the equation f (x) = 1 In fact, Dirichlet function refers to: (1) when x is a rational number, f (x) = 1; (2) when x is an irrational number, f (x) = 0
10. Why can't Dirichlet higher numbers be expressed as limit functions of continuous functions
11. How to find the minimum positive period in positive metaphysical function?
12. 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
13. 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
14. 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)
15. 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?
16. 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?
17. Properties of periodic function
18. 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?
19. 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
20. 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?