Who knows the strategy to prove the periodicity of function

Who knows the strategy to prove the periodicity of function

The period can be obtained by transforming the sum difference product into a trigonometric function
My father is a year old and my son is B years old. My father is () years older than my son. My father is () times older than my son this year
A-B years older, B times a
Proof of periodic function
As shown in the picture, question 14,
Wood has a picture
In high school, there is a common way to prove periodic function
Let a constant α be nonzero, and prove that f (x + α) = f (x) is constant
This system of equations, α, generally has multiple solutions, because in a periodic function, the integral multiple of the period is still a period
My son is 12 years old and my father is 39 years old
A. Three years later B. three years ago C. nine years later D. impossible
After X years, the father's age is four times that of his son. According to the meaning of the question: 39 + x = 4 (12 + x), the solution is: x = - 3, that is, three years ago, the father's age was four times that of his son
If there is a constant P such that f (x) satisfies f (PX) = f (px-p / 2) (x belongs to R), then a positive period of F (x) is?
If f (2x + 1) is an even function, then the equation of symmetry axis of the image of y = f (2x) is?
F (PX) = f (px-p / 2) f (px-1 / 2p + 1 / 2P) = f (px-1 / 2p + 1 / 2p-1 / 2P) = f (px-1 / 2P), let px-1 / 2p be t, then f (T) = P (t-1 / 2P), then a positive period of F (x) is p / 22
1/2
f(px)=f(px-p/2)=f ( p( x - 1/2) )
If f (2x + 1) is an even function, then it is symmetric with respect to x = 0
F (2x) = f (2 (x - 1 / 2) + 1) is to shift the whole right by 1 / 2
So about x = 1 / 2 symmetry
1. For the function f (x) = f (x + a), the period of the function is a
So for the function f (PX), the period T = (P / 2)
Let x = Px, then the period of function f (x) is t = (P / 2) / P
That is, t = 1 / 2
(it can be verified by assuming a periodic function)
2. F (2x + 1) is symmetric about the y-axis, i.e. x = 0 (* denotes a multiplication sign)
2*(x+1/2)=2x+1
That is to say, f (2x) is obtained by translating f (2x + 1) 1 / 2 units to the right
... unfold
1. For the function f (x) = f (x + a), the period of the function is a
So for the function f (PX), the period T = (P / 2)
Let x = Px, then the period of function f (x) is t = (P / 2) / P
That is, t = 1 / 2
(it can be verified by assuming a periodic function)
2. F (2x + 1) is symmetric about the y-axis, i.e. x = 0 (* denotes a multiplication sign)
2*(x+1/2)=2x+1
That is to say, f (2x) is obtained by translating f (2x + 1) 1 / 2 units to the right
That is, the image of F (2x) is folded symmetrically about x = 1 / 2
My son is 13 years old. My father is 40 years old. Is there a year when my father is four times as old as his son?
In the year of X, the father's age is exactly four times of his son's. The answer is: x = - 4. Four years ago, the father's age was exactly four times of his son's
Given that the function defined on R is both periodic and odd, why f (- t / 2) = f (T / 2) = f (0) = 0?
Because it's an odd function
F（-T/2）=-F(T/2)
F(0)=0
Because it's a periodic function, the period is t
F(-T/2)=F[(-T/2)+T]=F(T/2)
So there are
F（-T/2）=-F(T/2)
F(-T/2)=F(T/2)
At the same time, it holds that - f (T / 2) = f (T / 2) = 0
So f (- t / 2) = f (T / 2) = f (0) = 0
My father is 49 years old and my son is 21 years old______ Five years ago, the father was five times as old as his son
Suppose that the age of a father is five times the age of his son x years ago, (21-x) × 5 = 49-x & nbsp; & nbsp; 105-5x = 49-x & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; 4x = 56 & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; X = 14 A: the age of a father 14 years ago is five times the age of his son
Let there be a constant P > 0 such that f (PX) = f (px-p / 2) and X is a real number
1. Find a period of F (x)
2. Find a positive period of F (PX)
Let there be a constant P > 0, such that f (PX) = f (px-p / 2), X belongs to a real number. 1. Find a period of F (x). 2. Find a positive period of F (PX). From the trigonometric function, we know that the period of sin2x = sin (2x-2 π) = = > SiNx is 2 π. F (PX) = f (px-p / 2) = = > F (x) is p / 2 (2) ∵ the period of SiNx is 2 π = = > sin2x
f(px)=f(px-2/p)
Change PX to X
f(x)=f(x-2/p)
A positive period of F (x) is 2 / P
If we find the positive period of F (PX)
Let g (x) = f (PX) = f (px-2 / P) = f (P (X-2 / P ^ 2) = g (X-2 / P ^ 2)
The positive period of F (PX) is 2 / P ^ 2
Let t = Px, then px-p / 2 = T-P / 2
f(t)=f(t-p/2)
Because x ∈ R, P > 0 constant, so t ∈ R, so: F (x) = f (X-P / 2)
F (x) period P / 2
f(px)=f(px-p/2)
F (PX) period P / 2
1 * 1 / 2 + 2 * 1 / 3 + 3 * 1 / 4 +. + 39 * 1 / 40=
1 * 1 / 2 + 2 * 1 / 3 + 3 * 1 / 4 +. + 39 * 1 / 40
=1-1/2+1/2-1/3+...+1/39-1/40
=1-1/40
=39/40;
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